
Citation: Ahmed M. Gemeay, Najwan Alsadat, Christophe Chesneau, Mohammed Elgarhy. Power unit inverse Lindley distribution with different measures of uncertainty, estimation and applications[J]. AIMS Mathematics, 2024, 9(8): 20976-21024. doi: 10.3934/math.20241021
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In the context of Bayesian statistics, Lindley (L) [1] explored the idea of the one-parameter L distribution (LD) as a counterexample to fiducial distributions. Reference [2] provides a comprehensive explanation of the statistical properties of the LD. In particular, it is emphasized that the LD outperforms the well-known exponential distribution in several important aspects. One of these is that by simply adjusting its unique parameter, the LD can fit data with an increasing failure rate quite efficiently. Logically, exploring more options and possibilities for this distribution would increase its modeling flexibility. Over the last few decades, researchers have therefore proposed several extensions of the LD under different circumstances.
Some mathematical ingredients that define the LD are now recalled. First, the LD can be viewed as a combination of exponential (β) and gamma (2, β) distributions. It has the cumulative distribution function (CDF) and probability density function (PDF) given by the following equations:
G(w;β)=(1+w1+β)e−βw,w>0,β>0, | (1.1) |
with G(w;β)=0 for w≤0, and
g(w;β)=β21+β(1+w)e−βw,w>0,β>0, | (1.2) |
with g(w;β)=0 for w≤0, respectively.
The following are some of the most important generalizations of this distribution: discrete Poisson-LD [3], zero-truncated Poisson-LD [4], three-parameter generalization of the LD [5], generalized LD [6], weighted LD [7], extended LD [8], exponential-Poisson-LD [9], special two-parameter LD [10], power LD [11], novel weighted LD [12], type Ⅰ half logistic LD [13], and alpha power transformed power LD [14]. As evidenced by these works, extensions of the LD are still a hot topic, and efforts to construct flexible models based on it continue.
As suggested in reference [15], the inverse LD (ILD) is obtained by using the transformation Y=1/W, where W denotes a random variable with the LD. After standard manipulations of the CDF and PDF in Eqs (1.1) and (1.2), respectively, the CDF and PDF of Y are as follows:
G(y;β)=(1+β(1+β)y)e−βy,y>0,β>0, | (1.3) |
with G(y;β)=0 for y≤0, and
g(y;β)=β21+βy−3(1+y)e−βy,y>0,β>0, | (1.4) |
with g(y;β)=0 for y≤0, respectively. In fact, the ILD is only one member of the family of inverse distributions. Indeed, the literature on this family of inverse random variables is extensive and includes the inverse Kumaraswamy distribution [16], inverse power LD [17], inverse Xgamma distribution [18], inverse exponentiated Weibull distribution [19], inverse power Lomax distribution [20], inverse exponentiated Lomax distribution [21], inverse Lomax-Rayleigh distribution [22], inverse Nadarajah-Haghighi distribution [23], inverse Nakagami-m distribution [24], inverse Topp-Leone distribution [25], inverse power Muth distribution [26], inverse Maxwell distribution [27], inverse unit Teissier distribution [28], and inverse power Cauchy distribution [29].
On the other hand, beyond the distributions with support (0,∞), it is necessary to create new flexible distributions capable of offering models adapted to the analysis of datasets with values in [0,1]. Indeed, many disciplines, including medical, actuarial, and financial sciences, are in need of such "unit distributions". A classic approach is to modify distributions with support (0,∞) to fit the unit interval [0,1]. The resulting distributions generally provide more flexibility throughout [0,1] without changing the properties of the base distribution. With this in mind, many unit distributions have been created. For example, there is the unit Birnbaum-Saunders distribution [30], unit Weibull distribution [31,32,33], unit Gompertz distribution [34], unit LD [35], unit inverse Gaussian distribution [36], unit Burr XII distribution [37], unit exponentiated half-logistic distribution [38], unit half-logistic geometric distribution [39], unit omega distribution [40], unit power Burr X distribution [41], unit inverse exponentiated Weibull distribution [42], and generalized unit half-logistic geometric distribution [43].
With the above state of the art in mind, some mathematical elements and motivations are now developed to define the scope of this paper. Starting from a random variable Y with the ILD, we consider X=Y/(1+Y). Then, by construction, the support of X is [0,1]. The distribution of X thus defines a unit distribution, which we logically call the unit ILD (UILD). After some standard developments based on Eqs (1.1) and (1.2), we can prove that it has the following CDF and PDF:
G(x;β)=eβ1+β(1+βx)e−βx,0<x≤1,β>0, | (1.5) |
with G(x;β)=0 for x≤0 and G(x;β)=1 for x>1, and
g(x;β)=β2eβ1+βx−3e−βx,0<x≤1,β>0, | (1.6) |
with g(x;β)=0 for x≤0 or x>1, respectively. Given this information, the focus of this study is on a derived two-parameter unit distribution, which we call the power unit ILD (PUILD). We are particularly interested in it for the following reasons:
ⅰ) It is very simple, with only two parameters to adjust, one being the scale parameter and the other the shape parameter.
ⅱ) The PDF of the PUILD is characterized by being possibly unimodal, decreasing, increasing, and right-skewed. In addition, the hazard rate function (HRF) may be increasing, U-shaped, or N-shaped.
ⅲ) It is possible to express in closed form the corresponding quantile and median.
ⅳ) In fact, beyond quantile analysis, many important measures can be determined in closed form, including mode, moments, mean, variance, coefficient of variation, index of dispersion, inverse moments, harmonic mean, incomplete moments, and Lorenz and Bonferroni curves. All moment-type measures involve the incomplete gamma function, which is implemented in all mathematical software, making them easy to determine.
ⅴ) Thanks to the manageability of PUILD, some measures of uncertainty can be calculated, such as Shannon entropy, Rényi entropy, exponential entropy, Havrda and Charvat entropy, Arimoto entropy, Awad and Alawneh 1 entropy, Awad and Alawneh 2 entropy, extropy, and weighted extropy.
ⅵ) Most of the existing parametric estimation approaches can be applied to the PUILD. In particular, this paper considers fifteen of them, which are the maximum likelihood, Anderson-Darling, Cramér-von-Mises, maximum product of spacings, least squares, right-tail Anderson-Darling, weighted least squares, left-tail Anderson-Darling, minimum spacing absolute distance, minimum spacing absolute-log distance, Anderson-Darling left-tail second order, Kolmogorov, minimum spacing square distance, minimum spacing square-log distance, and minimum spacing Linex distance approaches.
ⅶ) As usual, the invariance property of maximum likelihood estimation can be used to estimate the different measures of uncertainty. The PUILD is of interest in this respect because of the simple expressions of these measures.
ⅷ) Due to its flexible features, the PUILD is competitive in fitting unit data compared to the direct candidates. This importance is highlighted in this paper by considering several current statistical models, including the unit inverse Lindley, exponentiated Topp-Leone, Kumaraswamy, and beta and transformed gamma models, as well as two applications using real datasets.
There are nine parts to the paper. The creation of the PUILD is detailed in Section 2. Section 3 calculates some of its key measures. Section 4 is devoted to a wide panel of its uncertainty measures. Fifteen different estimation approaches are covered in Section 5. Section 6 discusses the results of the simulation. The practicality and adaptability of the PUILD are demonstrated in Section 7 through two real datasets. Finally, Section 8 presents the conclusion.
This section introduces the mathematical basis of the PUILD. It is based on the transformation Z=X1/δ, where X is a random variable with the UILD. Based on Eqs (1.5) and (1.6), the CDF and PDF of the PUILD are given by
G(z;β,δ)=eβ1+β(1+βzδ)e−βzδ,0<z≤1,β,δ>0, | (2.1) |
with G(z;β,δ)=0 for z≤0 and G(z;β,δ)=1 for z>1, and
g(z;β,δ)=δβ2eβ1+βz−2δ−1e−βzδ,0<z≤1,β,δ>0, | (2.2) |
with g(z;β,δ)=0 for z≤0 or z>1, respectively. We can check that limz→0g(z;β,δ)=0 and limz→1g(z;β,δ)=g(1;β,δ)=δβ2/(1+β), which gives some indication of the limit possibilities of the corresponding model. More important aspects related to g(z;β,δ) will be revealed later.
In addition, the reliability function, the HRF, the reversed HRF, and the cumulative HRF are given, respectively, as follows:
S(z;β,δ)=1−eβ1+β(1+βzδ)e−βzδ, |
h(z;β,δ)=δβ2eβz−2δ−1e−βzδ1+β−eβ(1+βzδ)e−βzδ, |
τ(z;β,δ)=δβ2z−2δ−11+βzδ, |
and
H(z;β,δ)=−log[1−eβ(1+β)(1+βzδ)e−βzδ], |
all being valid for 0<z<1 and the standard complementary functions for the other values of z.
Figure 1 illustrates the plots of the two most important functions in terms of modeling significance: the PDF and the HRF.
From this figure, it is clear that the PDF can be decreasing, increasing, unimodal, and right-skewed, and the HRF can be N-shaped, U-shaped, or increasing. This demonstrates a high degree of adaptability required for different unit data analyses.
On the other hand, the odd ratio (OR), failure rate average (FRA), and Mills ratio (MR) of the PUILD are
OR(z;β,δ)=[(1+β)e−β(1+βzδ)−1eβzδ−1]−1, |
FRA(z;β,δ)=−log[1−eβ1+β(1+βzδ)e−βzδ]z, |
and
MR(z;β,δ)=1+β−eβ(1+βzδ)e−βzδδβ2eβz−2δ−1e−βzδ, |
both valid for 0<z<1. These expressions are quite manageable; these functions can be used for various purposes beyond those developed in this paper.
In this section, we examine some important measures of the PUILD. These help to understand its probabilistic properties.
If it is unique, the mode of the PUILD corresponds to the maximum point of the PDF into the support [0,1]. It can be determined by equating dlog[g(z;β,δ)]dz with 0, as follows:
dlog[g(z;β,δ)]dz=−2δ+1z+βδzδ+1=0. | (3.1) |
After some reductions in complexity, Eq (3.1) becomes −(2δ+1)zδ+βδ=0, from which we derive a solution which is given as
z∗=(βδ2δ+1)1δ, | (3.2) |
provided that βδ≤2δ+1. Under this condition, if z<z∗, then we have dlog[g(z;β,δ)]/dz>0, and if z>z∗, then we have dlog[g(z;β,δ)]/dz<0. As a result, z∗ is the unique mode of the PUILD.
If βδ>2δ+1, the unique mode is immediately given as z∗=1. The PUILD is thus inherently unimodal, and the closed form expression of its mode is a valuable indicator of the modeling power of the PUILD.
The quantile function of the PUILD is given as Q(u;β,δ)=F−1(u;β,δ), with 0<u<1. It is thus calculated by inverting the CDF in Eq (2.1) as follows:
(eβ1+β)(1+β(Q(u;β,δ))−δ)e−β(Q(u;β,δ))−δ=u, |
that provides
(1+β(Q(u;β,δ))−δ)e−β(Q(u;β,δ))−δ=(1+β)e−βu. |
Multiplying each side of the previous equation by −e−1 gives the following Lambert-type equation:
−(1+β(Q(u;β,δ))−δ)e−(1+β(Q(u;β,δ))−δ)=−(1+β)e−(1+β)u. |
By introducing the negative Lambert W function of the real argument, denoted as W−1(.), we find that
Q(u;β,δ)={−1β−1βW−1[−(1+β)e−(1+β)u]}−1δ. |
As the Lambert W function is implemented in most scientific software, we can easily manipulate this quantile function for calculation purposes. Plugging u=0.25, 0.5, and 0.75 into this quantile function gives us the first, second (median), and third quantiles. Determining these quantiles facilitates statistical analysis and probabilistic modeling.
Let Z be a random variable with the PUILD. For any nonnegative integer r, since Z has a bounded support, the rth moment of Z always exists and is also bounded. Let us compute it by considering its integral expression. We have
μ′r=E(Zr)=∫10zrg(z;β,δ)dz=δβ2eβ1+β∫10zr−2δ−1e−βzδdz. |
If we apply the change of the variable v=βzδ, we get
μ′r=βrδeβ1+β∫∞βv1−rδe−vdv. |
Then, by introducing the upper incomplete gamma function Γ(u,t)=∫∞tzu−1e−zdz, with u>0 and t≥0, we get
μ′r=βrδeβ1+βΓ(2−rδ,β). | (3.3) |
Note that, since β>0, this expression is valid without restriction on the parameters.
Having a closed-form expression for the moments of all orders allows a precise analytical characterization of the properties of the PUILD. It facilitates efficient computation of important measures such as those presented below.
The mean and variance of the PUILD can be calculated by inserting r = 1 and 2 in Eq (3.3), as follows:
E(Z)=μ′1=β1δeβ1+βΓ(2−1δ,β), |
and
var(Z)=μ′2−μ′12=β2δeβ1+βΓ(2−2δ,β)−β2δe2β(1+β)2[Γ(2−1δ,β)]2. |
Similarly, the corresponding skewness is given as E{[Z−E(Z)]3/[var(Z)]3/2}, the kurtosis is specified as E{[Z−E(Z)]4/[var(Z)]2}, the coefficient of variation (CV) is indicated as [var(Z)]1/2/E(Z), and the index of dispersion (ID) is expressed as var(Z)/E(Z). Figure 2 shows the 3D plots of these measures for different values of β and δ.
From this figure, for the values of the parameters considered, it can be seen that the skewness varies approximately from 0.5 to 4.5, indicating a wide range of possibilities. Furthermore, the corresponding kurtosis can be small or large. The PUILD thus reaches the three established kurtosis states: it can be leptokurtic, mesokurtic, and platykurtic. These facts complete the already observed shape flexibility of the PDF and HRD of the PUILD.
To complete this study of moments, for any nonnegative integer ω, let us express the ωth lower incomplete moment (LIM) of Z. After some integral manipulations, we find that
ϱω(t)=E(Zω1{Z<t})=∫t0zωg(z;β,δ)dz=δβ2eβ1+β∫t0zω−2δ−1e−βzδdz=βωδeβ1+βΓ(2−ωδ,βt−δ). |
By taking t=1 and ω=r, as expected, we refind ϱω(t)=μ′r.
For any nonnegative integer r, the inverse rth moment of Z can be calculated as follows:
μ′−r=E(Z−r)=∫10z−rg(z;β,δ)dz=δβ2eβ1+β∫10zr−2δ−1e−βzδdz. |
Again, by applying v=βzδ, we obtain
μ′−r=β−rδeβ1+β∫∞βv1+rδe−vdv. |
Then, using the incomplete gamma function, we find that
μ′−r=β−rδeβ1+βΓ(2+rδ,β). | (3.4) |
By substituting r = 1 in Eq (3.4), the harmonic mean of Z can be calculated as follows:
ε=β−1δeβ1+βΓ(2+1δ,β). |
These inverse moments complete the classical moment analysis of PUILD. These simple expressions show how they can be used in various probabilistic and statistical scenarios involving moments of various kinds.
The Lorenz (LOR) and Bonferroni (BON) curves are essential in reliability, economics, medicine, demography, and insurance. They can also be interpreted in a unit data analysis scenario. For this reason, we express them in the context of the PUILD. The LOR and BON curves are simply calculated as
LOR=ϱ1(t)E(Z)=Γ(2−1δ,βt−δ)Γ(2−1δ,β), |
and
BON=LORG(t;β,δ)=(1+β)Γ(2−1δ,βt−δ)eβtδΓ(2−1δ,β)eβ(1+βtδ), |
respectively. They are easily implemented for various statistical purposes.
There are several useful measures of uncertainty for a given distribution. In this section, we examine the most famous of these in the context of PUILD. Namely, there is the Shannon entropy, the Rényi entropy, the exponential entropy, the Havrda and Charvat entropy, the Arimoto entropy, the Tsallis, Awad and Alawneh 1 entropy, the Awad and Alawneh 2 entropy, the extropy, and the weighted extropy.
The two propositions below show that some sophisticated integrals using the PDF of the PUILD can be written using the incomplete gamma function. Later, we will see how these integrals relate to the entropy measures under consideration.
Proposition 1. Let g(z;β,δ) be given in Eq (2.2) and
Q(β,δ)=∫10g(z;β,δ)log[g(z;β,δ)]dz. |
Then, Q(β,δ) exists and is expressed as
Q(β,δ)=eβ1+β[(1+β)e−βlog(δβ2eβ1+β)−(2+1δ)[(β+1)e−βlog(β)−Γ(2,β)2]−Γ(3,β)], |
where Γn(.,.) denotes the nth derivative of the incomplete gamma function, that is, Γn(u,t)=∫∞tzu−1(log(z))ne−zdz, with u>0 and t≥0.
Proof. Thanks to Eq (2.2), we have
Q(β,δ)=∫10g(z;β,δ)log[g(z;β,δ)]dz=δβ2eβ1+β∫10z−2δ−1e−βzδlog[δβ2eβ1+βz−2δ−1e−βzδ]dz. |
Then, we have
Q(β,δ)=δβ2eβ1+β[I1−(2δ+1)I2−βI3], | (4.1) |
where
I1=log(δβ2eβ1+β)∫10z−2δ−1e−βzδdz,I2=∫10z−2δ−1log(z)e−βzδdz, |
and
I3=∫10z−3δ−1e−βzδdz. |
For I1, by the change of variables v=βzδ, we have
I1=1δβ2log(δβ2eβ1+β)∫∞βve−vdv=(1+β)e−βδβ2log(δβ2eβ1+β) |
and
I2=1δβ2∫∞βvlog(β1δv−1δ)e−vdv=1δ2β2∫∞βv[log(β)−log(v)]e−vdv, |
which implies that
I2=1δ2β2[(β+1)e−βlog(β)−Γ(2,β)2]. |
Also, with the same technique, we get
I3=1β3δ∫∞βv2e−vdv=Γ(3,β)β3δ. |
By inserting these expressions of I1, I2 and I3 in Eq (4.1), we get
Q(β,δ)=eβ1+β[(1+β)e−βlog(δβ2eβ1+β)−(2+1δ)[(β+1)e−βlog(β)−Γ(2,β)2]−Γ(3,β)]. |
This ends the proof of Proposition 1.
Proposition 2. Let κ>0, κ≠1, g(z;β,δ) be given in Eq (2.2) and
Iκ(β,δ)=∫10g(z;β,δ)κdz. |
Then, Iκ(β,δ) exists if, and only if, (2δ+1)κ>1, and it is expressed as
Iκ(β,δ)=1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδΓ(2κ+κ−1δ,κβ). |
Proof. Owing to Eq (2.2), we have
Iκ(β,δ)=∫10g(z;β,δ)κdz=(δβ2eβ1+β)κ∫10z−2κδ−κe−κβzδdz. |
By performing the change of variables v=κβzδ, we get
Iκ(β,δ)=1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδ∫∞κβv2κ+κ−1δ−1e−vdv, |
which implies that
Iκ(β,δ)=1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδΓ(2κ+κ−1δ,κβ). |
This ends the proof of Proposition 2.
In this study, Propositions 1 and 2 are of interest since Q(β,δ) and Iκ(β,δ) are the major components of various entropy measures defined in the setting of the PUILD. This is discussed in more detail in the next subsection.
As sketched in the introduction, the entropy of the PUILD can be measured in different manners. examining multiple measures of entropy for this distribution provides a comprehensive understanding of its uncertainty and complexity. This multi-faceted analysis is crucial in various fields such as information theory and machine learning. The most useful entropy measures from the literature are recalled in Table 1 for a general distribution with PDF denoted by g(z;β,δ). Also, we suppose that κ>0 and κ≠1 are basic assumptions.
Name of the entropy | Reference | Expression |
Shannon | [44] | S(β,δ)=−∫10g(z;β,δ)log[g(z;β,δ)]dz |
Rényi | [45] | Rκ(β,δ)=11−κlog[∫10g(z;β,δ)κdz] |
Exponential | [46] | Eκ(β,δ)=[∫10g(z;β,δ)κdz]11−κ |
Havrda and Charvat | [47] | HCκ(β,δ)=121−κ−1[∫10g(z;β,δ)κdz−1] |
Arimoto | [48] | Aκ(β,δ)=κ1−κ{[∫10g(z;β,δ)κdz]1κ−1} |
Tsallis | [49] | Tκ(β,δ)=1κ−1[1−∫10g(z;β,δ)κdz] |
Awad and Alawneh 1 | [50] | AA1κ(β,δ)=1κ−1log{[supz∈Rg(z;β,δ)]1−κ∫10g(z;β,δ)κdz} |
Awad and Alawneh 2 | [50] | AA2κ(β,δ)=121−κ−1[{[supz∈Rg(z;β,δ)]1−κ∫10g(z;β,δ)κdz}−1] |
It is assumed that supz∈Rg(z;β,δ) is finite and well-identified for the two entropy measures proposed in [50].
Shannon entropy. Based on Table 1, Eq (2.2), and Proposition 1, the Shannon entropy of the PUILD is obtained as
S(β,δ)=−Q(β,δ)=eβ1+β[(2+1δ)[(β+1)e−βlog(β)−Γ(2,β)2]+Γ(3,β)−(1+β)e−βlog(δβ2eβ1+β)]. |
Rényi entropy. Based on Table 1, Eq (2.2), and Proposition 2, the Rényi entropy of the PUILD can be expressed as
Rκ(β,δ)=11−κlog[Iκ(β,δ)]=11−κlog{1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδΓ(2κ+κ−1δ,κβ)}. |
Exponential entropy. Based on Table 1, Eq (2.2), and Proposition 2, the exponential entropy of the PUILD is specified by
Eκ(β,δ)=[Iκ(β,δ)]11−κ={1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδΓ(2κ+κ−1δ,κβ)}11−κ. |
Havrda and Charvát entropy. From Table 1, Eq (2.2), and Proposition 2, the Havrda and Charvát entropy of the PUILD can be expressed as
HCκ(β,δ)=121−κ−1[Iκ(β,δ)−1]=121−κ−1[1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδΓ(2κ+κ−1δ,κβ)−1]. |
Arimoto entropy. Again, from Table 1, Eq (2.2), and Proposition 2, the Arimoto entropy of the PUILD is specified by
Aκ(β,δ)=κ1−κ[Iκ(β,δ)1κ−1]=κ1−κ{[1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδΓ(2κ+κ−1δ,κβ)]1κ−1}. |
Tsallis entropy. Based on Table 1, Eq (2.2), and Proposition 2, the Tsallis entropy of the PUILD can be expressed as
Tκ(β,δ)=1κ−1[1−Iκ(β,δ)]=1κ−1[1−1δ(δβ2eβ1+β)κ(κβ)−2κ+1−κδΓ(2κ+κ−1δ,κβ)]. |
Awad and Alawneh 1 entropy. From Table 1, Eq (2.2), and Proposition 2, the Awad and Alawneh 1 entropy of the PUILD is given as
AA1κ(β,δ)=1κ−1log{[sup0<z<1g(z;β,δ)]1−κIκ(β,δ)}. | (4.2) |
Before going any further, we need to find sup0<z≤1g(z;β,δ). The following formula will do the necessary. For βδ<2δ+1, based on the equation (3.2), we have
s(β,δ)=sup0<z≤1g(z;β,δ)=g(z∗;β,δ)=(δeβ−1δ+2(1+β)β1δ)(2+1δ)2+1δ. | (4.3) |
Otherwise, we have z∗=1 and
s(β,δ)=g(z∗;β,δ)=δβ21+β. | (4.4) |
Based on Eq (4.3) or Eq (4.4), Eq (4.2) becomes
AA1κ(β,δ)=1κ−1log{[s(β,δ)]1−κ1δ(δβ2eβ1+β)κΓ(2κ+κ−1δ,κβ)(κβ)2κ−1−κδ}. |
Awad and Alawneh 2 entropy. From Table 1, Eq (2.2), Proposition 2, and Eq (4.3) or Eq (4.4), the Awad and Alawneh 2 entropy of the PUILD is given as
AA2κ=121−κ−1[{[sup0<z<1g(z;β,δ)]1−κIκ(β,δ)}−1]=121−κ−1[{[s(β,δ)]1−κ1δ(δβ2eβ1+β)κΓ(2κ+κ−1δ,κβ)(κβ)2κ−1−κδ}−1]. |
Theoretically, it is complicated to study the behavior of these entropy measures. For this reason, a numerical study is proposed in the next section.
Lad et al. [51] introduced a new measure of uncertainty: the extropy, which can be represented as the double complement of the entropy [44]. The extropy can be used statistically to assess the accuracy of predicting distributions using the total log scoring method. The definition of the extropy of the PUILD is
Φ=−12∫10g(z;β,δ)2dz. | (4.5) |
By inserting Eq (2.2) into Eq (4.5), we obtain
Φ=−12[δ2β4e2β(1+β)2∫10z−4δ−2e−2βzδdz]. |
After some simplifications, we get
Φ=−132[δe2βΓ(4+1δ,2β)(2β)1δ(1+β)2]. |
Another analogue to the weighted entropy proposed in [52] is the weighted extropy. It can be expressed as
Φw=−121∫0zg(z;β,δ)2dz. | (4.6) |
By inserting Eq (2.2) into Eq (4.6), we get
Φw=−132[δe2βΓ(4,2β)(1+β)2]. |
The simple expressions of Φ and Φw are an advantage of the PUILD. They allow a deeper extropy analysis without computational effort.
In this section, we examine the conventional approaches to estimating the two parameters of the PUILD. In these estimation methods, an objective function is optimized by maximization or minimization to obtain the most appropriate estimates.
This part calculates the maximum likelihood estimates (MLEs) (^β1,^δ1) of (β,δ) based on a simple random sample. To detail this procedure, assume that z1,z2,…,zn is an observed simple random sample of size n drawn from the PUILD. Then, the log-likelihood function (L-LF) of β and δ is given by
logL=nlog(δ)+2nlog(β)+nβ−nlog(1+β)−(1+2δ)n∑i=1log(zi)−n∑i=1βziδ. | (5.1) |
The desired MLEs are obtained by maximizing this L-LF. In this sense, differentiating Eq (5.1) with respect to the parameters β and δ, we obtain
∂logL∂β=n+2nβ−n1+β−n∑i=11ziδ, | (5.2) |
and
∂logL∂δ=nδ−2n∑i=1log(zi)+βn∑i=1log(zi)ziδ. | (5.3) |
Since finding the exact solution to Eqs (5.2) and (5.3) equal to 0 is difficult, we will use optimization techniques such as the Newton-Raphson approach using the Mathematica software program to maximize it.
Let (z1:n,z2:n,…,zn:n) represent the ordered simple random sample of (z1,z2,…,zn). Then the Anderson-Darling estimates (ADEs) (^β2,^δ2) are obtained by minimizing the following function:
A=−n−1nn∑i=1(2i−1){log[G(zi:n,β,δ)]+log[S(zn−i−1:n,β,δ)]}=−n−1nn∑i=1(2i−1){log[eβ1+β(1+βzδi:n)e−βzδi:n]+log[1−eβ1+β(1+βzδn−i−1:n)e−βzδn−i−1:n]}. |
The Cramér-von Mises estimates (CVMEs) (^β3,^δ3) are determined by minimizing the following function:
C=112n+n∑i=1[G(zi:n,β,δ)−2i−12n]2=112n+n∑i=1[eβ1+β(1+βzδi:n)e−βzδi:n−2i−12n]2. |
The method of maximum product of spacings estimates (MPSEs) (^β4,^δ4) are obtained by maximizing the following function:
MPS=1n+1n+1∑i=1log(ξi,n), |
where
ξi,n=G(zi:n,β,δ)−G(zi−1:n,β,δ)=eβ1+β(1+βzδi:n)e−βzδi:n−eβ1+β(1+βzδi−1:n)e−βzδi−1:n, | (5.4) |
completed by G(z0:n,β,δ)=0 and G(zn+1:n,β,δ)=1.
The ordinary least squares estimates (OLSEs) (^β5,^δ5) are calculated by minimizing the following function:
V=n∑i=1[G(zi:n,β,δ)−in+1]2=n∑i=1[eβ1+β(1+βzδi:n)e−βzδi:n−in+1]2. |
The right-tail ADEs (RTADEs) (\hat {\beta_6}, \hat {\delta_6}) are determined by minimizing the following function:
\begin{align*} R & = \frac{n}{2}-2\sum\limits_{i = 1}^{n} G(z_{i:n},\beta, \delta) -\frac{1}{n}\sum\limits_{i = 1}^{n}(2i-1)\log \left[S(z_{i:n},\beta, \delta)\right]\\ & = \frac{n}{2}-2\sum\limits_{i = 1}^{n} \frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}} -\frac{1}{n}\sum\limits_{i = 1}^{n}(2i-1)\log \left[1-\frac{{{e^\beta }}}{{ {1 + \beta }}}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}}\right]. \end{align*} |
The weighted least squares estimates (WLSEs) (\hat {\beta_7}, \hat {\delta_7}) are obtained by minimizing the following function:
\begin{align*} W & = \sum\limits_{i = 1}^{n}\frac{(n+1)^2(n+2)}{i(n-i+1)}\left[G(z_{i:n},\beta, \delta)-\frac{i}{n+1}\right]^2\\ & = \sum\limits_{i = 1}^{n}\frac{(n+1)^2(n+2)}{i(n-i+1)}\left[\frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}}-\frac{i}{n+1}\right]^2. \end{align*} |
The left-tail ADEs (LTADEs) (\hat {\beta_8}, \hat {\delta_8}) are computed by minimizing the following function:
\begin{align*} L & = -\frac{3}{2}n+2\sum\limits_{i = 1}^{n}G(z_{i:n},\beta, \delta)-\frac{1}{n}\sum\limits_{i = 1}^{n}(2i-1)\log \left[G(z_{i:n},\beta, \delta) \right]\\ & = -\frac{3}{2}n+2\sum\limits_{i = 1}^{n}\frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}}-\frac{1}{n}\sum\limits_{i = 1}^{n}(2i-1)\log \left[\frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}}\right]. \end{align*} |
The minimum spacing absolute distance estimates (MSADEs) (\hat {\beta_9}, \hat {\delta_9}) are obtained by minimizing the following function:
\begin{align*} \zeta & = \sum\limits_{i = 1}^{n+1}\left|\xi_{i,n}-\frac{1}{n+1}\right|, \end{align*} |
where \xi_{i, n} is given in Eq (5.4).
The minimum spacing absolute-log distance estimates (MSALDEs) (\hat {\beta_{10}}, \hat {\delta_{10}}) are obtained by minimizing the following function:
\begin{align*} \varUpsilon = \sum\limits_{i = 1}^{n+1}\left|\log (\xi_{i,n})-\log\left(\frac{1}{n+1}\right)\right|, \end{align*} |
where \xi_{i, n} is given in Eq (5.4).
The Anderson-Darling left-tail second order estimates (ADSOEs) (\hat {\beta_{11}}, \hat {\delta_{11}}) are determined by minimizing the following function:
\begin{align*} LTS& = 2\sum\limits_{i = 1}^{n}\log \left[ G(z_{i:n},\beta, \delta) \right]+\frac{1}{n}\sum\limits_{i = 1}^{n}\frac{(2i-1)}{G(z_{i:n},\beta, \delta)}\\ & = 2\sum\limits_{i = 1}^{n}\log \left[\frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}}\right]+\frac{1}{n}\sum\limits_{i = 1}^{n}\frac{(2i-1)}{ \frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}} }. \end{align*} |
The Kolmogorov estimates (KEs) (\hat {\beta_{12}}, \hat {\delta_{12}}) are obtained by minimizing the following function:
\begin{align*} KM& = \underset{i = 1, \ldots,n}{\max}\left[\frac{i}{n}-G(z_{i:n},\beta, \delta),G(z_{i:n},\beta, \delta)-\frac{i-1}{n}\right]\\ & = \underset{i = 1, \ldots,n}{ \max}\left[\frac{i}{n}- \frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}} , \frac{{{e^\beta }}}{{ {1 + \beta } }}\left( {1 + \frac{\beta }{{z_{i:n}^\delta }}} \right){e^{ - \frac{\beta }{{z_{i:n}^\delta }}}} -\frac{i-1}{n}\right]. \end{align*} |
The minimum spacing square distance estimates (MSSDEs) (\hat {\beta_{13}}, \hat {\delta_{13}}) are calculated by minimizing the following function:
\begin{align*} \phi & = \sum\limits_{i = 1}^{n+1}\left(\xi_{i,n}-\frac{1}{n+1}\right)^2, \end{align*} |
where \xi_{i, n} is given in Eq (5.4).
The minimum spacing square-log distance estimates (MSSLDEs) (\hat {\beta_{14}}, \hat {\delta_{14}}) are obtained by minimizing the following function:
\begin{align*} \delta & = \sum\limits_{i = 1}^{n+1}\left[\log (\xi_{i,n})-\log \left(\frac{1}{n+1}\right)\right]^2, \end{align*} |
where \xi_{i, n} is given in Eq (5.4).
The minimum spacing Linex distance estimates (MSLDEs) (\hat {\beta_{15}}, \hat {\delta_{15}}) are determined by minimizing the following function:
\begin{align*} \varDelta & = \sum\limits_{i = 1}^{n+1}\left[e^{\xi_{i,n}-\frac{1}{n+1}}-\left(\xi_{i,n}-\frac{1}{n+1}\right)-1\right], \end{align*} |
where \xi_{i, n} is given in Eq (5.4).
This section evaluates the effectiveness of the estimation techniques presented in Section 5. Simulated datasets were generated according to the proposed model, and the estimation techniques that were considered were applied to estimate the unknown parameters. The associated performance was evaluated using five different metrics described below. For a\in \{\delta, \beta\} , these metrics are as follows:
ⅰ) Average bias (BIAS) given as |Bias(\hat{a})| = (1/M)\sum_{j = 1}^{M}|\hat{a_j}-a| , where j refers to the label of the considered sample, among M samples of size n ,
ⅱ) Mean squared error (MSE) indicated as MSE = (1/M)\sum_{j = 1}^{M}(\hat{a_j}-a)^2 ,
ⅲ) Mean relative error (MRE) defined as MRE = (1/M)\sum_{j = 1}^{M}|\hat{a_j}-a|/a ,
ⅳ) Average absolute difference ( D_{abs} ) indicated as D_{abs} = [1/(nM)]\sum_{j = 1}^{M}\sum_{u = 1}^{n}|G(x_{j, u}; \beta, \delta)-G(x_{j, u};\hat{ \beta_j}, \hat{ \delta_j})| , where x_{j, u} denotes the values obtained at the sample labeled j and its u_{th} component.
ⅴ) Maximum absolute difference ( D_{max} ) expressed as D_{max} = (1/M)\sum_{j = 1}^{M}\max\limits_{u = 1, \ldots, n} |G(x_{j, u}; \beta, \delta)-G(x_{j, u};\hat{ \beta_j}, \hat{ \delta_j})| .
The purpose of the simulation study is to determine the optimal estimation strategy for the proposed model.
The simulation results are presented in Tables 2–6. Furthermore, the partial and total ranks for the estimates are given in Table 7.
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.22373 ^{\{ 7 \}} | 0.22577 ^{\{ 8 \}} | 0.25312 ^{\{ 14 \}} | 0.20754 ^{\{ 5 \}} | 0.25032 ^{\{ 13 \}} | 0.2571 ^{\{ 15 \}} | 0.2331 ^{\{ 11 \}} | 0.23345 ^{\{ 12 \}} | 0.13758 ^{\{ 2 \}} | 0.18677 ^{\{ 3 \}} | 0.22978 ^{\{ 10 \}} | 0.08802 ^{\{ 1 \}} | 0.22914 ^{\{ 9 \}} | 0.2176 ^{\{ 6 \}} | 0.20036 ^{\{ 4 \}} |
BIAS( \hat{\beta} ) | 0.7744 ^{\{ 7 \}} | 0.80526 ^{\{ 9 \}} | 0.75617 ^{\{ 6 \}} | 0.82699 ^{\{ 11 \}} | 0.77464 ^{\{ 8 \}} | 0.80846 ^{\{ 10 \}} | 0.84149 ^{\{ 13 \}} | 0.83473 ^{\{ 12 \}} | 0.29217 ^{\{ 2 \}} | 0.7215 ^{\{ 4 \}} | 0.85207 ^{\{ 15 \}} | 0.04434 ^{\{ 1 \}} | 0.72321 ^{\{ 5 \}} | 0.84278 ^{\{ 14 \}} | 0.64726 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.07932 ^{\{ 8 \}} | 0.07826 ^{\{ 7 \}} | 0.09935 ^{\{ 14 \}} | 0.06438 ^{\{ 4 \}} | 0.09656 ^{\{ 13 \}} | 0.09952 ^{\{ 15 \}} | 0.08196 ^{\{ 9 \}} | 0.08382 ^{\{ 11 \}} | 0.03691 ^{\{ 2 \}} | 0.05693 ^{\{ 3 \}} | 0.08258 ^{\{ 10 \}} | 0.01303 ^{\{ 1 \}} | 0.08619 ^{\{ 12 \}} | 0.07251 ^{\{ 6 \}} | 0.06923 ^{\{ 5 \}} | |
MSE( \hat{\beta} ) | 0.85507 ^{\{ 7 \}} | 0.93048 ^{\{ 10 \}} | 0.79375 ^{\{ 5 \}} | 1.00678 ^{\{ 13 \}} | 0.83857 ^{\{ 6 \}} | 0.87698 ^{\{ 8 \}} | 0.99586 ^{\{ 12 \}} | 0.98746 ^{\{ 11 \}} | 0.29807 ^{\{ 2 \}} | 0.88098 ^{\{ 9 \}} | 1.04622 ^{\{ 15 \}} | 0.00824 ^{\{ 1 \}} | 0.74237 ^{\{ 4 \}} | 1.02031 ^{\{ 14 \}} | 0.62886 ^{\{ 3 \}} | |
MRE( \hat{\delta} ) | 0.31961 ^{\{ 7 \}} | 0.32252 ^{\{ 8 \}} | 0.36159 ^{\{ 14 \}} | 0.29648 ^{\{ 5 \}} | 0.3576 ^{\{ 13 \}} | 0.36728 ^{\{ 15 \}} | 0.333 ^{\{ 11 \}} | 0.3335 ^{\{ 12 \}} | 0.19654 ^{\{ 2 \}} | 0.26682 ^{\{ 3 \}} | 0.32825 ^{\{ 10 \}} | 0.12574 ^{\{ 1 \}} | 0.32734 ^{\{ 9 \}} | 0.31086 ^{\{ 6 \}} | 0.28622 ^{\{ 4 \}} | |
MRE( \hat{\beta} ) | 0.30976 ^{\{ 7 \}} | 0.3221 ^{\{ 9 \}} | 0.30247 ^{\{ 6 \}} | 0.33079 ^{\{ 11 \}} | 0.30985 ^{\{ 8 \}} | 0.32339 ^{\{ 10 \}} | 0.33659 ^{\{ 13 \}} | 0.33389 ^{\{ 12 \}} | 0.11687 ^{\{ 2 \}} | 0.2886 ^{\{ 4 \}} | 0.34083 ^{\{ 15 \}} | 0.01774 ^{\{ 1 \}} | 0.28928 ^{\{ 5 \}} | 0.33711 ^{\{ 14 \}} | 0.2589 ^{\{ 3 \}} | |
D_{abs} | 0.04005 ^{\{ 1 \}} | 0.04126 ^{\{ 4 \}} | 0.04356 ^{\{ 10 \}} | 0.04111 ^{\{ 2 \}} | 0.04425 ^{\{ 11 \}} | 0.04479 ^{\{ 12 \}} | 0.04172 ^{\{ 6 \}} | 0.0414 ^{\{ 5 \}} | 0.04519 ^{\{ 13 \}} | 0.04312 ^{\{ 9 \}} | 0.04124 ^{\{ 3 \}} | 0.04269 ^{\{ 8 \}} | 0.06087 ^{\{ 15 \}} | 0.04232 ^{\{ 7 \}} | 0.05767 ^{\{ 14 \}} | |
D_{max} | 0.06512 ^{\{ 3 \}} | 0.06645 ^{\{ 4 \}} | 0.07113 ^{\{ 12 \}} | 0.06499 ^{\{ 2 \}} | 0.0708 ^{\{ 11 \}} | 0.07279 ^{\{ 13 \}} | 0.06712 ^{\{ 8 \}} | 0.06667 ^{\{ 5 \}} | 0.06964 ^{\{ 10 \}} | 0.06785 ^{\{ 9 \}} | 0.06693 ^{\{ 6 \}} | 0.06424 ^{\{ 1 \}} | 0.09209 ^{\{ 15 \}} | 0.06696 ^{\{ 7 \}} | 0.08699 ^{\{ 14 \}} | |
\sum Ranks | 47 ^{\{ 4 \}} | 59 ^{\{ 7 \}} | 81 ^{\{ 11 \}} | 53 ^{\{ 6 \}} | 83 ^{\{ 12.5 \}} | 98 ^{\{ 15 \}} | 83 ^{\{ 12.5 \}} | 80 ^{\{ 10 \}} | 35 ^{\{ 2 \}} | 44 ^{\{ 3 \}} | 84 ^{\{ 14 \}} | 15 ^{\{ 1 \}} | 74 ^{\{ 8.5 \}} | 74 ^{\{ 8.5 \}} | 50 ^{\{ 5 \}} | |
60 | BIAS( \hat{\delta} ) | 0.17259 ^{\{ 5 \}} | 0.19678 ^{\{ 11 \}} | 0.22989 ^{\{ 15 \}} | 0.17065 ^{\{ 4 \}} | 0.20041 ^{\{ 13 \}} | 0.22807 ^{\{ 14 \}} | 0.19883 ^{\{ 12 \}} | 0.19121 ^{\{ 9 \}} | 0.1085 ^{\{ 2 \}} | 0.16308 ^{\{ 3 \}} | 0.19184 ^{\{ 10 \}} | 0.05999 ^{\{ 1 \}} | 0.18004 ^{\{ 7 \}} | 0.18312 ^{\{ 8 \}} | 0.17361 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.68127 ^{\{ 5 \}} | 0.78854 ^{\{ 14 \}} | 0.7418 ^{\{ 8 \}} | 0.74548 ^{\{ 9 \}} | 0.71544 ^{\{ 7 \}} | 0.77753 ^{\{ 13 \}} | 0.7739 ^{\{ 12 \}} | 0.74844 ^{\{ 10 \}} | 0.27546 ^{\{ 2 \}} | 0.69382 ^{\{ 6 \}} | 0.75274 ^{\{ 11 \}} | 0.03594 ^{\{ 1 \}} | 0.58942 ^{\{ 3 \}} | 0.80753 ^{\{ 15 \}} | 0.59255 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.04735 ^{\{ 5 \}} | 0.05737 ^{\{ 9 \}} | 0.0815 ^{\{ 15 \}} | 0.04443 ^{\{ 4 \}} | 0.06477 ^{\{ 13 \}} | 0.08028 ^{\{ 14 \}} | 0.06309 ^{\{ 12 \}} | 0.05805 ^{\{ 10 \}} | 0.02402 ^{\{ 2 \}} | 0.04275 ^{\{ 3 \}} | 0.05944 ^{\{ 11 \}} | 0.00602 ^{\{ 1 \}} | 0.05663 ^{\{ 8 \}} | 0.05151 ^{\{ 6 \}} | 0.05354 ^{\{ 7 \}} | |
MSE( \hat{\beta} ) | 0.71243 ^{\{ 5 \}} | 0.92878 ^{\{ 14 \}} | 0.76945 ^{\{ 7 \}} | 0.87146 ^{\{ 12 \}} | 0.74299 ^{\{ 6 \}} | 0.85999 ^{\{ 10 \}} | 0.91218 ^{\{ 13 \}} | 0.84322 ^{\{ 9 \}} | 0.26265 ^{\{ 2 \}} | 0.81005 ^{\{ 8 \}} | 0.86264 ^{\{ 11 \}} | 0.00554 ^{\{ 1 \}} | 0.50974 ^{\{ 3 \}} | 0.99557 ^{\{ 15 \}} | 0.5388 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.24656 ^{\{ 5 \}} | 0.28111 ^{\{ 11 \}} | 0.32842 ^{\{ 15 \}} | 0.24378 ^{\{ 4 \}} | 0.2863 ^{\{ 13 \}} | 0.32581 ^{\{ 14 \}} | 0.28404 ^{\{ 12 \}} | 0.27316 ^{\{ 9 \}} | 0.155 ^{\{ 2 \}} | 0.23297 ^{\{ 3 \}} | 0.27405 ^{\{ 10 \}} | 0.08571 ^{\{ 1 \}} | 0.25719 ^{\{ 7 \}} | 0.26159 ^{\{ 8 \}} | 0.24802 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.27251 ^{\{ 5 \}} | 0.31542 ^{\{ 14 \}} | 0.29672 ^{\{ 8 \}} | 0.29819 ^{\{ 9 \}} | 0.28618 ^{\{ 7 \}} | 0.31101 ^{\{ 13 \}} | 0.30956 ^{\{ 12 \}} | 0.29938 ^{\{ 10 \}} | 0.11018 ^{\{ 2 \}} | 0.27753 ^{\{ 6 \}} | 0.3011 ^{\{ 11 \}} | 0.01437 ^{\{ 1 \}} | 0.23577 ^{\{ 3 \}} | 0.32301 ^{\{ 15 \}} | 0.23702 ^{\{ 4 \}} | |
D_{abs} | 0.0295 ^{\{ 2 \}} | 0.03057 ^{\{ 5 \}} | 0.03189 ^{\{ 13 \}} | 0.02811 ^{\{ 1 \}} | 0.03141 ^{\{ 11 \}} | 0.03108 ^{\{ 9 \}} | 0.03081 ^{\{ 6 \}} | 0.03091 ^{\{ 8 \}} | 0.03083 ^{\{ 7 \}} | 0.0314 ^{\{ 10 \}} | 0.02956 ^{\{ 3 \}} | 0.02965 ^{\{ 4 \}} | 0.04047 ^{\{ 14 \}} | 0.03166 ^{\{ 12 \}} | 0.04048 ^{\{ 15 \}} | |
D_{max} | 0.04787 ^{\{ 3 \}} | 0.04999 ^{\{ 6 \}} | 0.05351 ^{\{ 13 \}} | 0.04551 ^{\{ 2 \}} | 0.05135 ^{\{ 11 \}} | 0.05238 ^{\{ 12 \}} | 0.05037 ^{\{ 7 \}} | 0.05048 ^{\{ 9 \}} | 0.04829 ^{\{ 4 \}} | 0.05038 ^{\{ 8 \}} | 0.04885 ^{\{ 5 \}} | 0.04457 ^{\{ 1 \}} | 0.06312 ^{\{ 14 \}} | 0.05093 ^{\{ 10 \}} | 0.06349 ^{\{ 15 \}} | |
\sum Ranks | 35 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 94 ^{\{ 14 \}} | 45 ^{\{ 4 \}} | 81 ^{\{ 10 \}} | 99 ^{\{ 15 \}} | 86 ^{\{ 12 \}} | 74 ^{\{ 9 \}} | 23 ^{\{ 2 \}} | 47 ^{\{ 5 \}} | 72 ^{\{ 8 \}} | 11 ^{\{ 1 \}} | 59 ^{\{ 6 \}} | 89 ^{\{ 13 \}} | 61 ^{\{ 7 \}} | |
100 | BIAS( \hat{\delta} ) | 0.14625 ^{\{ 7 \}} | 0.16744 ^{\{ 12 \}} | 0.19457 ^{\{ 14 \}} | 0.14162 ^{\{ 5 \}} | 0.185 ^{\{ 13 \}} | 0.21524 ^{\{ 15 \}} | 0.16618 ^{\{ 11 \}} | 0.15543 ^{\{ 9 \}} | 0.0977 ^{\{ 2 \}} | 0.13859 ^{\{ 4 \}} | 0.15888 ^{\{ 10 \}} | 0.0488 ^{\{ 1 \}} | 0.13599 ^{\{ 3 \}} | 0.15167 ^{\{ 8 \}} | 0.14603 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.59238 ^{\{ 5 \}} | 0.68527 ^{\{ 13 \}} | 0.68361 ^{\{ 12 \}} | 0.67297 ^{\{ 9 \}} | 0.67575 ^{\{ 10 \}} | 0.75015 ^{\{ 15 \}} | 0.66326 ^{\{ 8 \}} | 0.62369 ^{\{ 6 \}} | 0.27316 ^{\{ 2 \}} | 0.64513 ^{\{ 7 \}} | 0.70079 ^{\{ 14 \}} | 0.03289 ^{\{ 1 \}} | 0.48933 ^{\{ 3 \}} | 0.67978 ^{\{ 11 \}} | 0.52629 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.03434 ^{\{ 6 \}} | 0.04318 ^{\{ 12 \}} | 0.0602 ^{\{ 14 \}} | 0.02976 ^{\{ 3 \}} | 0.05233 ^{\{ 13 \}} | 0.07238 ^{\{ 15 \}} | 0.04301 ^{\{ 11 \}} | 0.0378 ^{\{ 8 \}} | 0.02044 ^{\{ 2 \}} | 0.03082 ^{\{ 4 \}} | 0.03962 ^{\{ 10 \}} | 0.0038 ^{\{ 1 \}} | 0.03378 ^{\{ 5 \}} | 0.03602 ^{\{ 7 \}} | 0.03945 ^{\{ 9 \}} | |
MSE( \hat{\beta} ) | 0.56392 ^{\{ 5 \}} | 0.7543 ^{\{ 13 \}} | 0.69846 ^{\{ 9 \}} | 0.74207 ^{\{ 12 \}} | 0.66928 ^{\{ 7 \}} | 0.84187 ^{\{ 15 \}} | 0.68905 ^{\{ 8 \}} | 0.62999 ^{\{ 6 \}} | 0.25309 ^{\{ 2 \}} | 0.73522 ^{\{ 10 \}} | 0.78489 ^{\{ 14 \}} | 0.00524 ^{\{ 1 \}} | 0.37739 ^{\{ 3 \}} | 0.74173 ^{\{ 11 \}} | 0.48398 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.20892 ^{\{ 7 \}} | 0.2392 ^{\{ 12 \}} | 0.27796 ^{\{ 14 \}} | 0.20231 ^{\{ 5 \}} | 0.26429 ^{\{ 13 \}} | 0.30748 ^{\{ 15 \}} | 0.2374 ^{\{ 11 \}} | 0.22204 ^{\{ 9 \}} | 0.13957 ^{\{ 2 \}} | 0.19798 ^{\{ 4 \}} | 0.22697 ^{\{ 10 \}} | 0.06972 ^{\{ 1 \}} | 0.19427 ^{\{ 3 \}} | 0.21668 ^{\{ 8 \}} | 0.20861 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.23695 ^{\{ 5 \}} | 0.27411 ^{\{ 13 \}} | 0.27344 ^{\{ 12 \}} | 0.26919 ^{\{ 9 \}} | 0.2703 ^{\{ 10 \}} | 0.30006 ^{\{ 15 \}} | 0.26531 ^{\{ 8 \}} | 0.24948 ^{\{ 6 \}} | 0.10926 ^{\{ 2 \}} | 0.25805 ^{\{ 7 \}} | 0.28032 ^{\{ 14 \}} | 0.01315 ^{\{ 1 \}} | 0.19573 ^{\{ 3 \}} | 0.27191 ^{\{ 11 \}} | 0.21052 ^{\{ 4 \}} | |
D_{abs} | 0.02341 ^{\{ 2 \}} | 0.02377 ^{\{ 4 \}} | 0.02389 ^{\{ 5 \}} | 0.02292 ^{\{ 1 \}} | 0.02502 ^{\{ 8 \}} | 0.02535 ^{\{ 10 \}} | 0.02347 ^{\{ 3 \}} | 0.024 ^{\{ 6 \}} | 0.02561 ^{\{ 12 \}} | 0.02526 ^{\{ 9 \}} | 0.02537 ^{\{ 11 \}} | 0.02447 ^{\{ 7 \}} | 0.03101 ^{\{ 15 \}} | 0.02653 ^{\{ 13 \}} | 0.03041 ^{\{ 14 \}} | |
D_{max} | 0.03833 ^{\{ 3 \}} | 0.03924 ^{\{ 5 \}} | 0.04063 ^{\{ 9 \}} | 0.03709 ^{\{ 2 \}} | 0.0422 ^{\{ 11 \}} | 0.04366 ^{\{ 13 \}} | 0.03885 ^{\{ 4 \}} | 0.03946 ^{\{ 6 \}} | 0.04025 ^{\{ 7 \}} | 0.04062 ^{\{ 8 \}} | 0.04131 ^{\{ 10 \}} | 0.03704 ^{\{ 1 \}} | 0.0492 ^{\{ 15 \}} | 0.04265 ^{\{ 12 \}} | 0.04866 ^{\{ 14 \}} | |
\sum Ranks | 40 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 89 ^{\{ 13 \}} | 46 ^{\{ 4 \}} | 85 ^{\{ 12 \}} | 113 ^{\{ 15 \}} | 64 ^{\{ 9 \}} | 56 ^{\{ 7 \}} | 31 ^{\{ 2 \}} | 53 ^{\{ 6 \}} | 93 ^{\{ 14 \}} | 14 ^{\{ 1 \}} | 50 ^{\{ 5 \}} | 81 ^{\{ 10 \}} | 61 ^{\{ 8 \}} | |
200 | BIAS( \hat{\delta} ) | 0.09648 ^{\{ 3 \}} | 0.12482 ^{\{ 10 \}} | 0.15103 ^{\{ 14 \}} | 0.11384 ^{\{ 6 \}} | 0.1425 ^{\{ 13 \}} | 0.17269 ^{\{ 15 \}} | 0.13156 ^{\{ 12 \}} | 0.11484 ^{\{ 7 \}} | 0.07786 ^{\{ 2 \}} | 0.11322 ^{\{ 5 \}} | 0.12492 ^{\{ 11 \}} | 0.03359 ^{\{ 1 \}} | 0.11157 ^{\{ 4 \}} | 0.11946 ^{\{ 8 \}} | 0.12256 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.42875 ^{\{ 3 \}} | 0.5295 ^{\{ 7 \}} | 0.61287 ^{\{ 14 \}} | 0.54979 ^{\{ 8 \}} | 0.59034 ^{\{ 13 \}} | 0.63661 ^{\{ 15 \}} | 0.55697 ^{\{ 10 \}} | 0.50632 ^{\{ 6 \}} | 0.25968 ^{\{ 2 \}} | 0.55546 ^{\{ 9 \}} | 0.58785 ^{\{ 12 \}} | 0.02817 ^{\{ 1 \}} | 0.437 ^{\{ 4 \}} | 0.5818 ^{\{ 11 \}} | 0.45147 ^{\{ 5 \}} | |
MSE( \hat{\delta} ) | 0.01468 ^{\{ 3 \}} | 0.02419 ^{\{ 10 \}} | 0.03546 ^{\{ 14 \}} | 0.01995 ^{\{ 5 \}} | 0.03181 ^{\{ 13 \}} | 0.04731 ^{\{ 15 \}} | 0.02608 ^{\{ 11 \}} | 0.02091 ^{\{ 6 \}} | 0.01268 ^{\{ 2 \}} | 0.01992 ^{\{ 4 \}} | 0.02357 ^{\{ 9 \}} | 0.00198 ^{\{ 1 \}} | 0.023 ^{\{ 8 \}} | 0.02177 ^{\{ 7 \}} | 0.02768 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.31499 ^{\{ 3 \}} | 0.46709 ^{\{ 7 \}} | 0.61924 ^{\{ 14 \}} | 0.53484 ^{\{ 9 \}} | 0.58357 ^{\{ 12 \}} | 0.65048 ^{\{ 15 \}} | 0.5114 ^{\{ 8 \}} | 0.44203 ^{\{ 6 \}} | 0.2079 ^{\{ 2 \}} | 0.55213 ^{\{ 10 \}} | 0.56088 ^{\{ 11 \}} | 0.00473 ^{\{ 1 \}} | 0.34757 ^{\{ 4 \}} | 0.58871 ^{\{ 13 \}} | 0.37455 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.13782 ^{\{ 3 \}} | 0.17831 ^{\{ 10 \}} | 0.21575 ^{\{ 14 \}} | 0.16263 ^{\{ 6 \}} | 0.20357 ^{\{ 13 \}} | 0.2467 ^{\{ 15 \}} | 0.18794 ^{\{ 12 \}} | 0.16405 ^{\{ 7 \}} | 0.11122 ^{\{ 2 \}} | 0.16174 ^{\{ 5 \}} | 0.17846 ^{\{ 11 \}} | 0.04798 ^{\{ 1 \}} | 0.15939 ^{\{ 4 \}} | 0.17065 ^{\{ 8 \}} | 0.17509 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.1715 ^{\{ 3 \}} | 0.2118 ^{\{ 7 \}} | 0.24515 ^{\{ 14 \}} | 0.21992 ^{\{ 8 \}} | 0.23613 ^{\{ 13 \}} | 0.25465 ^{\{ 15 \}} | 0.22279 ^{\{ 10 \}} | 0.20253 ^{\{ 6 \}} | 0.10387 ^{\{ 2 \}} | 0.22218 ^{\{ 9 \}} | 0.23514 ^{\{ 12 \}} | 0.01127 ^{\{ 1 \}} | 0.1748 ^{\{ 4 \}} | 0.23272 ^{\{ 11 \}} | 0.18059 ^{\{ 5 \}} | |
D_{abs} | 0.01667 ^{\{ 2 \}} | 0.0172 ^{\{ 5 \}} | 0.01824 ^{\{ 9 \}} | 0.01714 ^{\{ 4 \}} | 0.01785 ^{\{ 8 \}} | 0.01886 ^{\{ 11 \}} | 0.01762 ^{\{ 6 \}} | 0.01696 ^{\{ 3 \}} | 0.01783 ^{\{ 7 \}} | 0.02031 ^{\{ 13 \}} | 0.01851 ^{\{ 10 \}} | 0.01629 ^{\{ 1 \}} | 0.02275 ^{\{ 15 \}} | 0.01899 ^{\{ 12 \}} | 0.02256 ^{\{ 14 \}} | |
D_{max} | 0.0269 ^{\{ 2 \}} | 0.02854 ^{\{ 6 \}} | 0.03108 ^{\{ 11 \}} | 0.02791 ^{\{ 4 \}} | 0.03005 ^{\{ 8 \}} | 0.03272 ^{\{ 13 \}} | 0.0292 ^{\{ 7 \}} | 0.02788 ^{\{ 3 \}} | 0.02821 ^{\{ 5 \}} | 0.03259 ^{\{ 12 \}} | 0.03021 ^{\{ 9 \}} | 0.02461 ^{\{ 1 \}} | 0.03658 ^{\{ 15 \}} | 0.03087 ^{\{ 10 \}} | 0.03649 ^{\{ 14 \}} | |
\sum Ranks | 22 ^{\{ 2 \}} | 62 ^{\{ 7 \}} | 104 ^{\{ 14 \}} | 50 ^{\{ 5 \}} | 93 ^{\{ 13 \}} | 114 ^{\{ 15 \}} | 76 ^{\{ 10 \}} | 44 ^{\{ 4 \}} | 24 ^{\{ 3 \}} | 67 ^{\{ 8 \}} | 85 ^{\{ 12 \}} | 8 ^{\{ 1 \}} | 58 ^{\{ 6 \}} | 80 ^{\{ 11 \}} | 73 ^{\{ 9 \}} | |
300 | BIAS( \hat{\delta} ) | 0.08469 ^{\{ 3 \}} | 0.10079 ^{\{ 10 \}} | 0.1252 ^{\{ 14 \}} | 0.09076 ^{\{ 4 \}} | 0.12392 ^{\{ 13 \}} | 0.14567 ^{\{ 15 \}} | 0.10608 ^{\{ 12 \}} | 0.09495 ^{\{ 6 \}} | 0.06793 ^{\{ 2 \}} | 0.09395 ^{\{ 5 \}} | 0.10567 ^{\{ 11 \}} | 0.02746 ^{\{ 1 \}} | 0.09748 ^{\{ 7 \}} | 0.10026 ^{\{ 8 \}} | 0.10057 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.36972 ^{\{ 3 \}} | 0.43786 ^{\{ 8 \}} | 0.50879 ^{\{ 13 \}} | 0.43499 ^{\{ 7 \}} | 0.53725 ^{\{ 14 \}} | 0.57736 ^{\{ 15 \}} | 0.45602 ^{\{ 10 \}} | 0.42746 ^{\{ 6 \}} | 0.24286 ^{\{ 2 \}} | 0.44069 ^{\{ 9 \}} | 0.5055 ^{\{ 12 \}} | 0.0244 ^{\{ 1 \}} | 0.38919 ^{\{ 5 \}} | 0.46693 ^{\{ 11 \}} | 0.3866 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.01148 ^{\{ 3 \}} | 0.01591 ^{\{ 8 \}} | 0.02484 ^{\{ 14 \}} | 0.01253 ^{\{ 4 \}} | 0.02397 ^{\{ 13 \}} | 0.03315 ^{\{ 15 \}} | 0.01778 ^{\{ 10 \}} | 0.01434 ^{\{ 6 \}} | 0.00976 ^{\{ 2 \}} | 0.01432 ^{\{ 5 \}} | 0.01768 ^{\{ 9 \}} | 0.00126 ^{\{ 1 \}} | 0.01805 ^{\{ 11 \}} | 0.01511 ^{\{ 7 \}} | 0.01817 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.24338 ^{\{ 3 \}} | 0.33897 ^{\{ 7 \}} | 0.44168 ^{\{ 12 \}} | 0.32176 ^{\{ 6 \}} | 0.51632 ^{\{ 14 \}} | 0.54731 ^{\{ 15 \}} | 0.35494 ^{\{ 9 \}} | 0.34003 ^{\{ 8 \}} | 0.17342 ^{\{ 2 \}} | 0.36794 ^{\{ 10 \}} | 0.44555 ^{\{ 13 \}} | 0.00288 ^{\{ 1 \}} | 0.29013 ^{\{ 5 \}} | 0.37214 ^{\{ 11 \}} | 0.26368 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.12099 ^{\{ 3 \}} | 0.14399 ^{\{ 10 \}} | 0.17886 ^{\{ 14 \}} | 0.12966 ^{\{ 4 \}} | 0.17703 ^{\{ 13 \}} | 0.2081 ^{\{ 15 \}} | 0.15154 ^{\{ 12 \}} | 0.13565 ^{\{ 6 \}} | 0.09705 ^{\{ 2 \}} | 0.13422 ^{\{ 5 \}} | 0.15096 ^{\{ 11 \}} | 0.03922 ^{\{ 1 \}} | 0.13925 ^{\{ 7 \}} | 0.14323 ^{\{ 8 \}} | 0.14368 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.14789 ^{\{ 3 \}} | 0.17514 ^{\{ 8 \}} | 0.20352 ^{\{ 13 \}} | 0.174 ^{\{ 7 \}} | 0.2149 ^{\{ 14 \}} | 0.23094 ^{\{ 15 \}} | 0.18241 ^{\{ 10 \}} | 0.17099 ^{\{ 6 \}} | 0.09715 ^{\{ 2 \}} | 0.17628 ^{\{ 9 \}} | 0.2022 ^{\{ 12 \}} | 0.00976 ^{\{ 1 \}} | 0.15568 ^{\{ 5 \}} | 0.18677 ^{\{ 11 \}} | 0.15464 ^{\{ 4 \}} | |
D_{abs} | 0.01342 ^{\{ 1 \}} | 0.0141 ^{\{ 5 \}} | 0.01471 ^{\{ 7 \}} | 0.01402 ^{\{ 4 \}} | 0.01492 ^{\{ 8 \}} | 0.01569 ^{\{ 12 \}} | 0.015 ^{\{ 9 \}} | 0.01378 ^{\{ 3 \}} | 0.01434 ^{\{ 6 \}} | 0.01559 ^{\{ 11 \}} | 0.01595 ^{\{ 13 \}} | 0.01376 ^{\{ 2 \}} | 0.01845 ^{\{ 15 \}} | 0.01547 ^{\{ 10 \}} | 0.0179 ^{\{ 14 \}} | |
D_{max} | 0.02182 ^{\{ 2 \}} | 0.02339 ^{\{ 6 \}} | 0.025 ^{\{ 8 \}} | 0.0228 ^{\{ 5 \}} | 0.02526 ^{\{ 10 \}} | 0.02734 ^{\{ 13 \}} | 0.02465 ^{\{ 7 \}} | 0.02256 ^{\{ 3 \}} | 0.02277 ^{\{ 4 \}} | 0.02521 ^{\{ 9 \}} | 0.02581 ^{\{ 12 \}} | 0.02079 ^{\{ 1 \}} | 0.03004 ^{\{ 15 \}} | 0.02528 ^{\{ 11 \}} | 0.029 ^{\{ 14 \}} | |
\sum Ranks | 21 ^{\{ 2 \}} | 62 ^{\{ 6 \}} | 95 ^{\{ 13 \}} | 41 ^{\{ 4 \}} | 99 ^{\{ 14 \}} | 115 ^{\{ 15 \}} | 79 ^{\{ 11 \}} | 44 ^{\{ 5 \}} | 22 ^{\{ 3 \}} | 63 ^{\{ 7 \}} | 93 ^{\{ 12 \}} | 9 ^{\{ 1 \}} | 70 ^{\{ 8.5 \}} | 77 ^{\{ 10 \}} | 70 ^{\{ 8.5 \}} | |
400 | BIAS( \hat{\delta} ) | 0.07174 ^{\{ 3 \}} | 0.09293 ^{\{ 11 \}} | 0.10943 ^{\{ 13 \}} | 0.07801 ^{\{ 4 \}} | 0.11145 ^{\{ 14 \}} | 0.13305 ^{\{ 15 \}} | 0.09175 ^{\{ 10 \}} | 0.08463 ^{\{ 6 \}} | 0.0625 ^{\{ 2 \}} | 0.08834 ^{\{ 8 \}} | 0.09467 ^{\{ 12 \}} | 0.02457 ^{\{ 1 \}} | 0.08264 ^{\{ 5 \}} | 0.08637 ^{\{ 7 \}} | 0.09147 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.31908 ^{\{ 3 \}} | 0.40392 ^{\{ 9 \}} | 0.47059 ^{\{ 14 \}} | 0.35775 ^{\{ 5 \}} | 0.46071 ^{\{ 13 \}} | 0.53833 ^{\{ 15 \}} | 0.40286 ^{\{ 8 \}} | 0.38381 ^{\{ 7 \}} | 0.2324 ^{\{ 2 \}} | 0.4171 ^{\{ 11 \}} | 0.45466 ^{\{ 12 \}} | 0.02287 ^{\{ 1 \}} | 0.33301 ^{\{ 4 \}} | 0.40844 ^{\{ 10 \}} | 0.38044 ^{\{ 6 \}} | |
MSE( \hat{\delta} ) | 0.00819 ^{\{ 3 \}} | 0.01379 ^{\{ 10 \}} | 0.01888 ^{\{ 13 \}} | 0.00948 ^{\{ 4 \}} | 0.01913 ^{\{ 14 \}} | 0.02688 ^{\{ 15 \}} | 0.01303 ^{\{ 9 \}} | 0.01121 ^{\{ 5 \}} | 0.00806 ^{\{ 2 \}} | 0.01219 ^{\{ 7 \}} | 0.01427 ^{\{ 12 \}} | 0.00096 ^{\{ 1 \}} | 0.01264 ^{\{ 8 \}} | 0.01176 ^{\{ 6 \}} | 0.01408 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.17003 ^{\{ 3 \}} | 0.27561 ^{\{ 8 \}} | 0.39787 ^{\{ 14 \}} | 0.22918 ^{\{ 5 \}} | 0.3543 ^{\{ 12 \}} | 0.49644 ^{\{ 15 \}} | 0.28197 ^{\{ 9 \}} | 0.25617 ^{\{ 7 \}} | 0.15577 ^{\{ 2 \}} | 0.30793 ^{\{ 11 \}} | 0.38113 ^{\{ 13 \}} | 0.00249 ^{\{ 1 \}} | 0.21587 ^{\{ 4 \}} | 0.30226 ^{\{ 10 \}} | 0.24672 ^{\{ 6 \}} | |
MRE( \hat{\delta} ) | 0.10249 ^{\{ 3 \}} | 0.13276 ^{\{ 11 \}} | 0.15633 ^{\{ 13 \}} | 0.11145 ^{\{ 4 \}} | 0.15921 ^{\{ 14 \}} | 0.19008 ^{\{ 15 \}} | 0.13108 ^{\{ 10 \}} | 0.1209 ^{\{ 6 \}} | 0.08929 ^{\{ 2 \}} | 0.1262 ^{\{ 8 \}} | 0.13524 ^{\{ 12 \}} | 0.0351 ^{\{ 1 \}} | 0.11806 ^{\{ 5 \}} | 0.12339 ^{\{ 7 \}} | 0.13067 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.12763 ^{\{ 3 \}} | 0.16157 ^{\{ 9 \}} | 0.18824 ^{\{ 14 \}} | 0.1431 ^{\{ 5 \}} | 0.18428 ^{\{ 13 \}} | 0.21533 ^{\{ 15 \}} | 0.16114 ^{\{ 8 \}} | 0.15352 ^{\{ 7 \}} | 0.09296 ^{\{ 2 \}} | 0.16684 ^{\{ 11 \}} | 0.18186 ^{\{ 12 \}} | 0.00915 ^{\{ 1 \}} | 0.1332 ^{\{ 4 \}} | 0.16337 ^{\{ 10 \}} | 0.15218 ^{\{ 6 \}} | |
D_{abs} | 0.01202 ^{\{ 1 \}} | 0.01289 ^{\{ 5 \}} | 0.01357 ^{\{ 9 \}} | 0.01216 ^{\{ 3 \}} | 0.01333 ^{\{ 8 \}} | 0.01381 ^{\{ 11 \}} | 0.01248 ^{\{ 4 \}} | 0.01302 ^{\{ 7 \}} | 0.01293 ^{\{ 6 \}} | 0.01404 ^{\{ 12 \}} | 0.01407 ^{\{ 13 \}} | 0.01203 ^{\{ 2 \}} | 0.01585 ^{\{ 15 \}} | 0.01372 ^{\{ 10 \}} | 0.01554 ^{\{ 14 \}} | |
D_{max} | 0.01947 ^{\{ 2 \}} | 0.02129 ^{\{ 7 \}} | 0.02291 ^{\{ 12 \}} | 0.01985 ^{\{ 3 \}} | 0.0227 ^{\{ 9 \}} | 0.02414 ^{\{ 13 \}} | 0.02063 ^{\{ 5 \}} | 0.02113 ^{\{ 6 \}} | 0.02061 ^{\{ 4 \}} | 0.02288 ^{\{ 10 \}} | 0.02289 ^{\{ 11 \}} | 0.01819 ^{\{ 1 \}} | 0.02568 ^{\{ 15 \}} | 0.02227 ^{\{ 8 \}} | 0.02537 ^{\{ 14 \}} | |
\sum Ranks | 21 ^{\{ 2 \}} | 70 ^{\{ 9 \}} | 102 ^{\{ 14 \}} | 33 ^{\{ 4 \}} | 97 ^{\{ 12.5 \}} | 114 ^{\{ 15 \}} | 63 ^{\{ 7 \}} | 51 ^{\{ 5 \}} | 22 ^{\{ 3 \}} | 78 ^{\{ 11 \}} | 97 ^{\{ 12.5 \}} | 9 ^{\{ 1 \}} | 60 ^{\{ 6 \}} | 68 ^{\{ 8 \}} | 75 ^{\{ 10 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.03792 ^{\{ 2 \}} | 0.04631 ^{\{ 7 \}} | 0.05819 ^{\{ 14 \}} | 0.04377 ^{\{ 5 \}} | 0.05299 ^{\{ 10 \}} | 0.06162 ^{\{ 15 \}} | 0.04975 ^{\{ 9 \}} | 0.0476 ^{\{ 8 \}} | 0.03992 ^{\{ 3 \}} | 0.04343 ^{\{ 4 \}} | 0.05571 ^{\{ 11.5 \}} | 0.0272 ^{\{ 1 \}} | 0.05571 ^{\{ 11.5 \}} | 0.04408 ^{\{ 6 \}} | 0.05653 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.20078 ^{\{ 3 \}} | 0.22619 ^{\{ 6 \}} | 0.24363 ^{\{ 12 \}} | 0.23866 ^{\{ 10 \}} | 0.23765 ^{\{ 9 \}} | 0.23982 ^{\{ 11 \}} | 0.23231 ^{\{ 7 \}} | 0.21986 ^{\{ 5 \}} | 0.16668 ^{\{ 2 \}} | 0.20972 ^{\{ 4 \}} | 0.25183 ^{\{ 14 \}} | 0.09301 ^{\{ 1 \}} | 0.26527 ^{\{ 15 \}} | 0.23274 ^{\{ 8 \}} | 0.24674 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00219 ^{\{ 2 \}} | 0.00339 ^{\{ 7 \}} | 0.00555 ^{\{ 14 \}} | 0.00291 ^{\{ 4 \}} | 0.00433 ^{\{ 10 \}} | 0.00612 ^{\{ 15 \}} | 0.00397 ^{\{ 9 \}} | 0.00366 ^{\{ 8 \}} | 0.00287 ^{\{ 3 \}} | 0.00308 ^{\{ 5 \}} | 0.00528 ^{\{ 13 \}} | 0.0014 ^{\{ 1 \}} | 0.00479 ^{\{ 11 \}} | 0.00312 ^{\{ 6 \}} | 0.00498 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.0644 ^{\{ 3 \}} | 0.07884 ^{\{ 6 \}} | 0.08607 ^{\{ 10 \}} | 0.08968 ^{\{ 12 \}} | 0.08623 ^{\{ 11 \}} | 0.08533 ^{\{ 8 \}} | 0.08135 ^{\{ 7 \}} | 0.07377 ^{\{ 4 \}} | 0.05803 ^{\{ 2 \}} | 0.07457 ^{\{ 5 \}} | 0.09151 ^{\{ 14 \}} | 0.02381 ^{\{ 1 \}} | 0.10724 ^{\{ 15 \}} | 0.08546 ^{\{ 9 \}} | 0.08994 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.15167 ^{\{ 2 \}} | 0.18523 ^{\{ 7 \}} | 0.23277 ^{\{ 14 \}} | 0.17509 ^{\{ 5 \}} | 0.21197 ^{\{ 10 \}} | 0.24649 ^{\{ 15 \}} | 0.199 ^{\{ 9 \}} | 0.19039 ^{\{ 8 \}} | 0.15967 ^{\{ 3 \}} | 0.1737 ^{\{ 4 \}} | 0.22284 ^{\{ 11.5 \}} | 0.10879 ^{\{ 1 \}} | 0.22284 ^{\{ 11.5 \}} | 0.17631 ^{\{ 6 \}} | 0.22614 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.26771 ^{\{ 3 \}} | 0.30159 ^{\{ 6 \}} | 0.32484 ^{\{ 12 \}} | 0.31822 ^{\{ 10 \}} | 0.31687 ^{\{ 9 \}} | 0.31975 ^{\{ 11 \}} | 0.30975 ^{\{ 7 \}} | 0.29315 ^{\{ 5 \}} | 0.22223 ^{\{ 2 \}} | 0.27962 ^{\{ 4 \}} | 0.33577 ^{\{ 14 \}} | 0.12402 ^{\{ 1 \}} | 0.35369 ^{\{ 15 \}} | 0.31032 ^{\{ 8 \}} | 0.32898 ^{\{ 13 \}} | |
D_{abs} | 0.03978 ^{\{ 1 \}} | 0.04375 ^{\{ 5 \}} | 0.04853 ^{\{ 12 \}} | 0.04421 ^{\{ 7 \}} | 0.04553 ^{\{ 8 \}} | 0.04857 ^{\{ 13 \}} | 0.04381 ^{\{ 6 \}} | 0.04289 ^{\{ 2 \}} | 0.04619 ^{\{ 9 \}} | 0.04303 ^{\{ 3 \}} | 0.04796 ^{\{ 11 \}} | 0.0432 ^{\{ 4 \}} | 0.05475 ^{\{ 15 \}} | 0.04735 ^{\{ 10 \}} | 0.05436 ^{\{ 14 \}} | |
D_{max} | 0.06553 ^{\{ 1 \}} | 0.07131 ^{\{ 6 \}} | 0.08238 ^{\{ 12 \}} | 0.07085 ^{\{ 4 \}} | 0.07553 ^{\{ 10 \}} | 0.08325 ^{\{ 13 \}} | 0.0722 ^{\{ 7 \}} | 0.07117 ^{\{ 5 \}} | 0.07504 ^{\{ 8 \}} | 0.07035 ^{\{ 3 \}} | 0.07991 ^{\{ 11 \}} | 0.06891 ^{\{ 2 \}} | 0.08996 ^{\{ 15 \}} | 0.07547 ^{\{ 9 \}} | 0.0896 ^{\{ 14 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 50 ^{\{ 6 \}} | 100 ^{\{ 11.5 \}} | 57 ^{\{ 7 \}} | 77 ^{\{ 10 \}} | 101 ^{\{ 13 \}} | 61 ^{\{ 8 \}} | 45 ^{\{ 5 \}} | 32 ^{\{ 3.5 \}} | 32 ^{\{ 3.5 \}} | 100 ^{\{ 11.5 \}} | 12 ^{\{ 1 \}} | 109 ^{\{ 15 \}} | 62 ^{\{ 9 \}} | 105 ^{\{ 14 \}} | |
60 | BIAS( \hat{\delta} ) | 0.02673 ^{\{ 2 \}} | 0.03656 ^{\{ 8 \}} | 0.04072 ^{\{ 12 \}} | 0.03364 ^{\{ 4 \}} | 0.04024 ^{\{ 11 \}} | 0.04471 ^{\{ 14 \}} | 0.03721 ^{\{ 9 \}} | 0.03459 ^{\{ 6 \}} | 0.02974 ^{\{ 3 \}} | 0.03401 ^{\{ 5 \}} | 0.03937 ^{\{ 10 \}} | 0.02163 ^{\{ 1 \}} | 0.04522 ^{\{ 15 \}} | 0.03572 ^{\{ 7 \}} | 0.04219 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.15368 ^{\{ 3 \}} | 0.1865 ^{\{ 8 \}} | 0.19022 ^{\{ 9 \}} | 0.18644 ^{\{ 7 \}} | 0.20018 ^{\{ 11 \}} | 0.20036 ^{\{ 12 \}} | 0.18376 ^{\{ 6 \}} | 0.17252 ^{\{ 4 \}} | 0.13065 ^{\{ 2 \}} | 0.17919 ^{\{ 5 \}} | 0.20222 ^{\{ 14 \}} | 0.08068 ^{\{ 1 \}} | 0.22493 ^{\{ 15 \}} | 0.19677 ^{\{ 10 \}} | 0.20108 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00113 ^{\{ 2 \}} | 0.00214 ^{\{ 8 \}} | 0.00273 ^{\{ 12 \}} | 0.00169 ^{\{ 4 \}} | 0.00252 ^{\{ 10 \}} | 0.00318 ^{\{ 15 \}} | 0.00217 ^{\{ 9 \}} | 0.00193 ^{\{ 7 \}} | 0.0016 ^{\{ 3 \}} | 0.00189 ^{\{ 6 \}} | 0.00253 ^{\{ 11 \}} | 0.00092 ^{\{ 1 \}} | 0.00314 ^{\{ 14 \}} | 0.00187 ^{\{ 5 \}} | 0.00278 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.04175 ^{\{ 3 \}} | 0.05644 ^{\{ 6 \}} | 0.05686 ^{\{ 7 \}} | 0.05842 ^{\{ 9 \}} | 0.06424 ^{\{ 14 \}} | 0.06282 ^{\{ 13 \}} | 0.05305 ^{\{ 5 \}} | 0.04927 ^{\{ 4 \}} | 0.03699 ^{\{ 2 \}} | 0.05786 ^{\{ 8 \}} | 0.06276 ^{\{ 12 \}} | 0.01743 ^{\{ 1 \}} | 0.07899 ^{\{ 15 \}} | 0.06166 ^{\{ 10 \}} | 0.06264 ^{\{ 11 \}} | |
MRE( \hat{\delta} ) | 0.10693 ^{\{ 2 \}} | 0.14623 ^{\{ 8 \}} | 0.16287 ^{\{ 12 \}} | 0.13457 ^{\{ 4 \}} | 0.16097 ^{\{ 11 \}} | 0.17885 ^{\{ 14 \}} | 0.14882 ^{\{ 9 \}} | 0.13837 ^{\{ 6 \}} | 0.11896 ^{\{ 3 \}} | 0.13604 ^{\{ 5 \}} | 0.15748 ^{\{ 10 \}} | 0.08653 ^{\{ 1 \}} | 0.18086 ^{\{ 15 \}} | 0.14289 ^{\{ 7 \}} | 0.16877 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.20491 ^{\{ 3 \}} | 0.24867 ^{\{ 8 \}} | 0.25362 ^{\{ 9 \}} | 0.24859 ^{\{ 7 \}} | 0.2669 ^{\{ 11 \}} | 0.26715 ^{\{ 12 \}} | 0.24501 ^{\{ 6 \}} | 0.23003 ^{\{ 4 \}} | 0.17419 ^{\{ 2 \}} | 0.23892 ^{\{ 5 \}} | 0.26963 ^{\{ 14 \}} | 0.10757 ^{\{ 1 \}} | 0.2999 ^{\{ 15 \}} | 0.26236 ^{\{ 10 \}} | 0.2681 ^{\{ 13 \}} | |
D_{abs} | 0.02865 ^{\{ 1 \}} | 0.03235 ^{\{ 3 \}} | 0.03403 ^{\{ 10 \}} | 0.03266 ^{\{ 6 \}} | 0.0338 ^{\{ 8 \}} | 0.03443 ^{\{ 11 \}} | 0.03265 ^{\{ 5 \}} | 0.03189 ^{\{ 2 \}} | 0.03385 ^{\{ 9 \}} | 0.03286 ^{\{ 7 \}} | 0.03487 ^{\{ 12 \}} | 0.03252 ^{\{ 4 \}} | 0.04035 ^{\{ 14 \}} | 0.03501 ^{\{ 13 \}} | 0.04078 ^{\{ 15 \}} | |
D_{max} | 0.04675 ^{\{ 1 \}} | 0.05334 ^{\{ 5 \}} | 0.0571 ^{\{ 11 \}} | 0.05293 ^{\{ 4 \}} | 0.05628 ^{\{ 9 \}} | 0.05832 ^{\{ 13 \}} | 0.05371 ^{\{ 7 \}} | 0.05256 ^{\{ 3 \}} | 0.05479 ^{\{ 8 \}} | 0.05366 ^{\{ 6 \}} | 0.05734 ^{\{ 12 \}} | 0.05239 ^{\{ 2 \}} | 0.06677 ^{\{ 14 \}} | 0.05649 ^{\{ 10 \}} | 0.06719 ^{\{ 15 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 54 ^{\{ 7 \}} | 82 ^{\{ 10 \}} | 45 ^{\{ 5 \}} | 85 ^{\{ 11 \}} | 104 ^{\{ 13 \}} | 56 ^{\{ 8 \}} | 36 ^{\{ 4 \}} | 32 ^{\{ 3 \}} | 47 ^{\{ 6 \}} | 95 ^{\{ 12 \}} | 12 ^{\{ 1 \}} | 117 ^{\{ 15 \}} | 72 ^{\{ 9 \}} | 106 ^{\{ 14 \}} | |
100 | BIAS( \hat{\delta} ) | 0.0231 ^{\{ 2 \}} | 0.02843 ^{\{ 6 \}} | 0.03137 ^{\{ 10 \}} | 0.02728 ^{\{ 4 \}} | 0.0316 ^{\{ 11 \}} | 0.03625 ^{\{ 15 \}} | 0.02813 ^{\{ 5 \}} | 0.02848 ^{\{ 7 \}} | 0.02475 ^{\{ 3 \}} | 0.02932 ^{\{ 9 \}} | 0.03253 ^{\{ 12 \}} | 0.01602 ^{\{ 1 \}} | 0.03265 ^{\{ 13 \}} | 0.029 ^{\{ 8 \}} | 0.03311 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.1315 ^{\{ 3 \}} | 0.14273 ^{\{ 5 \}} | 0.14892 ^{\{ 7 \}} | 0.15193 ^{\{ 8 \}} | 0.15411 ^{\{ 9 \}} | 0.16605 ^{\{ 14 \}} | 0.14096 ^{\{ 4 \}} | 0.14515 ^{\{ 6 \}} | 0.1185 ^{\{ 2 \}} | 0.15562 ^{\{ 10 \}} | 0.1731 ^{\{ 15 \}} | 0.06434 ^{\{ 1 \}} | 0.16197 ^{\{ 12 \}} | 0.16008 ^{\{ 11 \}} | 0.16502 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00086 ^{\{ 2 \}} | 0.00126 ^{\{ 5 \}} | 0.00157 ^{\{ 11 \}} | 0.00112 ^{\{ 4 \}} | 0.00156 ^{\{ 10 \}} | 0.00208 ^{\{ 15 \}} | 0.00127 ^{\{ 6 \}} | 0.00128 ^{\{ 7 \}} | 0.00107 ^{\{ 3 \}} | 0.00136 ^{\{ 9 \}} | 0.00166 ^{\{ 12.5 \}} | 0.00051 ^{\{ 1 \}} | 0.00166 ^{\{ 12.5 \}} | 0.00129 ^{\{ 8 \}} | 0.00177 ^{\{ 14 \}} | |
MSE( \hat{\beta} ) | 0.03144 ^{\{ 3 \}} | 0.03254 ^{\{ 4 \}} | 0.03644 ^{\{ 7 \}} | 0.03847 ^{\{ 8 \}} | 0.03959 ^{\{ 9 \}} | 0.04527 ^{\{ 14 \}} | 0.03344 ^{\{ 5 \}} | 0.03458 ^{\{ 6 \}} | 0.0301 ^{\{ 2 \}} | 0.04335 ^{\{ 12 \}} | 0.04773 ^{\{ 15 \}} | 0.01114 ^{\{ 1 \}} | 0.04165 ^{\{ 10 \}} | 0.04281 ^{\{ 11 \}} | 0.04409 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.09238 ^{\{ 2 \}} | 0.11374 ^{\{ 6 \}} | 0.12547 ^{\{ 10 \}} | 0.10911 ^{\{ 4 \}} | 0.12639 ^{\{ 11 \}} | 0.14502 ^{\{ 15 \}} | 0.1125 ^{\{ 5 \}} | 0.11391 ^{\{ 7 \}} | 0.09899 ^{\{ 3 \}} | 0.11729 ^{\{ 9 \}} | 0.13012 ^{\{ 12 \}} | 0.0641 ^{\{ 1 \}} | 0.13062 ^{\{ 13 \}} | 0.11599 ^{\{ 8 \}} | 0.13244 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.17533 ^{\{ 3 \}} | 0.1903 ^{\{ 5 \}} | 0.19856 ^{\{ 7 \}} | 0.20257 ^{\{ 8 \}} | 0.20548 ^{\{ 9 \}} | 0.2214 ^{\{ 14 \}} | 0.18795 ^{\{ 4 \}} | 0.19354 ^{\{ 6 \}} | 0.158 ^{\{ 2 \}} | 0.20749 ^{\{ 10 \}} | 0.2308 ^{\{ 15 \}} | 0.08578 ^{\{ 1 \}} | 0.21596 ^{\{ 12 \}} | 0.21345 ^{\{ 11 \}} | 0.22003 ^{\{ 13 \}} | |
D_{abs} | 0.02498 ^{\{ 2 \}} | 0.02503 ^{\{ 3 \}} | 0.02656 ^{\{ 7 \}} | 0.02588 ^{\{ 5 \}} | 0.02596 ^{\{ 6 \}} | 0.02722 ^{\{ 11 \}} | 0.02554 ^{\{ 4 \}} | 0.02663 ^{\{ 8 \}} | 0.02832 ^{\{ 12 \}} | 0.02699 ^{\{ 9 \}} | 0.02868 ^{\{ 13 \}} | 0.02409 ^{\{ 1 \}} | 0.03054 ^{\{ 14 \}} | 0.02703 ^{\{ 10 \}} | 0.03077 ^{\{ 15 \}} | |
D_{max} | 0.04055 ^{\{ 2 \}} | 0.04137 ^{\{ 3 \}} | 0.04456 ^{\{ 10 \}} | 0.04216 ^{\{ 5 \}} | 0.04346 ^{\{ 6 \}} | 0.04638 ^{\{ 12 \}} | 0.04211 ^{\{ 4 \}} | 0.04374 ^{\{ 7 \}} | 0.04578 ^{\{ 11 \}} | 0.04428 ^{\{ 9 \}} | 0.04702 ^{\{ 13 \}} | 0.03867 ^{\{ 1 \}} | 0.0502 ^{\{ 14 \}} | 0.04408 ^{\{ 8 \}} | 0.05067 ^{\{ 15 \}} | |
\sum Ranks | 19 ^{\{ 2 \}} | 37 ^{\{ 3.5 \}} | 69 ^{\{ 8 \}} | 46 ^{\{ 6 \}} | 71 ^{\{ 9 \}} | 110 ^{\{ 14 \}} | 37 ^{\{ 3.5 \}} | 54 ^{\{ 7 \}} | 38 ^{\{ 5 \}} | 77 ^{\{ 11 \}} | 107.5 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 100.5 ^{\{ 12 \}} | 75 ^{\{ 10 \}} | 111 ^{\{ 15 \}} | |
200 | BIAS( \hat{\delta} ) | 0.0173 ^{\{ 2 \}} | 0.01938 ^{\{ 7 \}} | 0.02284 ^{\{ 11 \}} | 0.019 ^{\{ 4 \}} | 0.02223 ^{\{ 10 \}} | 0.02627 ^{\{ 15 \}} | 0.02055 ^{\{ 9 \}} | 0.01934 ^{\{ 6 \}} | 0.01838 ^{\{ 3 \}} | 0.01911 ^{\{ 5 \}} | 0.02522 ^{\{ 13 \}} | 0.01175 ^{\{ 1 \}} | 0.02527 ^{\{ 14 \}} | 0.01943 ^{\{ 8 \}} | 0.02293 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.09413 ^{\{ 3 \}} | 0.09871 ^{\{ 5 \}} | 0.10837 ^{\{ 11 \}} | 0.1018 ^{\{ 7 \}} | 0.10854 ^{\{ 12 \}} | 0.11908 ^{\{ 13 \}} | 0.10176 ^{\{ 6 \}} | 0.0986 ^{\{ 4 \}} | 0.08773 ^{\{ 2 \}} | 0.1022 ^{\{ 8 \}} | 0.14097 ^{\{ 15 \}} | 0.04869 ^{\{ 1 \}} | 0.13008 ^{\{ 14 \}} | 0.10478 ^{\{ 9 \}} | 0.10707 ^{\{ 10 \}} | |
MSE( \hat{\delta} ) | 0.00047 ^{\{ 2 \}} | 0.00059 ^{\{ 5.5 \}} | 0.00082 ^{\{ 11 \}} | 0.00057 ^{\{ 3 \}} | 0.00078 ^{\{ 10 \}} | 0.00108 ^{\{ 15 \}} | 0.00065 ^{\{ 9 \}} | 0.00058 ^{\{ 4 \}} | 0.00061 ^{\{ 7 \}} | 0.00062 ^{\{ 8 \}} | 0.00098 ^{\{ 13.5 \}} | 0.00029 ^{\{ 1 \}} | 0.00098 ^{\{ 13.5 \}} | 0.00059 ^{\{ 5.5 \}} | 0.00091 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.01471 ^{\{ 2 \}} | 0.01633 ^{\{ 5 \}} | 0.01863 ^{\{ 9 \}} | 0.01774 ^{\{ 7 \}} | 0.01979 ^{\{ 11 \}} | 0.0231 ^{\{ 13 \}} | 0.01716 ^{\{ 6 \}} | 0.01534 ^{\{ 3 \}} | 0.01628 ^{\{ 4 \}} | 0.0195 ^{\{ 10 \}} | 0.03311 ^{\{ 15 \}} | 0.00672 ^{\{ 1 \}} | 0.02761 ^{\{ 14 \}} | 0.01843 ^{\{ 8 \}} | 0.02024 ^{\{ 12 \}} | |
MRE( \hat{\delta} ) | 0.06921 ^{\{ 2 \}} | 0.07753 ^{\{ 7 \}} | 0.09134 ^{\{ 11 \}} | 0.07601 ^{\{ 4 \}} | 0.08891 ^{\{ 10 \}} | 0.10508 ^{\{ 15 \}} | 0.0822 ^{\{ 9 \}} | 0.07736 ^{\{ 6 \}} | 0.07351 ^{\{ 3 \}} | 0.07643 ^{\{ 5 \}} | 0.10088 ^{\{ 13 \}} | 0.04701 ^{\{ 1 \}} | 0.10108 ^{\{ 14 \}} | 0.07774 ^{\{ 8 \}} | 0.09172 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.1255 ^{\{ 3 \}} | 0.13161 ^{\{ 5 \}} | 0.1445 ^{\{ 11 \}} | 0.13574 ^{\{ 7 \}} | 0.14471 ^{\{ 12 \}} | 0.15878 ^{\{ 13 \}} | 0.13568 ^{\{ 6 \}} | 0.13146 ^{\{ 4 \}} | 0.11697 ^{\{ 2 \}} | 0.13626 ^{\{ 8 \}} | 0.18796 ^{\{ 15 \}} | 0.06491 ^{\{ 1 \}} | 0.17344 ^{\{ 14 \}} | 0.1397 ^{\{ 9 \}} | 0.14275 ^{\{ 10 \}} | |
D_{abs} | 0.01803 ^{\{ 3 \}} | 0.01795 ^{\{ 2 \}} | 0.01849 ^{\{ 6 \}} | 0.01827 ^{\{ 4 \}} | 0.01852 ^{\{ 7 \}} | 0.01943 ^{\{ 9 \}} | 0.01842 ^{\{ 5 \}} | 0.01853 ^{\{ 8 \}} | 0.01976 ^{\{ 12 \}} | 0.01957 ^{\{ 10 \}} | 0.02149 ^{\{ 13 \}} | 0.0176 ^{\{ 1 \}} | 0.0237 ^{\{ 15 \}} | 0.01974 ^{\{ 11 \}} | 0.02287 ^{\{ 14 \}} | |
D_{max} | 0.02919 ^{\{ 2 \}} | 0.02952 ^{\{ 3 \}} | 0.0311 ^{\{ 8 \}} | 0.02977 ^{\{ 4 \}} | 0.03109 ^{\{ 7 \}} | 0.03335 ^{\{ 12 \}} | 0.0306 ^{\{ 6 \}} | 0.03031 ^{\{ 5 \}} | 0.03239 ^{\{ 11 \}} | 0.03179 ^{\{ 9 \}} | 0.03524 ^{\{ 13 \}} | 0.02807 ^{\{ 1 \}} | 0.03898 ^{\{ 15 \}} | 0.03199 ^{\{ 10 \}} | 0.03733 ^{\{ 14 \}} | |
\sum Ranks | 19 ^{\{ 2 \}} | 39.5 ^{\{ 3 \}} | 78 ^{\{ 10 \}} | 40 ^{\{ 4.5 \}} | 79 ^{\{ 11 \}} | 105 ^{\{ 13 \}} | 56 ^{\{ 7 \}} | 40 ^{\{ 4.5 \}} | 44 ^{\{ 6 \}} | 63 ^{\{ 8 \}} | 110.5 ^{\{ 14 \}} | 8 ^{\{ 1 \}} | 113.5 ^{\{ 15 \}} | 68.5 ^{\{ 9 \}} | 96 ^{\{ 12 \}} | |
300 | BIAS( \hat{\delta} ) | 0.01419 ^{\{ 2 \}} | 0.01736 ^{\{ 9 \}} | 0.01883 ^{\{ 13 \}} | 0.01549 ^{\{ 4 \}} | 0.01842 ^{\{ 12 \}} | 0.02076 ^{\{ 15 \}} | 0.01633 ^{\{ 7 \}} | 0.01636 ^{\{ 8 \}} | 0.01534 ^{\{ 3 \}} | 0.01629 ^{\{ 6 \}} | 0.01982 ^{\{ 14 \}} | 0.00983 ^{\{ 1 \}} | 0.01821 ^{\{ 11 \}} | 0.01577 ^{\{ 5 \}} | 0.01778 ^{\{ 10 \}} |
BIAS( \hat{\beta} ) | 0.0731 ^{\{ 2 \}} | 0.08699 ^{\{ 9 \}} | 0.08814 ^{\{ 11 \}} | 0.08227 ^{\{ 4 \}} | 0.09008 ^{\{ 12 \}} | 0.09593 ^{\{ 14 \}} | 0.08242 ^{\{ 5 \}} | 0.08312 ^{\{ 6 \}} | 0.07557 ^{\{ 3 \}} | 0.08758 ^{\{ 10 \}} | 0.10944 ^{\{ 15 \}} | 0.03688 ^{\{ 1 \}} | 0.09063 ^{\{ 13 \}} | 0.08574 ^{\{ 8 \}} | 0.08526 ^{\{ 7 \}} | |
MSE( \hat{\delta} ) | 0.00031 ^{\{ 2 \}} | 0.00048 ^{\{ 9 \}} | 0.00056 ^{\{ 12 \}} | 0.00038 ^{\{ 3 \}} | 0.00054 ^{\{ 10.5 \}} | 0.00067 ^{\{ 15 \}} | 0.00044 ^{\{ 8 \}} | 0.00041 ^{\{ 5.5 \}} | 0.00041 ^{\{ 5.5 \}} | 0.00043 ^{\{ 7 \}} | 0.00062 ^{\{ 14 \}} | 2e-04 ^{\{ 1 \}} | 0.00057 ^{\{ 13 \}} | 4e-04 ^{\{ 4 \}} | 0.00054 ^{\{ 10.5 \}} | |
MSE( \hat{\beta} ) | 0.00848 ^{\{ 2 \}} | 0.01219 ^{\{ 8 \}} | 0.01264 ^{\{ 10 \}} | 0.01114 ^{\{ 4 \}} | 0.01321 ^{\{ 11 \}} | 0.01472 ^{\{ 14 \}} | 0.01129 ^{\{ 5 \}} | 0.0111 ^{\{ 3 \}} | 0.01187 ^{\{ 6 \}} | 0.01322 ^{\{ 12 \}} | 0.02008 ^{\{ 15 \}} | 0.00412 ^{\{ 1 \}} | 0.01441 ^{\{ 13 \}} | 0.01218 ^{\{ 7 \}} | 0.01225 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.05677 ^{\{ 2 \}} | 0.06944 ^{\{ 9 \}} | 0.07531 ^{\{ 13 \}} | 0.06196 ^{\{ 4 \}} | 0.07367 ^{\{ 12 \}} | 0.08303 ^{\{ 15 \}} | 0.06531 ^{\{ 7 \}} | 0.06545 ^{\{ 8 \}} | 0.06137 ^{\{ 3 \}} | 0.06515 ^{\{ 6 \}} | 0.07927 ^{\{ 14 \}} | 0.03933 ^{\{ 1 \}} | 0.07285 ^{\{ 11 \}} | 0.06308 ^{\{ 5 \}} | 0.07112 ^{\{ 10 \}} | |
MRE( \hat{\beta} ) | 0.09747 ^{\{ 2 \}} | 0.11599 ^{\{ 9 \}} | 0.11753 ^{\{ 11 \}} | 0.10969 ^{\{ 4 \}} | 0.12011 ^{\{ 12 \}} | 0.12791 ^{\{ 14 \}} | 0.1099 ^{\{ 5 \}} | 0.11082 ^{\{ 6 \}} | 0.10076 ^{\{ 3 \}} | 0.11678 ^{\{ 10 \}} | 0.14592 ^{\{ 15 \}} | 0.04918 ^{\{ 1 \}} | 0.12084 ^{\{ 13 \}} | 0.11432 ^{\{ 8 \}} | 0.11368 ^{\{ 7 \}} | |
D_{abs} | 0.01447 ^{\{ 2 \}} | 0.01529 ^{\{ 7 \}} | 0.01535 ^{\{ 8 \}} | 0.01484 ^{\{ 3 \}} | 0.01519 ^{\{ 6 \}} | 0.01601 ^{\{ 9 \}} | 0.01493 ^{\{ 4 \}} | 0.01504 ^{\{ 5 \}} | 0.01637 ^{\{ 11 \}} | 0.01677 ^{\{ 12 \}} | 0.01742 ^{\{ 13 \}} | 0.01433 ^{\{ 1 \}} | 0.01906 ^{\{ 15 \}} | 0.0163 ^{\{ 10 \}} | 0.0188 ^{\{ 14 \}} | |
D_{max} | 0.02349 ^{\{ 2 \}} | 0.02522 ^{\{ 6 \}} | 0.02584 ^{\{ 8 \}} | 0.02421 ^{\{ 3 \}} | 0.02533 ^{\{ 7 \}} | 0.02724 ^{\{ 12 \}} | 0.02471 ^{\{ 4 \}} | 0.02483 ^{\{ 5 \}} | 0.02669 ^{\{ 10 \}} | 0.02712 ^{\{ 11 \}} | 0.02847 ^{\{ 13 \}} | 0.02302 ^{\{ 1 \}} | 0.03095 ^{\{ 15 \}} | 0.02641 ^{\{ 9 \}} | 0.0306 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 1 \}} | 64 ^{\{ 7 \}} | 84 ^{\{ 12 \}} | 28 ^{\{ 3 \}} | 80.5 ^{\{ 11 \}} | 106 ^{\{ 14 \}} | 43 ^{\{ 5 \}} | 44.5 ^{\{ 6 \}} | 42.5 ^{\{ 4 \}} | 72 ^{\{ 9 \}} | 111 ^{\{ 15 \}} | 21 ^{\{ 2 \}} | 102 ^{\{ 13 \}} | 67 ^{\{ 8 \}} | 79.5 ^{\{ 10 \}} | |
400 | BIAS( \hat{\delta} ) | 0.0123 ^{\{ 2 \}} | 0.01445 ^{\{ 7 \}} | 0.01582 ^{\{ 12 \}} | 0.01337 ^{\{ 4 \}} | 0.01565 ^{\{ 11 \}} | 0.01778 ^{\{ 14 \}} | 0.01429 ^{\{ 6 \}} | 0.01336 ^{\{ 3 \}} | 0.01397 ^{\{ 5 \}} | 0.01482 ^{\{ 9 \}} | 0.01814 ^{\{ 15 \}} | 0.0086 ^{\{ 1 \}} | 0.01606 ^{\{ 13 \}} | 0.01477 ^{\{ 8 \}} | 0.01543 ^{\{ 10 \}} |
BIAS( \hat{\beta} ) | 0.06611 ^{\{ 2 \}} | 0.07122 ^{\{ 6 \}} | 0.07645 ^{\{ 9 \}} | 0.07072 ^{\{ 5 \}} | 0.07554 ^{\{ 8 \}} | 0.08205 ^{\{ 14 \}} | 0.07194 ^{\{ 7 \}} | 0.06821 ^{\{ 3 \}} | 0.07008 ^{\{ 4 \}} | 0.07952 ^{\{ 13 \}} | 0.10105 ^{\{ 15 \}} | 0.03501 ^{\{ 1 \}} | 0.07876 ^{\{ 10 \}} | 0.07925 ^{\{ 12 \}} | 0.07879 ^{\{ 11 \}} | |
MSE( \hat{\delta} ) | 0.00023 ^{\{ 2 \}} | 0.00032 ^{\{ 6 \}} | 4e-04 ^{\{ 12 \}} | 0.00027 ^{\{ 3.5 \}} | 0.00039 ^{\{ 11 \}} | 0.00049 ^{\{ 14 \}} | 0.00032 ^{\{ 6 \}} | 0.00027 ^{\{ 3.5 \}} | 0.00035 ^{\{ 8 \}} | 0.00036 ^{\{ 9 \}} | 5e-04 ^{\{ 15 \}} | 0.00016 ^{\{ 1 \}} | 0.00044 ^{\{ 13 \}} | 0.00032 ^{\{ 6 \}} | 0.00037 ^{\{ 10 \}} | |
MSE( \hat{\beta} ) | 0.00715 ^{\{ 2 \}} | 0.00794 ^{\{ 5 \}} | 0.00928 ^{\{ 8 \}} | 0.00783 ^{\{ 4 \}} | 0.00904 ^{\{ 7 \}} | 0.01058 ^{\{ 12 \}} | 0.00843 ^{\{ 6 \}} | 0.00727 ^{\{ 3 \}} | 0.00986 ^{\{ 11 \}} | 0.011 ^{\{ 14 \}} | 0.0167 ^{\{ 15 \}} | 0.00343 ^{\{ 1 \}} | 0.01096 ^{\{ 13 \}} | 0.00975 ^{\{ 10 \}} | 0.00967 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.0492 ^{\{ 2 \}} | 0.05779 ^{\{ 7 \}} | 0.06329 ^{\{ 12 \}} | 0.05346 ^{\{ 4 \}} | 0.06258 ^{\{ 11 \}} | 0.07111 ^{\{ 14 \}} | 0.05715 ^{\{ 6 \}} | 0.05345 ^{\{ 3 \}} | 0.05587 ^{\{ 5 \}} | 0.05929 ^{\{ 9 \}} | 0.07256 ^{\{ 15 \}} | 0.0344 ^{\{ 1 \}} | 0.06425 ^{\{ 13 \}} | 0.0591 ^{\{ 8 \}} | 0.06173 ^{\{ 10 \}} | |
MRE( \hat{\beta} ) | 0.08815 ^{\{ 2 \}} | 0.09495 ^{\{ 6 \}} | 0.10194 ^{\{ 9 \}} | 0.0943 ^{\{ 5 \}} | 0.10072 ^{\{ 8 \}} | 0.1094 ^{\{ 14 \}} | 0.09591 ^{\{ 7 \}} | 0.09095 ^{\{ 3 \}} | 0.09344 ^{\{ 4 \}} | 0.10603 ^{\{ 13 \}} | 0.13473 ^{\{ 15 \}} | 0.04668 ^{\{ 1 \}} | 0.10501 ^{\{ 10 \}} | 0.10567 ^{\{ 12 \}} | 0.10505 ^{\{ 11 \}} | |
D_{abs} | 0.01235 ^{\{ 2 \}} | 0.01243 ^{\{ 3 \}} | 0.0136 ^{\{ 8 \}} | 0.01295 ^{\{ 6 \}} | 0.01324 ^{\{ 7 \}} | 0.0137 ^{\{ 9 \}} | 0.01286 ^{\{ 5 \}} | 0.01258 ^{\{ 4 \}} | 0.01459 ^{\{ 11 \}} | 0.01484 ^{\{ 12 \}} | 0.01561 ^{\{ 13 \}} | 0.0121 ^{\{ 1 \}} | 0.01697 ^{\{ 15 \}} | 0.01421 ^{\{ 10 \}} | 0.01637 ^{\{ 14 \}} | |
D_{max} | 0.02003 ^{\{ 2 \}} | 0.02073 ^{\{ 4 \}} | 0.02267 ^{\{ 8 \}} | 0.0211 ^{\{ 5 \}} | 0.02216 ^{\{ 7 \}} | 0.02335 ^{\{ 10 \}} | 0.02123 ^{\{ 6 \}} | 0.02064 ^{\{ 3 \}} | 0.02367 ^{\{ 11 \}} | 0.02415 ^{\{ 12 \}} | 0.02563 ^{\{ 13 \}} | 0.01949 ^{\{ 1 \}} | 0.02748 ^{\{ 15 \}} | 0.02321 ^{\{ 9 \}} | 0.02659 ^{\{ 14 \}} | |
\sum Ranks | 16 ^{\{ 2 \}} | 44 ^{\{ 5 \}} | 80 ^{\{ 10 \}} | 36.5 ^{\{ 4 \}} | 70 ^{\{ 8 \}} | 100 ^{\{ 13 \}} | 49 ^{\{ 6 \}} | 25.5 ^{\{ 3 \}} | 59 ^{\{ 7 \}} | 91 ^{\{ 12 \}} | 116 ^{\{ 15 \}} | 8 ^{\{ 1 \}} | 101 ^{\{ 14 \}} | 75 ^{\{ 9 \}} | 89 ^{\{ 11 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.36362 ^{\{ 4 \}} | 0.41591 ^{\{ 10 \}} | 0.46651 ^{\{ 14 \}} | 0.3611 ^{\{ 3 \}} | 0.43973 ^{\{ 13 \}} | 0.47679 ^{\{ 15 \}} | 0.42017 ^{\{ 11 \}} | 0.3771 ^{\{ 7 \}} | 0.31372 ^{\{ 2 \}} | 0.37161 ^{\{ 6 \}} | 0.42351 ^{\{ 12 \}} | 0.18844 ^{\{ 1 \}} | 0.41564 ^{\{ 9 \}} | 0.3691 ^{\{ 5 \}} | 0.40653 ^{\{ 8 \}} |
BIAS( \hat{\beta} ) | 0.44791 ^{\{ 3 \}} | 0.47888 ^{\{ 6 \}} | 0.50494 ^{\{ 11 \}} | 0.48893 ^{\{ 8 \}} | 0.50806 ^{\{ 12 \}} | 0.49815 ^{\{ 10 \}} | 0.50822 ^{\{ 13 \}} | 0.46815 ^{\{ 4 \}} | 0.37554 ^{\{ 2 \}} | 0.47535 ^{\{ 5 \}} | 0.51634 ^{\{ 15 \}} | 0.26964 ^{\{ 1 \}} | 0.51083 ^{\{ 14 \}} | 0.48372 ^{\{ 7 \}} | 0.49761 ^{\{ 9 \}} | |
MSE( \hat{\delta} ) | 0.21037 ^{\{ 4 \}} | 0.26991 ^{\{ 10 \}} | 0.33182 ^{\{ 14 \}} | 0.20202 ^{\{ 3 \}} | 0.29159 ^{\{ 13 \}} | 0.35861 ^{\{ 15 \}} | 0.27624 ^{\{ 11 \}} | 0.22664 ^{\{ 7 \}} | 0.17428 ^{\{ 2 \}} | 0.21641 ^{\{ 5 \}} | 0.2846 ^{\{ 12 \}} | 0.05936 ^{\{ 1 \}} | 0.26835 ^{\{ 9 \}} | 0.21794 ^{\{ 6 \}} | 0.25878 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.29849 ^{\{ 3 \}} | 0.33823 ^{\{ 5 \}} | 0.36564 ^{\{ 11 \}} | 0.36137 ^{\{ 10 \}} | 0.37711 ^{\{ 13 \}} | 0.3567 ^{\{ 8 \}} | 0.3741 ^{\{ 12 \}} | 0.32343 ^{\{ 4 \}} | 0.25591 ^{\{ 2 \}} | 0.35154 ^{\{ 7 \}} | 0.38344 ^{\{ 15 \}} | 0.12377 ^{\{ 1 \}} | 0.38059 ^{\{ 14 \}} | 0.35106 ^{\{ 6 \}} | 0.35683 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.24241 ^{\{ 4 \}} | 0.27727 ^{\{ 10 \}} | 0.31101 ^{\{ 14 \}} | 0.24073 ^{\{ 3 \}} | 0.29315 ^{\{ 13 \}} | 0.31786 ^{\{ 15 \}} | 0.28011 ^{\{ 11 \}} | 0.2514 ^{\{ 7 \}} | 0.20915 ^{\{ 2 \}} | 0.24774 ^{\{ 6 \}} | 0.28234 ^{\{ 12 \}} | 0.12563 ^{\{ 1 \}} | 0.27709 ^{\{ 9 \}} | 0.24606 ^{\{ 5 \}} | 0.27102 ^{\{ 8 \}} | |
MRE( \hat{\beta} ) | 0.29861 ^{\{ 3 \}} | 0.31926 ^{\{ 6 \}} | 0.33662 ^{\{ 11 \}} | 0.32595 ^{\{ 8 \}} | 0.33871 ^{\{ 12 \}} | 0.3321 ^{\{ 10 \}} | 0.33882 ^{\{ 13 \}} | 0.3121 ^{\{ 4 \}} | 0.25036 ^{\{ 2 \}} | 0.3169 ^{\{ 5 \}} | 0.34423 ^{\{ 15 \}} | 0.17976 ^{\{ 1 \}} | 0.34056 ^{\{ 14 \}} | 0.32248 ^{\{ 7 \}} | 0.33174 ^{\{ 9 \}} | |
D_{abs} | 0.03944 ^{\{ 1 \}} | 0.04453 ^{\{ 8 \}} | 0.04527 ^{\{ 10 \}} | 0.04151 ^{\{ 3 \}} | 0.04361 ^{\{ 5 \}} | 0.04561 ^{\{ 12 \}} | 0.04232 ^{\{ 4 \}} | 0.04094 ^{\{ 2 \}} | 0.04552 ^{\{ 11 \}} | 0.04852 ^{\{ 13 \}} | 0.04426 ^{\{ 7 \}} | 0.04386 ^{\{ 6 \}} | 0.05483 ^{\{ 15 \}} | 0.04475 ^{\{ 9 \}} | 0.05458 ^{\{ 14 \}} | |
D_{max} | 0.06565 ^{\{ 1 \}} | 0.07283 ^{\{ 8 \}} | 0.07644 ^{\{ 11 \}} | 0.06681 ^{\{ 2 \}} | 0.0725 ^{\{ 7 \}} | 0.07674 ^{\{ 12 \}} | 0.06966 ^{\{ 5 \}} | 0.06724 ^{\{ 3 \}} | 0.07288 ^{\{ 9 \}} | 0.07755 ^{\{ 13 \}} | 0.07318 ^{\{ 10 \}} | 0.06774 ^{\{ 4 \}} | 0.0875 ^{\{ 15 \}} | 0.07206 ^{\{ 6 \}} | 0.08689 ^{\{ 14 \}} | |
\sum Ranks | 23 ^{\{ 2 \}} | 63 ^{\{ 8 \}} | 96 ^{\{ 12 \}} | 40 ^{\{ 5 \}} | 88 ^{\{ 11 \}} | 97 ^{\{ 13 \}} | 80 ^{\{ 10 \}} | 38 ^{\{ 4 \}} | 32 ^{\{ 3 \}} | 60 ^{\{ 7 \}} | 98 ^{\{ 14 \}} | 16 ^{\{ 1 \}} | 99 ^{\{ 15 \}} | 51 ^{\{ 6 \}} | 79 ^{\{ 9 \}} | |
60 | BIAS( \hat{\delta} ) | 0.25193 ^{\{ 2 \}} | 0.31471 ^{\{ 8 \}} | 0.35735 ^{\{ 14 \}} | 0.27976 ^{\{ 4 \}} | 0.34812 ^{\{ 12 \}} | 0.40958 ^{\{ 15 \}} | 0.32601 ^{\{ 10 \}} | 0.28841 ^{\{ 5 \}} | 0.26064 ^{\{ 3 \}} | 0.29235 ^{\{ 7 \}} | 0.32535 ^{\{ 9 \}} | 0.17636 ^{\{ 1 \}} | 0.35405 ^{\{ 13 \}} | 0.28887 ^{\{ 6 \}} | 0.34765 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.34938 ^{\{ 2 \}} | 0.40316 ^{\{ 5 \}} | 0.41962 ^{\{ 7 \}} | 0.412 ^{\{ 6 \}} | 0.42457 ^{\{ 9 \}} | 0.45051 ^{\{ 14 \}} | 0.42184 ^{\{ 8 \}} | 0.38627 ^{\{ 4 \}} | 0.34945 ^{\{ 3 \}} | 0.42834 ^{\{ 10 \}} | 0.44339 ^{\{ 12 \}} | 0.24504 ^{\{ 1 \}} | 0.46935 ^{\{ 15 \}} | 0.43185 ^{\{ 11 \}} | 0.44997 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.10131 ^{\{ 2 \}} | 0.14878 ^{\{ 8 \}} | 0.20227 ^{\{ 14 \}} | 0.12128 ^{\{ 4 \}} | 0.19801 ^{\{ 13 \}} | 0.25959 ^{\{ 15 \}} | 0.16855 ^{\{ 10 \}} | 0.12991 ^{\{ 5 \}} | 0.12121 ^{\{ 3 \}} | 0.13012 ^{\{ 6 \}} | 0.1646 ^{\{ 9 \}} | 0.05318 ^{\{ 1 \}} | 0.19114 ^{\{ 11 \}} | 0.13018 ^{\{ 7 \}} | 0.1939 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.19887 ^{\{ 2 \}} | 0.24543 ^{\{ 5 \}} | 0.2617 ^{\{ 6 \}} | 0.28333 ^{\{ 9 \}} | 0.28214 ^{\{ 8 \}} | 0.30154 ^{\{ 13 \}} | 0.27339 ^{\{ 7 \}} | 0.23646 ^{\{ 4 \}} | 0.23071 ^{\{ 3 \}} | 0.29922 ^{\{ 11 \}} | 0.30099 ^{\{ 12 \}} | 0.1013 ^{\{ 1 \}} | 0.3262 ^{\{ 15 \}} | 0.29724 ^{\{ 10 \}} | 0.31345 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.16795 ^{\{ 2 \}} | 0.20981 ^{\{ 8 \}} | 0.23823 ^{\{ 14 \}} | 0.18651 ^{\{ 4 \}} | 0.23208 ^{\{ 12 \}} | 0.27306 ^{\{ 15 \}} | 0.21734 ^{\{ 10 \}} | 0.19227 ^{\{ 5 \}} | 0.17376 ^{\{ 3 \}} | 0.1949 ^{\{ 7 \}} | 0.2169 ^{\{ 9 \}} | 0.11757 ^{\{ 1 \}} | 0.23603 ^{\{ 13 \}} | 0.19258 ^{\{ 6 \}} | 0.23177 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.23292 ^{\{ 2 \}} | 0.26877 ^{\{ 5 \}} | 0.27975 ^{\{ 7 \}} | 0.27467 ^{\{ 6 \}} | 0.28305 ^{\{ 9 \}} | 0.30034 ^{\{ 14 \}} | 0.28122 ^{\{ 8 \}} | 0.25751 ^{\{ 4 \}} | 0.23297 ^{\{ 3 \}} | 0.28556 ^{\{ 10 \}} | 0.2956 ^{\{ 12 \}} | 0.16336 ^{\{ 1 \}} | 0.3129 ^{\{ 15 \}} | 0.2879 ^{\{ 11 \}} | 0.29998 ^{\{ 13 \}} | |
D_{abs} | 0.02961 ^{\{ 1 \}} | 0.03051 ^{\{ 2 \}} | 0.03132 ^{\{ 6 \}} | 0.03103 ^{\{ 5 \}} | 0.03319 ^{\{ 9 \}} | 0.03268 ^{\{ 7 \}} | 0.03101 ^{\{ 4 \}} | 0.03063 ^{\{ 3 \}} | 0.03576 ^{\{ 13 \}} | 0.03372 ^{\{ 11 \}} | 0.03366 ^{\{ 10 \}} | 0.03275 ^{\{ 8 \}} | 0.03914 ^{\{ 14 \}} | 0.03378 ^{\{ 12 \}} | 0.03955 ^{\{ 15 \}} | |
D_{max} | 0.04842 ^{\{ 1 \}} | 0.05047 ^{\{ 4 \}} | 0.05302 ^{\{ 7 \}} | 0.04997 ^{\{ 2 \}} | 0.05541 ^{\{ 11 \}} | 0.05606 ^{\{ 12 \}} | 0.05152 ^{\{ 6 \}} | 0.05036 ^{\{ 3 \}} | 0.0573 ^{\{ 13 \}} | 0.05469 ^{\{ 9 \}} | 0.05517 ^{\{ 10 \}} | 0.0514 ^{\{ 5 \}} | 0.06451 ^{\{ 14 \}} | 0.05445 ^{\{ 8 \}} | 0.06483 ^{\{ 15 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 45 ^{\{ 6 \}} | 75 ^{\{ 10 \}} | 40 ^{\{ 4 \}} | 83 ^{\{ 11.5 \}} | 105 ^{\{ 14 \}} | 63 ^{\{ 7 \}} | 33 ^{\{ 3 \}} | 44 ^{\{ 5 \}} | 71 ^{\{ 8.5 \}} | 83 ^{\{ 11.5 \}} | 19 ^{\{ 2 \}} | 110 ^{\{ 15 \}} | 71 ^{\{ 8.5 \}} | 104 ^{\{ 13 \}} | |
100 | BIAS( \hat{\delta} ) | 0.20656 ^{\{ 2 \}} | 0.25391 ^{\{ 8 \}} | 0.30172 ^{\{ 13 \}} | 0.23433 ^{\{ 5 \}} | 0.29934 ^{\{ 12 \}} | 0.327 ^{\{ 15 \}} | 0.25672 ^{\{ 9 \}} | 0.23765 ^{\{ 6 \}} | 0.2224 ^{\{ 3 \}} | 0.24672 ^{\{ 7 \}} | 0.26161 ^{\{ 10 \}} | 0.16346 ^{\{ 1 \}} | 0.30254 ^{\{ 14 \}} | 0.23314 ^{\{ 4 \}} | 0.29396 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.28605 ^{\{ 2 \}} | 0.34848 ^{\{ 6 \}} | 0.37222 ^{\{ 10 \}} | 0.35822 ^{\{ 8 \}} | 0.38959 ^{\{ 13 \}} | 0.38082 ^{\{ 12 \}} | 0.34956 ^{\{ 7 \}} | 0.32677 ^{\{ 4 \}} | 0.30141 ^{\{ 3 \}} | 0.36475 ^{\{ 9 \}} | 0.37444 ^{\{ 11 \}} | 0.22382 ^{\{ 1 \}} | 0.40691 ^{\{ 15 \}} | 0.34202 ^{\{ 5 \}} | 0.40473 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.06736 ^{\{ 2 \}} | 0.10079 ^{\{ 8 \}} | 0.14079 ^{\{ 14 \}} | 0.08235 ^{\{ 3 \}} | 0.13993 ^{\{ 13 \}} | 0.17078 ^{\{ 15 \}} | 0.10453 ^{\{ 9 \}} | 0.09235 ^{\{ 6 \}} | 0.08912 ^{\{ 5 \}} | 0.09267 ^{\{ 7 \}} | 0.10813 ^{\{ 10 \}} | 0.0469 ^{\{ 1 \}} | 0.13908 ^{\{ 12 \}} | 0.08462 ^{\{ 4 \}} | 0.13209 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.14544 ^{\{ 2 \}} | 0.19976 ^{\{ 6 \}} | 0.22305 ^{\{ 9 \}} | 0.21696 ^{\{ 8 \}} | 0.23768 ^{\{ 13 \}} | 0.23128 ^{\{ 12 \}} | 0.20607 ^{\{ 7 \}} | 0.18067 ^{\{ 4 \}} | 0.1778 ^{\{ 3 \}} | 0.22367 ^{\{ 10 \}} | 0.22523 ^{\{ 11 \}} | 0.08729 ^{\{ 1 \}} | 0.26214 ^{\{ 15 \}} | 0.1935 ^{\{ 5 \}} | 0.25864 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.13771 ^{\{ 2 \}} | 0.16927 ^{\{ 8 \}} | 0.20115 ^{\{ 13 \}} | 0.15622 ^{\{ 5 \}} | 0.19956 ^{\{ 12 \}} | 0.218 ^{\{ 15 \}} | 0.17115 ^{\{ 9 \}} | 0.15844 ^{\{ 6 \}} | 0.14827 ^{\{ 3 \}} | 0.16448 ^{\{ 7 \}} | 0.17441 ^{\{ 10 \}} | 0.10897 ^{\{ 1 \}} | 0.2017 ^{\{ 14 \}} | 0.15543 ^{\{ 4 \}} | 0.19598 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.1907 ^{\{ 2 \}} | 0.23232 ^{\{ 6 \}} | 0.24815 ^{\{ 10 \}} | 0.23882 ^{\{ 8 \}} | 0.25972 ^{\{ 13 \}} | 0.25388 ^{\{ 12 \}} | 0.23304 ^{\{ 7 \}} | 0.21785 ^{\{ 4 \}} | 0.20094 ^{\{ 3 \}} | 0.24317 ^{\{ 9 \}} | 0.24963 ^{\{ 11 \}} | 0.14921 ^{\{ 1 \}} | 0.27127 ^{\{ 15 \}} | 0.22801 ^{\{ 5 \}} | 0.26982 ^{\{ 14 \}} | |
D_{abs} | 0.02235 ^{\{ 1 \}} | 0.02482 ^{\{ 3 \}} | 0.02569 ^{\{ 6 \}} | 0.02462 ^{\{ 2 \}} | 0.02622 ^{\{ 10 \}} | 0.02597 ^{\{ 8 \}} | 0.0254 ^{\{ 5 \}} | 0.02484 ^{\{ 4 \}} | 0.02732 ^{\{ 13 \}} | 0.02672 ^{\{ 12 \}} | 0.02663 ^{\{ 11 \}} | 0.02587 ^{\{ 7 \}} | 0.03105 ^{\{ 14 \}} | 0.02607 ^{\{ 9 \}} | 0.03134 ^{\{ 15 \}} | |
D_{max} | 0.03675 ^{\{ 1 \}} | 0.04106 ^{\{ 4 \}} | 0.04372 ^{\{ 10 \}} | 0.04 ^{\{ 2 \}} | 0.04399 ^{\{ 12 \}} | 0.04501 ^{\{ 13 \}} | 0.04203 ^{\{ 6 \}} | 0.04065 ^{\{ 3 \}} | 0.04393 ^{\{ 11 \}} | 0.04357 ^{\{ 8 \}} | 0.04363 ^{\{ 9 \}} | 0.0414 ^{\{ 5 \}} | 0.05119 ^{\{ 14 \}} | 0.04208 ^{\{ 7 \}} | 0.05143 ^{\{ 15 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 49 ^{\{ 7 \}} | 85 ^{\{ 11 \}} | 41 ^{\{ 4 \}} | 98 ^{\{ 12 \}} | 102 ^{\{ 13 \}} | 59 ^{\{ 8 \}} | 37 ^{\{ 3 \}} | 44 ^{\{ 6 \}} | 69 ^{\{ 9 \}} | 83 ^{\{ 10 \}} | 18 ^{\{ 2 \}} | 113 ^{\{ 15 \}} | 43 ^{\{ 5 \}} | 105 ^{\{ 14 \}} | |
200 | BIAS( \hat{\delta} ) | 0.15288 ^{\{ 2 \}} | 0.18956 ^{\{ 8 \}} | 0.21534 ^{\{ 11 \}} | 0.16331 ^{\{ 3 \}} | 0.21778 ^{\{ 12 \}} | 0.26177 ^{\{ 15 \}} | 0.1897 ^{\{ 9 \}} | 0.1732 ^{\{ 4 \}} | 0.17595 ^{\{ 5 \}} | 0.18315 ^{\{ 7 \}} | 0.20027 ^{\{ 10 \}} | 0.13041 ^{\{ 1 \}} | 0.2255 ^{\{ 14 \}} | 0.18271 ^{\{ 6 \}} | 0.22008 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.2161 ^{\{ 2 \}} | 0.25397 ^{\{ 6 \}} | 0.27965 ^{\{ 10 \}} | 0.24415 ^{\{ 4 \}} | 0.28824 ^{\{ 11 \}} | 0.32319 ^{\{ 15 \}} | 0.25894 ^{\{ 7 \}} | 0.23866 ^{\{ 3 \}} | 0.24499 ^{\{ 5 \}} | 0.27244 ^{\{ 9 \}} | 0.29863 ^{\{ 12 \}} | 0.1761 ^{\{ 1 \}} | 0.32229 ^{\{ 14 \}} | 0.26751 ^{\{ 8 \}} | 0.31447 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.03871 ^{\{ 2 \}} | 0.05626 ^{\{ 8 \}} | 0.07407 ^{\{ 12 \}} | 0.0409 ^{\{ 3 \}} | 0.07383 ^{\{ 11 \}} | 0.10743 ^{\{ 15 \}} | 0.05719 ^{\{ 9 \}} | 0.04689 ^{\{ 4 \}} | 0.0535 ^{\{ 7 \}} | 0.05235 ^{\{ 6 \}} | 0.06256 ^{\{ 10 \}} | 0.02939 ^{\{ 1 \}} | 0.07872 ^{\{ 14 \}} | 0.05169 ^{\{ 5 \}} | 0.07812 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.08188 ^{\{ 2 \}} | 0.11144 ^{\{ 5 \}} | 0.13504 ^{\{ 10 \}} | 0.10336 ^{\{ 4 \}} | 0.14447 ^{\{ 11 \}} | 0.17092 ^{\{ 14 \}} | 0.11355 ^{\{ 6 \}} | 0.09586 ^{\{ 3 \}} | 0.11854 ^{\{ 7 \}} | 0.13004 ^{\{ 9 \}} | 0.15599 ^{\{ 12 \}} | 0.0546 ^{\{ 1 \}} | 0.17622 ^{\{ 15 \}} | 0.12213 ^{\{ 8 \}} | 0.16875 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.10192 ^{\{ 2 \}} | 0.12638 ^{\{ 8 \}} | 0.14356 ^{\{ 11 \}} | 0.10888 ^{\{ 3 \}} | 0.14519 ^{\{ 12 \}} | 0.17452 ^{\{ 15 \}} | 0.12646 ^{\{ 9 \}} | 0.11547 ^{\{ 4 \}} | 0.1173 ^{\{ 5 \}} | 0.1221 ^{\{ 7 \}} | 0.13351 ^{\{ 10 \}} | 0.08694 ^{\{ 1 \}} | 0.15034 ^{\{ 14 \}} | 0.1218 ^{\{ 6 \}} | 0.14672 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.14407 ^{\{ 2 \}} | 0.16931 ^{\{ 6 \}} | 0.18643 ^{\{ 10 \}} | 0.16277 ^{\{ 4 \}} | 0.19216 ^{\{ 11 \}} | 0.21546 ^{\{ 15 \}} | 0.17263 ^{\{ 7 \}} | 0.1591 ^{\{ 3 \}} | 0.16333 ^{\{ 5 \}} | 0.18163 ^{\{ 9 \}} | 0.19909 ^{\{ 12 \}} | 0.1174 ^{\{ 1 \}} | 0.21486 ^{\{ 14 \}} | 0.17834 ^{\{ 8 \}} | 0.20965 ^{\{ 13 \}} | |
D_{abs} | 0.01708 ^{\{ 1 \}} | 0.01778 ^{\{ 4 \}} | 0.01841 ^{\{ 7 \}} | 0.01749 ^{\{ 2.5 \}} | 0.01868 ^{\{ 8 \}} | 0.01953 ^{\{ 9 \}} | 0.01788 ^{\{ 6 \}} | 0.01749 ^{\{ 2.5 \}} | 0.02059 ^{\{ 13 \}} | 0.02042 ^{\{ 12 \}} | 0.02035 ^{\{ 11 \}} | 0.01781 ^{\{ 5 \}} | 0.02273 ^{\{ 14 \}} | 0.01969 ^{\{ 10 \}} | 0.02289 ^{\{ 15 \}} | |
D_{max} | 0.02783 ^{\{ 1 \}} | 0.02966 ^{\{ 5 \}} | 0.03114 ^{\{ 7 \}} | 0.02835 ^{\{ 2 \}} | 0.03144 ^{\{ 8 \}} | 0.03378 ^{\{ 13 \}} | 0.02968 ^{\{ 6 \}} | 0.02876 ^{\{ 4 \}} | 0.03344 ^{\{ 12 \}} | 0.03311 ^{\{ 10 \}} | 0.03321 ^{\{ 11 \}} | 0.02872 ^{\{ 3 \}} | 0.03777 ^{\{ 15 \}} | 0.03208 ^{\{ 9 \}} | 0.03745 ^{\{ 14 \}} | |
\sum Ranks | 14 ^{\{ 1.5 \}} | 50 ^{\{ 5 \}} | 78 ^{\{ 10 \}} | 25.5 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 111 ^{\{ 14 \}} | 59 ^{\{ 6.5 \}} | 27.5 ^{\{ 4 \}} | 59 ^{\{ 6.5 \}} | 69 ^{\{ 9 \}} | 88 ^{\{ 12 \}} | 14 ^{\{ 1.5 \}} | 114 ^{\{ 15 \}} | 60 ^{\{ 8 \}} | 107 ^{\{ 13 \}} | |
300 | BIAS( \hat{\delta} ) | 0.12756 ^{\{ 2 \}} | 0.14717 ^{\{ 6 \}} | 0.17874 ^{\{ 12 \}} | 0.13713 ^{\{ 3 \}} | 0.17341 ^{\{ 11 \}} | 0.21053 ^{\{ 15 \}} | 0.16111 ^{\{ 9 \}} | 0.13772 ^{\{ 4 \}} | 0.1435 ^{\{ 5 \}} | 0.15587 ^{\{ 8 \}} | 0.16547 ^{\{ 10 \}} | 0.11423 ^{\{ 1 \}} | 0.19156 ^{\{ 14 \}} | 0.14743 ^{\{ 7 \}} | 0.18039 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.17764 ^{\{ 2 \}} | 0.19923 ^{\{ 4 \}} | 0.22686 ^{\{ 9 \}} | 0.20548 ^{\{ 6 \}} | 0.22944 ^{\{ 10 \}} | 0.26697 ^{\{ 13 \}} | 0.21258 ^{\{ 7 \}} | 0.18859 ^{\{ 3 \}} | 0.20381 ^{\{ 5 \}} | 0.23225 ^{\{ 11 \}} | 0.25014 ^{\{ 12 \}} | 0.14872 ^{\{ 1 \}} | 0.27192 ^{\{ 15 \}} | 0.21783 ^{\{ 8 \}} | 0.2676 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.02609 ^{\{ 2 \}} | 0.03343 ^{\{ 5 \}} | 0.04979 ^{\{ 12 \}} | 0.0295 ^{\{ 3 \}} | 0.0464 ^{\{ 11 \}} | 0.06938 ^{\{ 15 \}} | 0.04051 ^{\{ 9 \}} | 0.03057 ^{\{ 4 \}} | 0.03482 ^{\{ 7 \}} | 0.03819 ^{\{ 8 \}} | 0.0425 ^{\{ 10 \}} | 0.02264 ^{\{ 1 \}} | 0.05705 ^{\{ 14 \}} | 0.03431 ^{\{ 6 \}} | 0.05061 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.05098 ^{\{ 2 \}} | 0.06578 ^{\{ 4 \}} | 0.09046 ^{\{ 10 \}} | 0.0717 ^{\{ 5 \}} | 0.08987 ^{\{ 9 \}} | 0.12315 ^{\{ 13 \}} | 0.07547 ^{\{ 6 \}} | 0.06013 ^{\{ 3 \}} | 0.07747 ^{\{ 7 \}} | 0.09348 ^{\{ 11 \}} | 0.10805 ^{\{ 12 \}} | 0.04003 ^{\{ 1 \}} | 0.12352 ^{\{ 14 \}} | 0.08226 ^{\{ 8 \}} | 0.1243 ^{\{ 15 \}} | |
MRE( \hat{\delta} ) | 0.08504 ^{\{ 2 \}} | 0.09811 ^{\{ 6 \}} | 0.11916 ^{\{ 12 \}} | 0.09142 ^{\{ 3 \}} | 0.1156 ^{\{ 11 \}} | 0.14035 ^{\{ 15 \}} | 0.10741 ^{\{ 9 \}} | 0.09181 ^{\{ 4 \}} | 0.09567 ^{\{ 5 \}} | 0.10392 ^{\{ 8 \}} | 0.11031 ^{\{ 10 \}} | 0.07615 ^{\{ 1 \}} | 0.1277 ^{\{ 14 \}} | 0.09829 ^{\{ 7 \}} | 0.12026 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.11842 ^{\{ 2 \}} | 0.13282 ^{\{ 4 \}} | 0.15124 ^{\{ 9 \}} | 0.13699 ^{\{ 6 \}} | 0.15296 ^{\{ 10 \}} | 0.17798 ^{\{ 13 \}} | 0.14172 ^{\{ 7 \}} | 0.12573 ^{\{ 3 \}} | 0.13587 ^{\{ 5 \}} | 0.15484 ^{\{ 11 \}} | 0.16676 ^{\{ 12 \}} | 0.09915 ^{\{ 1 \}} | 0.18128 ^{\{ 15 \}} | 0.14522 ^{\{ 8 \}} | 0.1784 ^{\{ 14 \}} | |
D_{abs} | 0.01393 ^{\{ 1 \}} | 0.01439 ^{\{ 3 \}} | 0.01544 ^{\{ 8 \}} | 0.01449 ^{\{ 4 \}} | 0.01487 ^{\{ 6 \}} | 0.01613 ^{\{ 9 \}} | 0.0152 ^{\{ 7 \}} | 0.01395 ^{\{ 2 \}} | 0.0174 ^{\{ 13 \}} | 0.01678 ^{\{ 12 \}} | 0.01636 ^{\{ 11 \}} | 0.01461 ^{\{ 5 \}} | 0.01959 ^{\{ 15 \}} | 0.0163 ^{\{ 10 \}} | 0.01954 ^{\{ 14 \}} | |
D_{max} | 0.02275 ^{\{ 1 \}} | 0.02384 ^{\{ 5 \}} | 0.02607 ^{\{ 8 \}} | 0.02357 ^{\{ 3 \}} | 0.02515 ^{\{ 6 \}} | 0.0278 ^{\{ 12 \}} | 0.02531 ^{\{ 7 \}} | 0.02298 ^{\{ 2 \}} | 0.02812 ^{\{ 13 \}} | 0.02723 ^{\{ 11 \}} | 0.02671 ^{\{ 10 \}} | 0.02372 ^{\{ 4 \}} | 0.03222 ^{\{ 15 \}} | 0.02651 ^{\{ 9 \}} | 0.03184 ^{\{ 14 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 37 ^{\{ 5 \}} | 80 ^{\{ 10.5 \}} | 33 ^{\{ 4 \}} | 74 ^{\{ 9 \}} | 105 ^{\{ 13 \}} | 61 ^{\{ 7 \}} | 25 ^{\{ 3 \}} | 60 ^{\{ 6 \}} | 80 ^{\{ 10.5 \}} | 87 ^{\{ 12 \}} | 15 ^{\{ 2 \}} | 116 ^{\{ 15 \}} | 63 ^{\{ 8 \}} | 110 ^{\{ 14 \}} | |
400 | BIAS( \hat{\delta} ) | 0.10866 ^{\{ 2 \}} | 0.1345 ^{\{ 9 \}} | 0.15044 ^{\{ 11 \}} | 0.11647 ^{\{ 3 \}} | 0.15668 ^{\{ 13 \}} | 0.18648 ^{\{ 15 \}} | 0.13412 ^{\{ 8 \}} | 0.11906 ^{\{ 4 \}} | 0.13407 ^{\{ 7 \}} | 0.13152 ^{\{ 6 \}} | 0.1481 ^{\{ 10 \}} | 0.10545 ^{\{ 1 \}} | 0.16093 ^{\{ 14 \}} | 0.12007 ^{\{ 5 \}} | 0.15558 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.15352 ^{\{ 2 \}} | 0.18616 ^{\{ 8 \}} | 0.19179 ^{\{ 9 \}} | 0.16571 ^{\{ 4 \}} | 0.20261 ^{\{ 11 \}} | 0.23735 ^{\{ 15 \}} | 0.18354 ^{\{ 6 \}} | 0.16043 ^{\{ 3 \}} | 0.18538 ^{\{ 7 \}} | 0.19245 ^{\{ 10 \}} | 0.22086 ^{\{ 12 \}} | 0.1359 ^{\{ 1 \}} | 0.2328 ^{\{ 14 \}} | 0.17147 ^{\{ 5 \}} | 0.22366 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01904 ^{\{ 1 \}} | 0.02909 ^{\{ 8 \}} | 0.0361 ^{\{ 11 \}} | 0.02075 ^{\{ 3 \}} | 0.03897 ^{\{ 12 \}} | 0.05496 ^{\{ 15 \}} | 0.02796 ^{\{ 7 \}} | 0.02201 ^{\{ 4 \}} | 0.03078 ^{\{ 9 \}} | 0.02632 ^{\{ 6 \}} | 0.03496 ^{\{ 10 \}} | 0.01912 ^{\{ 2 \}} | 0.0414 ^{\{ 14 \}} | 0.02291 ^{\{ 5 \}} | 0.03957 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.03755 ^{\{ 2 \}} | 0.05973 ^{\{ 7 \}} | 0.06171 ^{\{ 9 \}} | 0.04387 ^{\{ 4 \}} | 0.07077 ^{\{ 11 \}} | 0.09576 ^{\{ 15 \}} | 0.05488 ^{\{ 6 \}} | 0.04111 ^{\{ 3 \}} | 0.06503 ^{\{ 10 \}} | 0.06059 ^{\{ 8 \}} | 0.08705 ^{\{ 13 \}} | 0.03291 ^{\{ 1 \}} | 0.09212 ^{\{ 14 \}} | 0.04832 ^{\{ 5 \}} | 0.08653 ^{\{ 12 \}} | |
MRE( \hat{\delta} ) | 0.07244 ^{\{ 2 \}} | 0.08967 ^{\{ 9 \}} | 0.10029 ^{\{ 11 \}} | 0.07765 ^{\{ 3 \}} | 0.10445 ^{\{ 13 \}} | 0.12432 ^{\{ 15 \}} | 0.08942 ^{\{ 8 \}} | 0.07938 ^{\{ 4 \}} | 0.08938 ^{\{ 7 \}} | 0.08768 ^{\{ 6 \}} | 0.09874 ^{\{ 10 \}} | 0.0703 ^{\{ 1 \}} | 0.10729 ^{\{ 14 \}} | 0.08004 ^{\{ 5 \}} | 0.10372 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.10235 ^{\{ 2 \}} | 0.1241 ^{\{ 8 \}} | 0.12786 ^{\{ 9 \}} | 0.11048 ^{\{ 4 \}} | 0.13507 ^{\{ 11 \}} | 0.15823 ^{\{ 15 \}} | 0.12236 ^{\{ 6 \}} | 0.10695 ^{\{ 3 \}} | 0.12359 ^{\{ 7 \}} | 0.1283 ^{\{ 10 \}} | 0.14724 ^{\{ 12 \}} | 0.0906 ^{\{ 1 \}} | 0.1552 ^{\{ 14 \}} | 0.11431 ^{\{ 5 \}} | 0.1491 ^{\{ 13 \}} | |
D_{abs} | 0.01246 ^{\{ 3 \}} | 0.01304 ^{\{ 6 \}} | 0.0132 ^{\{ 7 \}} | 0.01242 ^{\{ 1.5 \}} | 0.0134 ^{\{ 8 \}} | 0.0142 ^{\{ 10 \}} | 0.01274 ^{\{ 4 \}} | 0.01242 ^{\{ 1.5 \}} | 0.01486 ^{\{ 13 \}} | 0.01465 ^{\{ 11 \}} | 0.01477 ^{\{ 12 \}} | 0.01284 ^{\{ 5 \}} | 0.01692 ^{\{ 15 \}} | 0.01356 ^{\{ 9 \}} | 0.01666 ^{\{ 14 \}} | |
D_{max} | 0.0202 ^{\{ 1 \}} | 0.02154 ^{\{ 6 \}} | 0.02231 ^{\{ 8 \}} | 0.02028 ^{\{ 2 \}} | 0.02261 ^{\{ 9 \}} | 0.02465 ^{\{ 13 \}} | 0.02112 ^{\{ 5 \}} | 0.02043 ^{\{ 3 \}} | 0.02406 ^{\{ 11 \}} | 0.02377 ^{\{ 10 \}} | 0.02409 ^{\{ 12 \}} | 0.02086 ^{\{ 4 \}} | 0.02768 ^{\{ 15 \}} | 0.02207 ^{\{ 7 \}} | 0.02724 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 1 \}} | 61 ^{\{ 7 \}} | 75 ^{\{ 10 \}} | 24.5 ^{\{ 3 \}} | 88 ^{\{ 11 \}} | 113 ^{\{ 14 \}} | 50 ^{\{ 6 \}} | 25.5 ^{\{ 4 \}} | 71 ^{\{ 9 \}} | 67 ^{\{ 8 \}} | 91 ^{\{ 12 \}} | 16 ^{\{ 2 \}} | 114 ^{\{ 15 \}} | 46 ^{\{ 5 \}} | 103 ^{\{ 13 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.13624 ^{\{ 4 \}} | 0.15298 ^{\{ 10 \}} | 0.16819 ^{\{ 14 \}} | 0.13657 ^{\{ 5 \}} | 0.15175 ^{\{ 8 \}} | 0.18055 ^{\{ 15 \}} | 0.15251 ^{\{ 9 \}} | 0.14595 ^{\{ 6 \}} | 0.09288 ^{\{ 2 \}} | 0.12666 ^{\{ 3 \}} | 0.15422 ^{\{ 11 \}} | 0.05648 ^{\{ 1 \}} | 0.1601 ^{\{ 12 \}} | 0.14753 ^{\{ 7 \}} | 0.16209 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.57765 ^{\{ 4 \}} | 0.65842 ^{\{ 12 \}} | 0.63719 ^{\{ 9 \}} | 0.64979 ^{\{ 10 \}} | 0.60867 ^{\{ 7 \}} | 0.68047 ^{\{ 14 \}} | 0.65627 ^{\{ 11 \}} | 0.62937 ^{\{ 8 \}} | 0.25245 ^{\{ 2 \}} | 0.57701 ^{\{ 3 \}} | 0.66039 ^{\{ 13 \}} | 0.04775 ^{\{ 1 \}} | 0.5962 ^{\{ 6 \}} | 0.69113 ^{\{ 15 \}} | 0.58347 ^{\{ 5 \}} | |
MSE( \hat{\delta} ) | 0.02921 ^{\{ 5 \}} | 0.03635 ^{\{ 10 \}} | 0.04402 ^{\{ 14 \}} | 0.02809 ^{\{ 4 \}} | 0.03633 ^{\{ 9 \}} | 0.04928 ^{\{ 15 \}} | 0.03627 ^{\{ 8 \}} | 0.0338 ^{\{ 6 \}} | 0.01634 ^{\{ 2 \}} | 0.02598 ^{\{ 3 \}} | 0.03818 ^{\{ 11 \}} | 0.00526 ^{\{ 1 \}} | 0.04053 ^{\{ 12 \}} | 0.03407 ^{\{ 7 \}} | 0.0423 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.49086 ^{\{ 3 \}} | 0.62358 ^{\{ 11 \}} | 0.56339 ^{\{ 7 \}} | 0.62516 ^{\{ 12 \}} | 0.53029 ^{\{ 6 \}} | 0.6451 ^{\{ 14 \}} | 0.61704 ^{\{ 10 \}} | 0.58229 ^{\{ 9 \}} | 0.2075 ^{\{ 2 \}} | 0.56556 ^{\{ 8 \}} | 0.6369 ^{\{ 13 \}} | 0.01233 ^{\{ 1 \}} | 0.51078 ^{\{ 5 \}} | 0.68808 ^{\{ 15 \}} | 0.49454 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.27249 ^{\{ 4 \}} | 0.30596 ^{\{ 10 \}} | 0.33638 ^{\{ 14 \}} | 0.27314 ^{\{ 5 \}} | 0.3035 ^{\{ 8 \}} | 0.36111 ^{\{ 15 \}} | 0.30502 ^{\{ 9 \}} | 0.29189 ^{\{ 6 \}} | 0.18576 ^{\{ 2 \}} | 0.25331 ^{\{ 3 \}} | 0.30843 ^{\{ 11 \}} | 0.11296 ^{\{ 1 \}} | 0.32019 ^{\{ 12 \}} | 0.29507 ^{\{ 7 \}} | 0.32419 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.28882 ^{\{ 4 \}} | 0.32921 ^{\{ 12 \}} | 0.31859 ^{\{ 9 \}} | 0.3249 ^{\{ 10 \}} | 0.30434 ^{\{ 7 \}} | 0.34024 ^{\{ 14 \}} | 0.32814 ^{\{ 11 \}} | 0.31469 ^{\{ 8 \}} | 0.12623 ^{\{ 2 \}} | 0.28851 ^{\{ 3 \}} | 0.33019 ^{\{ 13 \}} | 0.02388 ^{\{ 1 \}} | 0.2981 ^{\{ 6 \}} | 0.34556 ^{\{ 15 \}} | 0.29173 ^{\{ 5 \}} | |
D_{abs} | 0.03768 ^{\{ 1 \}} | 0.04098 ^{\{ 5 \}} | 0.04468 ^{\{ 12 \}} | 0.04059 ^{\{ 2 \}} | 0.04585 ^{\{ 13 \}} | 0.04352 ^{\{ 10 \}} | 0.04069 ^{\{ 3 \}} | 0.04313 ^{\{ 9 \}} | 0.0425 ^{\{ 7 \}} | 0.04216 ^{\{ 6 \}} | 0.04287 ^{\{ 8 \}} | 0.04096 ^{\{ 4 \}} | 0.06514 ^{\{ 15 \}} | 0.04419 ^{\{ 11 \}} | 0.06029 ^{\{ 14 \}} | |
D_{max} | 0.06199 ^{\{ 1 \}} | 0.06671 ^{\{ 5 \}} | 0.07388 ^{\{ 13 \}} | 0.06517 ^{\{ 3 \}} | 0.07369 ^{\{ 12 \}} | 0.07286 ^{\{ 11 \}} | 0.06621 ^{\{ 4 \}} | 0.07003 ^{\{ 8 \}} | 0.06696 ^{\{ 6 \}} | 0.06706 ^{\{ 7 \}} | 0.07019 ^{\{ 9 \}} | 0.0623 ^{\{ 2 \}} | 0.10106 ^{\{ 15 \}} | 0.07066 ^{\{ 10 \}} | 0.09345 ^{\{ 14 \}} | |
\sum Ranks | 26 ^{\{ 3 \}} | 75 ^{\{ 9 \}} | 92 ^{\{ 14 \}} | 51 ^{\{ 5 \}} | 70 ^{\{ 8 \}} | 108 ^{\{ 15 \}} | 65 ^{\{ 7 \}} | 60 ^{\{ 6 \}} | 25 ^{\{ 2 \}} | 36 ^{\{ 4 \}} | 89 ^{\{ 13 \}} | 12 ^{\{ 1 \}} | 83 ^{\{ 11 \}} | 87 ^{\{ 12 \}} | 81 ^{\{ 10 \}} | |
60 | BIAS( \hat{\delta} ) | 0.10659 ^{\{ 3 \}} | 0.12172 ^{\{ 9 \}} | 0.14371 ^{\{ 14 \}} | 0.10716 ^{\{ 4 \}} | 0.13619 ^{\{ 13 \}} | 0.16304 ^{\{ 15 \}} | 0.13084 ^{\{ 11 \}} | 0.11699 ^{\{ 8 \}} | 0.07293 ^{\{ 2 \}} | 0.10752 ^{\{ 5 \}} | 0.12531 ^{\{ 10 \}} | 0.04248 ^{\{ 1 \}} | 0.13535 ^{\{ 12 \}} | 0.11105 ^{\{ 6 \}} | 0.11259 ^{\{ 7 \}} |
BIAS( \hat{\beta} ) | 0.50725 ^{\{ 5 \}} | 0.55301 ^{\{ 7 \}} | 0.6035 ^{\{ 12 \}} | 0.58515 ^{\{ 9 \}} | 0.5899 ^{\{ 11 \}} | 0.64435 ^{\{ 15 \}} | 0.60784 ^{\{ 14 \}} | 0.58714 ^{\{ 10 \}} | 0.23162 ^{\{ 2 \}} | 0.5467 ^{\{ 6 \}} | 0.60699 ^{\{ 13 \}} | 0.03925 ^{\{ 1 \}} | 0.49202 ^{\{ 4 \}} | 0.56928 ^{\{ 8 \}} | 0.4335 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.01774 ^{\{ 4 \}} | 0.02332 ^{\{ 9 \}} | 0.03276 ^{\{ 14 \}} | 0.01729 ^{\{ 3 \}} | 0.0283 ^{\{ 12 \}} | 0.04047 ^{\{ 15 \}} | 0.02684 ^{\{ 11 \}} | 0.0214 ^{\{ 7 \}} | 0.01089 ^{\{ 2 \}} | 0.01838 ^{\{ 5 \}} | 0.02516 ^{\{ 10 \}} | 0.00304 ^{\{ 1 \}} | 0.03231 ^{\{ 13 \}} | 0.02003 ^{\{ 6 \}} | 0.02258 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.41751 ^{\{ 5 \}} | 0.47226 ^{\{ 6 \}} | 0.53147 ^{\{ 10 \}} | 0.56899 ^{\{ 14 \}} | 0.50961 ^{\{ 8 \}} | 0.58852 ^{\{ 15 \}} | 0.55385 ^{\{ 12 \}} | 0.54235 ^{\{ 11 \}} | 0.18176 ^{\{ 2 \}} | 0.50391 ^{\{ 7 \}} | 0.56276 ^{\{ 13 \}} | 0.00761 ^{\{ 1 \}} | 0.36579 ^{\{ 4 \}} | 0.51548 ^{\{ 9 \}} | 0.28438 ^{\{ 3 \}} | |
MRE( \hat{\delta} ) | 0.21318 ^{\{ 3 \}} | 0.24344 ^{\{ 9 \}} | 0.28743 ^{\{ 14 \}} | 0.21432 ^{\{ 4 \}} | 0.27239 ^{\{ 13 \}} | 0.32609 ^{\{ 15 \}} | 0.26168 ^{\{ 11 \}} | 0.23398 ^{\{ 8 \}} | 0.14585 ^{\{ 2 \}} | 0.21505 ^{\{ 5 \}} | 0.25063 ^{\{ 10 \}} | 0.08496 ^{\{ 1 \}} | 0.27071 ^{\{ 12 \}} | 0.22211 ^{\{ 6 \}} | 0.22518 ^{\{ 7 \}} | |
MRE( \hat{\beta} ) | 0.25363 ^{\{ 5 \}} | 0.27651 ^{\{ 7 \}} | 0.30175 ^{\{ 12 \}} | 0.29257 ^{\{ 9 \}} | 0.29495 ^{\{ 11 \}} | 0.32217 ^{\{ 15 \}} | 0.30392 ^{\{ 14 \}} | 0.29357 ^{\{ 10 \}} | 0.11581 ^{\{ 2 \}} | 0.27335 ^{\{ 6 \}} | 0.30349 ^{\{ 13 \}} | 0.01962 ^{\{ 1 \}} | 0.24601 ^{\{ 4 \}} | 0.28464 ^{\{ 8 \}} | 0.21675 ^{\{ 3 \}} | |
D_{abs} | 0.02879 ^{\{ 1 \}} | 0.02974 ^{\{ 2 \}} | 0.03169 ^{\{ 9 \}} | 0.03041 ^{\{ 4 \}} | 0.03128 ^{\{ 7 \}} | 0.03293 ^{\{ 13 \}} | 0.03119 ^{\{ 6 \}} | 0.03093 ^{\{ 5 \}} | 0.03153 ^{\{ 8 \}} | 0.03238 ^{\{ 12 \}} | 0.03207 ^{\{ 10 \}} | 0.03029 ^{\{ 3 \}} | 0.04427 ^{\{ 15 \}} | 0.03231 ^{\{ 11 \}} | 0.04117 ^{\{ 14 \}} | |
D_{max} | 0.04743 ^{\{ 2 \}} | 0.04934 ^{\{ 4 \}} | 0.05375 ^{\{ 12 \}} | 0.0493 ^{\{ 3 \}} | 0.05256 ^{\{ 10 \}} | 0.0566 ^{\{ 13 \}} | 0.05167 ^{\{ 7 \}} | 0.05037 ^{\{ 6 \}} | 0.04949 ^{\{ 5 \}} | 0.05255 ^{\{ 9 \}} | 0.05272 ^{\{ 11 \}} | 0.04604 ^{\{ 1 \}} | 0.06992 ^{\{ 15 \}} | 0.05183 ^{\{ 8 \}} | 0.06489 ^{\{ 14 \}} | |
\sum Ranks | 28 ^{\{ 3 \}} | 53 ^{\{ 5 \}} | 97 ^{\{ 14 \}} | 50 ^{\{ 4 \}} | 85 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 86 ^{\{ 12 \}} | 65 ^{\{ 9 \}} | 25 ^{\{ 2 \}} | 55 ^{\{ 6 \}} | 90 ^{\{ 13 \}} | 10 ^{\{ 1 \}} | 79 ^{\{ 10 \}} | 62 ^{\{ 8 \}} | 59 ^{\{ 7 \}} | |
100 | BIAS( \hat{\delta} ) | 0.08286 ^{\{ 3 \}} | 0.10295 ^{\{ 11 \}} | 0.11725 ^{\{ 14 \}} | 0.08874 ^{\{ 4 \}} | 0.11484 ^{\{ 13 \}} | 0.14099 ^{\{ 15 \}} | 0.10506 ^{\{ 12 \}} | 0.09668 ^{\{ 9 \}} | 0.06209 ^{\{ 2 \}} | 0.09521 ^{\{ 8 \}} | 0.10061 ^{\{ 10 \}} | 0.0325 ^{\{ 1 \}} | 0.09485 ^{\{ 7 \}} | 0.09062 ^{\{ 6 \}} | 0.09056 ^{\{ 5 \}} |
BIAS( \hat{\beta} ) | 0.41544 ^{\{ 5 \}} | 0.51195 ^{\{ 9 \}} | 0.51848 ^{\{ 11 \}} | 0.50035 ^{\{ 7 \}} | 0.55605 ^{\{ 14 \}} | 0.57985 ^{\{ 15 \}} | 0.51445 ^{\{ 10 \}} | 0.48798 ^{\{ 6 \}} | 0.22274 ^{\{ 2 \}} | 0.52223 ^{\{ 12 \}} | 0.52272 ^{\{ 13 \}} | 0.03493 ^{\{ 1 \}} | 0.36852 ^{\{ 3 \}} | 0.5066 ^{\{ 8 \}} | 0.392 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.01073 ^{\{ 3 \}} | 0.01683 ^{\{ 10 \}} | 0.02226 ^{\{ 14 \}} | 0.01217 ^{\{ 4 \}} | 0.02079 ^{\{ 13 \}} | 0.03092 ^{\{ 15 \}} | 0.01737 ^{\{ 12 \}} | 0.01473 ^{\{ 7 \}} | 0.00763 ^{\{ 2 \}} | 0.01427 ^{\{ 6 \}} | 0.01587 ^{\{ 9 \}} | 0.00182 ^{\{ 1 \}} | 0.0169 ^{\{ 11 \}} | 0.0124 ^{\{ 5 \}} | 0.01525 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.29348 ^{\{ 5 \}} | 0.42649 ^{\{ 9 \}} | 0.43143 ^{\{ 11 \}} | 0.42826 ^{\{ 10 \}} | 0.49967 ^{\{ 14 \}} | 0.51432 ^{\{ 15 \}} | 0.42161 ^{\{ 8 \}} | 0.38818 ^{\{ 6 \}} | 0.15605 ^{\{ 2 \}} | 0.47725 ^{\{ 13 \}} | 0.43798 ^{\{ 12 \}} | 0.0066 ^{\{ 1 \}} | 0.21342 ^{\{ 3 \}} | 0.41474 ^{\{ 7 \}} | 0.23478 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.16573 ^{\{ 3 \}} | 0.2059 ^{\{ 11 \}} | 0.23451 ^{\{ 14 \}} | 0.17748 ^{\{ 4 \}} | 0.22968 ^{\{ 13 \}} | 0.28198 ^{\{ 15 \}} | 0.21011 ^{\{ 12 \}} | 0.19336 ^{\{ 9 \}} | 0.12419 ^{\{ 2 \}} | 0.19042 ^{\{ 8 \}} | 0.20121 ^{\{ 10 \}} | 0.065 ^{\{ 1 \}} | 0.18971 ^{\{ 7 \}} | 0.18125 ^{\{ 6 \}} | 0.18112 ^{\{ 5 \}} | |
MRE( \hat{\beta} ) | 0.20772 ^{\{ 5 \}} | 0.25598 ^{\{ 9 \}} | 0.25924 ^{\{ 11 \}} | 0.25018 ^{\{ 7 \}} | 0.27802 ^{\{ 14 \}} | 0.28993 ^{\{ 15 \}} | 0.25722 ^{\{ 10 \}} | 0.24399 ^{\{ 6 \}} | 0.11137 ^{\{ 2 \}} | 0.26111 ^{\{ 12 \}} | 0.26136 ^{\{ 13 \}} | 0.01746 ^{\{ 1 \}} | 0.18426 ^{\{ 3 \}} | 0.2533 ^{\{ 8 \}} | 0.196 ^{\{ 4 \}} | |
D_{abs} | 0.02231 ^{\{ 1 \}} | 0.02454 ^{\{ 6 \}} | 0.02511 ^{\{ 9 \}} | 0.02269 ^{\{ 3 \}} | 0.02562 ^{\{ 10 \}} | 0.02595 ^{\{ 12 \}} | 0.02449 ^{\{ 5 \}} | 0.02497 ^{\{ 7 \}} | 0.02396 ^{\{ 4 \}} | 0.02574 ^{\{ 11 \}} | 0.02509 ^{\{ 8 \}} | 0.02261 ^{\{ 2 \}} | 0.03104 ^{\{ 14 \}} | 0.02611 ^{\{ 13 \}} | 0.03167 ^{\{ 15 \}} | |
D_{max} | 0.03664 ^{\{ 2 \}} | 0.04054 ^{\{ 5.5 \}} | 0.0426 ^{\{ 11 \}} | 0.03686 ^{\{ 3 \}} | 0.04306 ^{\{ 12 \}} | 0.04533 ^{\{ 13 \}} | 0.04054 ^{\{ 5.5 \}} | 0.04099 ^{\{ 7 \}} | 0.03813 ^{\{ 4 \}} | 0.04213 ^{\{ 10 \}} | 0.04108 ^{\{ 8 \}} | 0.03463 ^{\{ 1 \}} | 0.0495 ^{\{ 14 \}} | 0.04206 ^{\{ 9 \}} | 0.05085 ^{\{ 15 \}} | |
\sum Ranks | 27 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 95 ^{\{ 13 \}} | 42 ^{\{ 4 \}} | 103 ^{\{ 14 \}} | 115 ^{\{ 15 \}} | 74.5 ^{\{ 10 \}} | 57 ^{\{ 5 \}} | 20 ^{\{ 2 \}} | 80 ^{\{ 11 \}} | 83 ^{\{ 12 \}} | 9 ^{\{ 1 \}} | 62 ^{\{ 7.5 \}} | 62 ^{\{ 7.5 \}} | 60 ^{\{ 6 \}} | |
200 | BIAS( \hat{\delta} ) | 0.06116 ^{\{ 3 \}} | 0.07841 ^{\{ 11 \}} | 0.08834 ^{\{ 13 \}} | 0.06582 ^{\{ 5 \}} | 0.08967 ^{\{ 14 \}} | 0.10318 ^{\{ 15 \}} | 0.07657 ^{\{ 10 \}} | 0.0701 ^{\{ 7 \}} | 0.05022 ^{\{ 2 \}} | 0.07169 ^{\{ 8 \}} | 0.0803 ^{\{ 12 \}} | 0.02285 ^{\{ 1 \}} | 0.06434 ^{\{ 4 \}} | 0.07255 ^{\{ 9 \}} | 0.06584 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.32425 ^{\{ 5 \}} | 0.40338 ^{\{ 10 \}} | 0.42002 ^{\{ 12 \}} | 0.37519 ^{\{ 7 \}} | 0.44639 ^{\{ 13 \}} | 0.47646 ^{\{ 15 \}} | 0.38899 ^{\{ 8 \}} | 0.36465 ^{\{ 6 \}} | 0.20622 ^{\{ 2 \}} | 0.39983 ^{\{ 9 \}} | 0.45097 ^{\{ 14 \}} | 0.02978 ^{\{ 1 \}} | 0.309 ^{\{ 3 \}} | 0.40885 ^{\{ 11 \}} | 0.32061 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.00603 ^{\{ 3 \}} | 0.0094 ^{\{ 11 \}} | 0.01227 ^{\{ 14 \}} | 0.00684 ^{\{ 4 \}} | 0.0121 ^{\{ 13 \}} | 0.01636 ^{\{ 15 \}} | 0.00916 ^{\{ 10 \}} | 0.00762 ^{\{ 6 \}} | 0.00531 ^{\{ 2 \}} | 0.00804 ^{\{ 8 \}} | 0.0099 ^{\{ 12 \}} | 0.00084 ^{\{ 1 \}} | 0.00725 ^{\{ 5 \}} | 0.00822 ^{\{ 9 \}} | 0.00789 ^{\{ 7 \}} | |
MSE( \hat{\beta} ) | 0.18213 ^{\{ 4 \}} | 0.27797 ^{\{ 9 \}} | 0.29886 ^{\{ 12 \}} | 0.25897 ^{\{ 8 \}} | 0.34454 ^{\{ 14 \}} | 0.37648 ^{\{ 15 \}} | 0.2585 ^{\{ 7 \}} | 0.23044 ^{\{ 6 \}} | 0.13188 ^{\{ 2 \}} | 0.2854 ^{\{ 10 \}} | 0.34418 ^{\{ 13 \}} | 0.00461 ^{\{ 1 \}} | 0.1536 ^{\{ 3 \}} | 0.29853 ^{\{ 11 \}} | 0.18333 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.12232 ^{\{ 3 \}} | 0.15682 ^{\{ 11 \}} | 0.17668 ^{\{ 13 \}} | 0.13164 ^{\{ 5 \}} | 0.17934 ^{\{ 14 \}} | 0.20636 ^{\{ 15 \}} | 0.15314 ^{\{ 10 \}} | 0.1402 ^{\{ 7 \}} | 0.10044 ^{\{ 2 \}} | 0.14338 ^{\{ 8 \}} | 0.16059 ^{\{ 12 \}} | 0.0457 ^{\{ 1 \}} | 0.12868 ^{\{ 4 \}} | 0.14509 ^{\{ 9 \}} | 0.13169 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.16212 ^{\{ 5 \}} | 0.20169 ^{\{ 10 \}} | 0.21001 ^{\{ 12 \}} | 0.18759 ^{\{ 7 \}} | 0.2232 ^{\{ 13 \}} | 0.23823 ^{\{ 15 \}} | 0.1945 ^{\{ 8 \}} | 0.18232 ^{\{ 6 \}} | 0.10311 ^{\{ 2 \}} | 0.19991 ^{\{ 9 \}} | 0.22548 ^{\{ 14 \}} | 0.01489 ^{\{ 1 \}} | 0.1545 ^{\{ 3 \}} | 0.20442 ^{\{ 11 \}} | 0.16031 ^{\{ 4 \}} | |
D_{abs} | 0.01724 ^{\{ 2 \}} | 0.01806 ^{\{ 7 \}} | 0.01844 ^{\{ 10 \}} | 0.01741 ^{\{ 3 \}} | 0.01838 ^{\{ 8 \}} | 0.01843 ^{\{ 9 \}} | 0.01779 ^{\{ 4 \}} | 0.01797 ^{\{ 6 \}} | 0.01789 ^{\{ 5 \}} | 0.02027 ^{\{ 13 \}} | 0.01959 ^{\{ 12 \}} | 0.01648 ^{\{ 1 \}} | 0.02114 ^{\{ 14 \}} | 0.0194 ^{\{ 11 \}} | 0.02249 ^{\{ 15 \}} | |
D_{max} | 0.02791 ^{\{ 2 \}} | 0.0299 ^{\{ 7 \}} | 0.03118 ^{\{ 9 \}} | 0.02827 ^{\{ 3 \}} | 0.03116 ^{\{ 8 \}} | 0.03221 ^{\{ 12 \}} | 0.02954 ^{\{ 6 \}} | 0.0293 ^{\{ 5 \}} | 0.02849 ^{\{ 4 \}} | 0.03291 ^{\{ 13 \}} | 0.03189 ^{\{ 11 \}} | 0.02515 ^{\{ 1 \}} | 0.03433 ^{\{ 14 \}} | 0.03138 ^{\{ 10 \}} | 0.03655 ^{\{ 15 \}} | |
\sum Ranks | 27 ^{\{ 3 \}} | 76 ^{\{ 9 \}} | 95 ^{\{ 12 \}} | 42 ^{\{ 4 \}} | 97 ^{\{ 13 \}} | 111 ^{\{ 15 \}} | 63 ^{\{ 8 \}} | 49 ^{\{ 5 \}} | 21 ^{\{ 2 \}} | 78 ^{\{ 10 \}} | 100 ^{\{ 14 \}} | 8 ^{\{ 1 \}} | 50 ^{\{ 6 \}} | 81 ^{\{ 11 \}} | 62 ^{\{ 7 \}} | |
300 | BIAS( \hat{\delta} ) | 0.04976 ^{\{ 3 \}} | 0.063 ^{\{ 10 \}} | 0.07722 ^{\{ 14 \}} | 0.05537 ^{\{ 5 \}} | 0.07222 ^{\{ 13 \}} | 0.08992 ^{\{ 15 \}} | 0.06346 ^{\{ 11 \}} | 0.05881 ^{\{ 7 \}} | 0.04405 ^{\{ 2 \}} | 0.06157 ^{\{ 9 \}} | 0.06661 ^{\{ 12 \}} | 0.01923 ^{\{ 1 \}} | 0.05374 ^{\{ 4 \}} | 0.05954 ^{\{ 8 \}} | 0.05577 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.26113 ^{\{ 3 \}} | 0.32348 ^{\{ 9 \}} | 0.37643 ^{\{ 14 \}} | 0.30537 ^{\{ 6 \}} | 0.36663 ^{\{ 12 \}} | 0.42261 ^{\{ 15 \}} | 0.32101 ^{\{ 8 \}} | 0.30553 ^{\{ 7 \}} | 0.18919 ^{\{ 2 \}} | 0.33927 ^{\{ 11 \}} | 0.37228 ^{\{ 13 \}} | 0.02946 ^{\{ 1 \}} | 0.27025 ^{\{ 5 \}} | 0.32572 ^{\{ 10 \}} | 0.2684 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.00405 ^{\{ 3 \}} | 0.00614 ^{\{ 10 \}} | 0.00941 ^{\{ 14 \}} | 0.00478 ^{\{ 4 \}} | 0.00824 ^{\{ 13 \}} | 0.01248 ^{\{ 15 \}} | 0.0063 ^{\{ 11 \}} | 0.00566 ^{\{ 7 \}} | 0.00375 ^{\{ 2 \}} | 0.00597 ^{\{ 9 \}} | 0.00674 ^{\{ 12 \}} | 0.00063 ^{\{ 1 \}} | 0.00503 ^{\{ 5 \}} | 0.00548 ^{\{ 6 \}} | 0.00588 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.12154 ^{\{ 3 \}} | 0.17435 ^{\{ 8 \}} | 0.24526 ^{\{ 14 \}} | 0.16177 ^{\{ 7 \}} | 0.2386 ^{\{ 13 \}} | 0.30176 ^{\{ 15 \}} | 0.17584 ^{\{ 9 \}} | 0.15982 ^{\{ 6 \}} | 0.10377 ^{\{ 2 \}} | 0.20948 ^{\{ 11 \}} | 0.23664 ^{\{ 12 \}} | 0.00414 ^{\{ 1 \}} | 0.13235 ^{\{ 4 \}} | 0.18403 ^{\{ 10 \}} | 0.14778 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.09951 ^{\{ 3 \}} | 0.12601 ^{\{ 10 \}} | 0.15444 ^{\{ 14 \}} | 0.11075 ^{\{ 5 \}} | 0.14445 ^{\{ 13 \}} | 0.17985 ^{\{ 15 \}} | 0.12692 ^{\{ 11 \}} | 0.11761 ^{\{ 7 \}} | 0.0881 ^{\{ 2 \}} | 0.12314 ^{\{ 9 \}} | 0.13323 ^{\{ 12 \}} | 0.03846 ^{\{ 1 \}} | 0.10748 ^{\{ 4 \}} | 0.11909 ^{\{ 8 \}} | 0.11153 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.13057 ^{\{ 3 \}} | 0.16174 ^{\{ 9 \}} | 0.18822 ^{\{ 14 \}} | 0.15269 ^{\{ 6 \}} | 0.18332 ^{\{ 12 \}} | 0.2113 ^{\{ 15 \}} | 0.1605 ^{\{ 8 \}} | 0.15276 ^{\{ 7 \}} | 0.09459 ^{\{ 2 \}} | 0.16963 ^{\{ 11 \}} | 0.18614 ^{\{ 13 \}} | 0.01473 ^{\{ 1 \}} | 0.13513 ^{\{ 5 \}} | 0.16286 ^{\{ 10 \}} | 0.1342 ^{\{ 4 \}} | |
D_{abs} | 0.01433 ^{\{ 2 \}} | 0.01475 ^{\{ 5.5 \}} | 0.0156 ^{\{ 10 \}} | 0.01439 ^{\{ 3 \}} | 0.01527 ^{\{ 7 \}} | 0.01648 ^{\{ 12 \}} | 0.01475 ^{\{ 5.5 \}} | 0.01471 ^{\{ 4 \}} | 0.01548 ^{\{ 8 \}} | 0.01669 ^{\{ 13 \}} | 0.01557 ^{\{ 9 \}} | 0.01377 ^{\{ 1 \}} | 0.01746 ^{\{ 14 \}} | 0.01569 ^{\{ 11 \}} | 0.01762 ^{\{ 15 \}} | |
D_{max} | 0.02313 ^{\{ 2 \}} | 0.02441 ^{\{ 5 \}} | 0.02642 ^{\{ 11 \}} | 0.0234 ^{\{ 3 \}} | 0.02566 ^{\{ 10 \}} | 0.02851 ^{\{ 14 \}} | 0.02444 ^{\{ 6 \}} | 0.02405 ^{\{ 4 \}} | 0.02467 ^{\{ 7 \}} | 0.02709 ^{\{ 12 \}} | 0.02545 ^{\{ 8 \}} | 0.02099 ^{\{ 1 \}} | 0.02842 ^{\{ 13 \}} | 0.02547 ^{\{ 9 \}} | 0.02869 ^{\{ 15 \}} | |
\sum Ranks | 22 ^{\{ 2 \}} | 66.5 ^{\{ 8 \}} | 105 ^{\{ 14 \}} | 39 ^{\{ 4 \}} | 93 ^{\{ 13 \}} | 116 ^{\{ 15 \}} | 69.5 ^{\{ 9 \}} | 49 ^{\{ 5 \}} | 27 ^{\{ 3 \}} | 85 ^{\{ 11 \}} | 91 ^{\{ 12 \}} | 8 ^{\{ 1 \}} | 54 ^{\{ 6 \}} | 72 ^{\{ 10 \}} | 63 ^{\{ 7 \}} | |
400 | BIAS( \hat{\delta} ) | 0.04586 ^{\{ 3 \}} | 0.05457 ^{\{ 10 \}} | 0.06718 ^{\{ 14 \}} | 0.04629 ^{\{ 4 \}} | 0.06491 ^{\{ 13 \}} | 0.07564 ^{\{ 15 \}} | 0.05516 ^{\{ 11 \}} | 0.05045 ^{\{ 8 \}} | 0.04056 ^{\{ 2 \}} | 0.05384 ^{\{ 9 \}} | 0.05857 ^{\{ 12 \}} | 0.01681 ^{\{ 1 \}} | 0.04911 ^{\{ 5 \}} | 0.0504 ^{\{ 7 \}} | 0.04983 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.24121 ^{\{ 4 \}} | 0.2717 ^{\{ 9 \}} | 0.32822 ^{\{ 14 \}} | 0.25116 ^{\{ 6 \}} | 0.31726 ^{\{ 12 \}} | 0.35759 ^{\{ 15 \}} | 0.2792 ^{\{ 10 \}} | 0.25723 ^{\{ 7 \}} | 0.17791 ^{\{ 2 \}} | 0.29726 ^{\{ 11 \}} | 0.3189 ^{\{ 13 \}} | 0.02741 ^{\{ 1 \}} | 0.2501 ^{\{ 5 \}} | 0.27113 ^{\{ 8 \}} | 0.23703 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.00335 ^{\{ 4 \}} | 0.00479 ^{\{ 10 \}} | 0.00713 ^{\{ 14 \}} | 0.00334 ^{\{ 3 \}} | 0.00654 ^{\{ 13 \}} | 0.00909 ^{\{ 15 \}} | 0.0047 ^{\{ 9 \}} | 0.00393 ^{\{ 5 \}} | 0.00319 ^{\{ 2 \}} | 0.00461 ^{\{ 8 \}} | 0.00534 ^{\{ 12 \}} | 0.00051 ^{\{ 1 \}} | 0.00433 ^{\{ 7 \}} | 0.00397 ^{\{ 6 \}} | 0.00484 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.09651 ^{\{ 3 \}} | 0.12678 ^{\{ 8 \}} | 0.18883 ^{\{ 14 \}} | 0.10551 ^{\{ 4 \}} | 0.17475 ^{\{ 13 \}} | 0.22525 ^{\{ 15 \}} | 0.12978 ^{\{ 10 \}} | 0.11072 ^{\{ 5 \}} | 0.08312 ^{\{ 2 \}} | 0.15568 ^{\{ 11 \}} | 0.17447 ^{\{ 12 \}} | 0.00395 ^{\{ 1 \}} | 0.12704 ^{\{ 9 \}} | 0.12604 ^{\{ 7 \}} | 0.11073 ^{\{ 6 \}} | |
MRE( \hat{\delta} ) | 0.09172 ^{\{ 3 \}} | 0.10915 ^{\{ 10 \}} | 0.13437 ^{\{ 14 \}} | 0.09257 ^{\{ 4 \}} | 0.12982 ^{\{ 13 \}} | 0.15127 ^{\{ 15 \}} | 0.11032 ^{\{ 11 \}} | 0.10089 ^{\{ 8 \}} | 0.08112 ^{\{ 2 \}} | 0.10768 ^{\{ 9 \}} | 0.11714 ^{\{ 12 \}} | 0.03363 ^{\{ 1 \}} | 0.09822 ^{\{ 5 \}} | 0.10079 ^{\{ 7 \}} | 0.09966 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.12061 ^{\{ 4 \}} | 0.13585 ^{\{ 9 \}} | 0.16411 ^{\{ 14 \}} | 0.12558 ^{\{ 6 \}} | 0.15863 ^{\{ 12 \}} | 0.1788 ^{\{ 15 \}} | 0.1396 ^{\{ 10 \}} | 0.12862 ^{\{ 7 \}} | 0.08896 ^{\{ 2 \}} | 0.14863 ^{\{ 11 \}} | 0.15945 ^{\{ 13 \}} | 0.01371 ^{\{ 1 \}} | 0.12505 ^{\{ 5 \}} | 0.13556 ^{\{ 8 \}} | 0.11852 ^{\{ 3 \}} | |
D_{abs} | 0.01235 ^{\{ 3 \}} | 0.01251 ^{\{ 4 \}} | 0.01316 ^{\{ 7 \}} | 0.01191 ^{\{ 2 \}} | 0.0133 ^{\{ 10 \}} | 0.01357 ^{\{ 11 \}} | 0.01283 ^{\{ 6 \}} | 0.01262 ^{\{ 5 \}} | 0.01327 ^{\{ 9 \}} | 0.01469 ^{\{ 13 \}} | 0.01384 ^{\{ 12 \}} | 0.0118 ^{\{ 1 \}} | 0.01612 ^{\{ 15 \}} | 0.01319 ^{\{ 8 \}} | 0.01595 ^{\{ 14 \}} | |
D_{max} | 0.02003 ^{\{ 3 \}} | 0.02068 ^{\{ 5 \}} | 0.02249 ^{\{ 10 \}} | 0.0194 ^{\{ 2 \}} | 0.02245 ^{\{ 9 \}} | 0.02345 ^{\{ 12 \}} | 0.0212 ^{\{ 6 \}} | 0.02064 ^{\{ 4 \}} | 0.02127 ^{\{ 7 \}} | 0.02382 ^{\{ 13 \}} | 0.02263 ^{\{ 11 \}} | 0.01806 ^{\{ 1 \}} | 0.02613 ^{\{ 15 \}} | 0.0215 ^{\{ 8 \}} | 0.02592 ^{\{ 14 \}} | |
\sum Ranks | 27 ^{\{ 2 \}} | 65 ^{\{ 8 \}} | 101 ^{\{ 14 \}} | 31 ^{\{ 4 \}} | 95 ^{\{ 12 \}} | 113 ^{\{ 15 \}} | 73 ^{\{ 10 \}} | 49 ^{\{ 5 \}} | 28 ^{\{ 3 \}} | 85 ^{\{ 11 \}} | 97 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 66 ^{\{ 9 \}} | 59 ^{\{ 6 \}} | 63 ^{\{ 7 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.24947 ^{\{ 2 \}} | 0.39247 ^{\{ 8 \}} | 0.45109 ^{\{ 13 \}} | 0.34898 ^{\{ 4 \}} | 0.4092 ^{\{ 10 \}} | 0.51093 ^{\{ 15 \}} | 0.3875 ^{\{ 7 \}} | 0.40032 ^{\{ 9 \}} | 0.2568 ^{\{ 3 \}} | 0.36632 ^{\{ 6 \}} | 0.47352 ^{\{ 14 \}} | 0.0773 ^{\{ 1 \}} | 0.42866 ^{\{ 12 \}} | 0.3597 ^{\{ 5 \}} | 0.40978 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.10849 ^{\{ 3 \}} | 0.12312 ^{\{ 5 \}} | 0.12419 ^{\{ 6 \}} | 0.12427 ^{\{ 7 \}} | 0.12917 ^{\{ 11 \}} | 0.13824 ^{\{ 12 \}} | 0.12135 ^{\{ 4 \}} | 0.12589 ^{\{ 8 \}} | 0.10086 ^{\{ 2 \}} | 0.12906 ^{\{ 10 \}} | 0.14169 ^{\{ 15 \}} | 0.05665 ^{\{ 1 \}} | 0.14068 ^{\{ 13 \}} | 0.12644 ^{\{ 9 \}} | 0.14073 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.1048 ^{\{ 2 \}} | 0.25062 ^{\{ 8 \}} | 0.35457 ^{\{ 13 \}} | 0.19139 ^{\{ 4 \}} | 0.26462 ^{\{ 10 \}} | 0.44465 ^{\{ 15 \}} | 0.24711 ^{\{ 7 \}} | 0.26728 ^{\{ 11 \}} | 0.13483 ^{\{ 3 \}} | 0.20392 ^{\{ 6 \}} | 0.39226 ^{\{ 14 \}} | 0.02074 ^{\{ 1 \}} | 0.27898 ^{\{ 12 \}} | 0.19815 ^{\{ 5 \}} | 0.25293 ^{\{ 9 \}} | |
MSE( \hat{\beta} ) | 0.02079 ^{\{ 3 \}} | 0.02301 ^{\{ 6 \}} | 0.02292 ^{\{ 5 \}} | 0.02435 ^{\{ 8 \}} | 0.0252 ^{\{ 10 \}} | 0.02791 ^{\{ 12 \}} | 0.02257 ^{\{ 4 \}} | 0.02416 ^{\{ 7 \}} | 0.01873 ^{\{ 2 \}} | 0.02604 ^{\{ 11 \}} | 0.02917 ^{\{ 13 \}} | 0.00597 ^{\{ 1 \}} | 0.03018 ^{\{ 15 \}} | 0.02461 ^{\{ 9 \}} | 0.02959 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.09979 ^{\{ 2 \}} | 0.15699 ^{\{ 8 \}} | 0.18043 ^{\{ 13 \}} | 0.13959 ^{\{ 4 \}} | 0.16368 ^{\{ 10 \}} | 0.20437 ^{\{ 15 \}} | 0.155 ^{\{ 7 \}} | 0.16013 ^{\{ 9 \}} | 0.10272 ^{\{ 3 \}} | 0.14653 ^{\{ 6 \}} | 0.18941 ^{\{ 14 \}} | 0.03092 ^{\{ 1 \}} | 0.17146 ^{\{ 12 \}} | 0.14388 ^{\{ 5 \}} | 0.16391 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.27123 ^{\{ 3 \}} | 0.3078 ^{\{ 5 \}} | 0.31047 ^{\{ 6 \}} | 0.31068 ^{\{ 7 \}} | 0.32292 ^{\{ 11 \}} | 0.34559 ^{\{ 12 \}} | 0.30337 ^{\{ 4 \}} | 0.31471 ^{\{ 8 \}} | 0.25215 ^{\{ 2 \}} | 0.32265 ^{\{ 10 \}} | 0.35423 ^{\{ 15 \}} | 0.14162 ^{\{ 1 \}} | 0.3517 ^{\{ 13 \}} | 0.31611 ^{\{ 9 \}} | 0.35182 ^{\{ 14 \}} | |
D_{abs} | 0.04159 ^{\{ 1 \}} | 0.0451 ^{\{ 3 \}} | 0.0482 ^{\{ 9 \}} | 0.0454 ^{\{ 5 \}} | 0.04619 ^{\{ 6 \}} | 0.0507 ^{\{ 13 \}} | 0.04517 ^{\{ 4 \}} | 0.04721 ^{\{ 8 \}} | 0.04986 ^{\{ 11.5 \}} | 0.04986 ^{\{ 11.5 \}} | 0.04865 ^{\{ 10 \}} | 0.04264 ^{\{ 2 \}} | 0.05413 ^{\{ 15 \}} | 0.0466 ^{\{ 7 \}} | 0.05336 ^{\{ 14 \}} | |
D_{max} | 0.066 ^{\{ 2 \}} | 0.07344 ^{\{ 4 \}} | 0.08102 ^{\{ 12 \}} | 0.07193 ^{\{ 3 \}} | 0.076 ^{\{ 7 \}} | 0.08624 ^{\{ 14 \}} | 0.07378 ^{\{ 5 \}} | 0.07752 ^{\{ 9 \}} | 0.07682 ^{\{ 8 \}} | 0.07921 ^{\{ 10 \}} | 0.08099 ^{\{ 11 \}} | 0.06259 ^{\{ 1 \}} | 0.08762 ^{\{ 15 \}} | 0.07444 ^{\{ 6 \}} | 0.0855 ^{\{ 13 \}} | |
\sum Ranks | 18 ^{\{ 2 \}} | 47 ^{\{ 6 \}} | 77 ^{\{ 11 \}} | 42 ^{\{ 4.5 \}} | 75 ^{\{ 10 \}} | 108 ^{\{ 15 \}} | 42 ^{\{ 4.5 \}} | 69 ^{\{ 8 \}} | 34.5 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 106 ^{\{ 13 \}} | 9 ^{\{ 1 \}} | 107 ^{\{ 14 \}} | 55 ^{\{ 7 \}} | 100 ^{\{ 12 \}} | |
60 | BIAS( \hat{\delta} ) | 0.20453 ^{\{ 2 \}} | 0.2779 ^{\{ 6 \}} | 0.29563 ^{\{ 10 \}} | 0.27622 ^{\{ 5 \}} | 0.31346 ^{\{ 11 \}} | 0.37466 ^{\{ 15 \}} | 0.28173 ^{\{ 7 \}} | 0.28733 ^{\{ 9 \}} | 0.24988 ^{\{ 3 \}} | 0.28363 ^{\{ 8 \}} | 0.35262 ^{\{ 14 \}} | 0.07199 ^{\{ 1 \}} | 0.33443 ^{\{ 13 \}} | 0.27509 ^{\{ 4 \}} | 0.32222 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.08487 ^{\{ 2 \}} | 0.09474 ^{\{ 5 \}} | 0.09425 ^{\{ 3 \}} | 0.10053 ^{\{ 9 \}} | 0.09961 ^{\{ 8 \}} | 0.10752 ^{\{ 12 \}} | 0.09656 ^{\{ 7 \}} | 0.09498 ^{\{ 6 \}} | 0.09445 ^{\{ 4 \}} | 0.10474 ^{\{ 11 \}} | 0.11801 ^{\{ 13 \}} | 0.04214 ^{\{ 1 \}} | 0.11894 ^{\{ 14 \}} | 0.10253 ^{\{ 10 \}} | 0.1193 ^{\{ 15 \}} | |
MSE( \hat{\delta} ) | 0.07403 ^{\{ 2 \}} | 0.12349 ^{\{ 6 \}} | 0.14155 ^{\{ 10 \}} | 0.11748 ^{\{ 4 \}} | 0.15224 ^{\{ 11 \}} | 0.22331 ^{\{ 15 \}} | 0.12956 ^{\{ 8 \}} | 0.13518 ^{\{ 9 \}} | 0.11795 ^{\{ 5 \}} | 0.1257 ^{\{ 7 \}} | 0.20342 ^{\{ 14 \}} | 0.01871 ^{\{ 1 \}} | 0.17116 ^{\{ 13 \}} | 0.1157 ^{\{ 3 \}} | 0.15442 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.01313 ^{\{ 2 \}} | 0.01465 ^{\{ 5 \}} | 0.0141 ^{\{ 3 \}} | 0.01638 ^{\{ 9 \}} | 0.01589 ^{\{ 8 \}} | 0.01792 ^{\{ 11 \}} | 0.01554 ^{\{ 7 \}} | 0.01446 ^{\{ 4 \}} | 0.01553 ^{\{ 6 \}} | 0.01793 ^{\{ 12 \}} | 0.0212 ^{\{ 13 \}} | 0.00382 ^{\{ 1 \}} | 0.02299 ^{\{ 15 \}} | 0.01724 ^{\{ 10 \}} | 0.02298 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.08181 ^{\{ 2 \}} | 0.11116 ^{\{ 6 \}} | 0.11825 ^{\{ 10 \}} | 0.11049 ^{\{ 5 \}} | 0.12538 ^{\{ 11 \}} | 0.14986 ^{\{ 15 \}} | 0.11269 ^{\{ 7 \}} | 0.11493 ^{\{ 9 \}} | 0.09995 ^{\{ 3 \}} | 0.11345 ^{\{ 8 \}} | 0.14105 ^{\{ 14 \}} | 0.0288 ^{\{ 1 \}} | 0.13377 ^{\{ 13 \}} | 0.11003 ^{\{ 4 \}} | 0.12889 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.21217 ^{\{ 2 \}} | 0.23684 ^{\{ 5 \}} | 0.23563 ^{\{ 3 \}} | 0.25133 ^{\{ 9 \}} | 0.24903 ^{\{ 8 \}} | 0.26879 ^{\{ 12 \}} | 0.24139 ^{\{ 7 \}} | 0.23744 ^{\{ 6 \}} | 0.23612 ^{\{ 4 \}} | 0.26185 ^{\{ 11 \}} | 0.29503 ^{\{ 13 \}} | 0.10535 ^{\{ 1 \}} | 0.29735 ^{\{ 14 \}} | 0.25632 ^{\{ 10 \}} | 0.29825 ^{\{ 15 \}} | |
D_{abs} | 0.03306 ^{\{ 3 \}} | 0.0336 ^{\{ 6 \}} | 0.03402 ^{\{ 7 \}} | 0.03258 ^{\{ 2 \}} | 0.03344 ^{\{ 5 \}} | 0.03621 ^{\{ 9 \}} | 0.03435 ^{\{ 8 \}} | 0.03312 ^{\{ 4 \}} | 0.03754 ^{\{ 12 \}} | 0.03649 ^{\{ 10 \}} | 0.03851 ^{\{ 13 \}} | 0.02949 ^{\{ 1 \}} | 0.03966 ^{\{ 14 \}} | 0.0365 ^{\{ 11 \}} | 0.04037 ^{\{ 15 \}} | |
D_{max} | 0.05248 ^{\{ 2 \}} | 0.05501 ^{\{ 5 \}} | 0.05639 ^{\{ 8 \}} | 0.05259 ^{\{ 3 \}} | 0.05553 ^{\{ 6 \}} | 0.06191 ^{\{ 12 \}} | 0.05603 ^{\{ 7 \}} | 0.05434 ^{\{ 4 \}} | 0.05941 ^{\{ 11 \}} | 0.05857 ^{\{ 10 \}} | 0.06353 ^{\{ 13 \}} | 0.04401 ^{\{ 1 \}} | 0.06477 ^{\{ 14 \}} | 0.05849 ^{\{ 9 \}} | 0.06561 ^{\{ 15 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 44 ^{\{ 3 \}} | 54 ^{\{ 7 \}} | 46 ^{\{ 4 \}} | 68 ^{\{ 10 \}} | 101 ^{\{ 12 \}} | 58 ^{\{ 8 \}} | 51 ^{\{ 6 \}} | 48 ^{\{ 5 \}} | 77 ^{\{ 11 \}} | 107 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 110 ^{\{ 14.5 \}} | 61 ^{\{ 9 \}} | 110 ^{\{ 14.5 \}} | |
100 | BIAS( \hat{\delta} ) | 0.16869 ^{\{ 2 \}} | 0.21955 ^{\{ 6 \}} | 0.25099 ^{\{ 11 \}} | 0.20658 ^{\{ 4 \}} | 0.23978 ^{\{ 10 \}} | 0.26936 ^{\{ 13 \}} | 0.22407 ^{\{ 7 \}} | 0.21601 ^{\{ 5 \}} | 0.20488 ^{\{ 3 \}} | 0.23686 ^{\{ 9 \}} | 0.2921 ^{\{ 15 \}} | 0.06504 ^{\{ 1 \}} | 0.27867 ^{\{ 14 \}} | 0.23071 ^{\{ 8 \}} | 0.26579 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.06441 ^{\{ 2 \}} | 0.0751 ^{\{ 5 \}} | 0.08048 ^{\{ 8 \}} | 0.07758 ^{\{ 6 \}} | 0.08158 ^{\{ 9 \}} | 0.08422 ^{\{ 10 \}} | 0.07438 ^{\{ 4 \}} | 0.07323 ^{\{ 3 \}} | 0.07816 ^{\{ 7 \}} | 0.0888 ^{\{ 12 \}} | 0.10363 ^{\{ 15 \}} | 0.03587 ^{\{ 1 \}} | 0.10299 ^{\{ 14 \}} | 0.08464 ^{\{ 11 \}} | 0.09863 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.04708 ^{\{ 2 \}} | 0.07537 ^{\{ 5 \}} | 0.09757 ^{\{ 11 \}} | 0.0662 ^{\{ 3 \}} | 0.08954 ^{\{ 10 \}} | 0.11225 ^{\{ 13 \}} | 0.07976 ^{\{ 8 \}} | 0.07321 ^{\{ 4 \}} | 0.07788 ^{\{ 6 \}} | 0.08763 ^{\{ 9 \}} | 0.13241 ^{\{ 15 \}} | 0.01303 ^{\{ 1 \}} | 0.11883 ^{\{ 14 \}} | 0.07847 ^{\{ 7 \}} | 0.10997 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.00749 ^{\{ 2 \}} | 0.00915 ^{\{ 4 \}} | 0.0102 ^{\{ 6 \}} | 0.01024 ^{\{ 7 \}} | 0.0109 ^{\{ 8 \}} | 0.01143 ^{\{ 9 \}} | 0.00918 ^{\{ 5 \}} | 0.00855 ^{\{ 3 \}} | 0.01159 ^{\{ 11 \}} | 0.0135 ^{\{ 12 \}} | 0.01684 ^{\{ 13 \}} | 0.00259 ^{\{ 1 \}} | 0.01761 ^{\{ 15 \}} | 0.01144 ^{\{ 10 \}} | 0.01689 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.06747 ^{\{ 2 \}} | 0.08782 ^{\{ 6 \}} | 0.10039 ^{\{ 11 \}} | 0.08263 ^{\{ 4 \}} | 0.09591 ^{\{ 10 \}} | 0.10774 ^{\{ 13 \}} | 0.08963 ^{\{ 7 \}} | 0.0864 ^{\{ 5 \}} | 0.08195 ^{\{ 3 \}} | 0.09474 ^{\{ 9 \}} | 0.11684 ^{\{ 15 \}} | 0.02602 ^{\{ 1 \}} | 0.11147 ^{\{ 14 \}} | 0.09228 ^{\{ 8 \}} | 0.10632 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.16103 ^{\{ 2 \}} | 0.18776 ^{\{ 5 \}} | 0.20121 ^{\{ 8 \}} | 0.19396 ^{\{ 6 \}} | 0.20394 ^{\{ 9 \}} | 0.21056 ^{\{ 10 \}} | 0.18594 ^{\{ 4 \}} | 0.18308 ^{\{ 3 \}} | 0.1954 ^{\{ 7 \}} | 0.22201 ^{\{ 12 \}} | 0.25908 ^{\{ 15 \}} | 0.08969 ^{\{ 1 \}} | 0.25746 ^{\{ 14 \}} | 0.2116 ^{\{ 11 \}} | 0.24657 ^{\{ 13 \}} | |
D_{abs} | 0.02344 ^{\{ 1 \}} | 0.0258 ^{\{ 4 \}} | 0.02729 ^{\{ 8 \}} | 0.02632 ^{\{ 5 \}} | 0.02689 ^{\{ 7 \}} | 0.02795 ^{\{ 9 \}} | 0.02543 ^{\{ 3 \}} | 0.02678 ^{\{ 6 \}} | 0.03037 ^{\{ 13 \}} | 0.02968 ^{\{ 12 \}} | 0.02967 ^{\{ 11 \}} | 0.02442 ^{\{ 2 \}} | 0.03344 ^{\{ 15 \}} | 0.02859 ^{\{ 10 \}} | 0.03216 ^{\{ 14 \}} | |
D_{max} | 0.03772 ^{\{ 2 \}} | 0.04225 ^{\{ 4 \}} | 0.04553 ^{\{ 8 \}} | 0.04245 ^{\{ 5 \}} | 0.0444 ^{\{ 7 \}} | 0.04705 ^{\{ 10 \}} | 0.04196 ^{\{ 3 \}} | 0.04374 ^{\{ 6 \}} | 0.0487 ^{\{ 12 \}} | 0.04793 ^{\{ 11 \}} | 0.04912 ^{\{ 13 \}} | 0.03646 ^{\{ 1 \}} | 0.05452 ^{\{ 15 \}} | 0.04636 ^{\{ 9 \}} | 0.05237 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 39 ^{\{ 4 \}} | 71 ^{\{ 9 \}} | 40 ^{\{ 5 \}} | 70 ^{\{ 8 \}} | 87 ^{\{ 12 \}} | 41 ^{\{ 6 \}} | 35 ^{\{ 3 \}} | 62 ^{\{ 7 \}} | 86 ^{\{ 11 \}} | 112 ^{\{ 14 \}} | 9 ^{\{ 1 \}} | 115 ^{\{ 15 \}} | 74 ^{\{ 10 \}} | 104 ^{\{ 13 \}} | |
200 | BIAS( \hat{\delta} ) | 0.12568 ^{\{ 2 \}} | 0.15802 ^{\{ 6 \}} | 0.16754 ^{\{ 10 \}} | 0.14131 ^{\{ 3 \}} | 0.17205 ^{\{ 11 \}} | 0.19869 ^{\{ 12 \}} | 0.1585 ^{\{ 8 \}} | 0.15009 ^{\{ 5 \}} | 0.14533 ^{\{ 4 \}} | 0.16686 ^{\{ 9 \}} | 0.215 ^{\{ 15 \}} | 0.05087 ^{\{ 1 \}} | 0.20091 ^{\{ 13 \}} | 0.15808 ^{\{ 7 \}} | 0.20236 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.04788 ^{\{ 2 \}} | 0.05346 ^{\{ 5 \}} | 0.05363 ^{\{ 6 \}} | 0.0502 ^{\{ 3 \}} | 0.05533 ^{\{ 9 \}} | 0.06203 ^{\{ 12 \}} | 0.05404 ^{\{ 7 \}} | 0.05068 ^{\{ 4 \}} | 0.05485 ^{\{ 8 \}} | 0.06093 ^{\{ 11 \}} | 0.07871 ^{\{ 15 \}} | 0.02619 ^{\{ 1 \}} | 0.07197 ^{\{ 13 \}} | 0.05806 ^{\{ 10 \}} | 0.07216 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.02442 ^{\{ 2 \}} | 0.0386 ^{\{ 5 \}} | 0.04604 ^{\{ 11 \}} | 0.03106 ^{\{ 3 \}} | 0.04597 ^{\{ 10 \}} | 0.06064 ^{\{ 13 \}} | 0.03893 ^{\{ 6 \}} | 0.03567 ^{\{ 4 \}} | 0.03895 ^{\{ 7 \}} | 0.04485 ^{\{ 9 \}} | 0.06901 ^{\{ 15 \}} | 0.00783 ^{\{ 1 \}} | 0.0603 ^{\{ 12 \}} | 0.03976 ^{\{ 8 \}} | 0.06373 ^{\{ 14 \}} | |
MSE( \hat{\beta} ) | 0.00378 ^{\{ 2 \}} | 0.00458 ^{\{ 5 \}} | 0.00482 ^{\{ 7 \}} | 0.00408 ^{\{ 3 \}} | 0.00482 ^{\{ 7 \}} | 0.00624 ^{\{ 12 \}} | 0.00482 ^{\{ 7 \}} | 0.00409 ^{\{ 4 \}} | 0.00537 ^{\{ 9 \}} | 0.00623 ^{\{ 11 \}} | 0.00978 ^{\{ 15 \}} | 0.00135 ^{\{ 1 \}} | 0.00858 ^{\{ 13 \}} | 0.00572 ^{\{ 10 \}} | 0.00898 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.05027 ^{\{ 2 \}} | 0.06321 ^{\{ 6 \}} | 0.06702 ^{\{ 10 \}} | 0.05652 ^{\{ 3 \}} | 0.06882 ^{\{ 11 \}} | 0.07948 ^{\{ 12 \}} | 0.0634 ^{\{ 8 \}} | 0.06003 ^{\{ 5 \}} | 0.05813 ^{\{ 4 \}} | 0.06675 ^{\{ 9 \}} | 0.086 ^{\{ 15 \}} | 0.02035 ^{\{ 1 \}} | 0.08036 ^{\{ 13 \}} | 0.06323 ^{\{ 7 \}} | 0.08094 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.1197 ^{\{ 2 \}} | 0.13366 ^{\{ 5 \}} | 0.13408 ^{\{ 6 \}} | 0.12549 ^{\{ 3 \}} | 0.13833 ^{\{ 9 \}} | 0.15508 ^{\{ 12 \}} | 0.13511 ^{\{ 7 \}} | 0.12671 ^{\{ 4 \}} | 0.13713 ^{\{ 8 \}} | 0.15232 ^{\{ 11 \}} | 0.19678 ^{\{ 15 \}} | 0.06548 ^{\{ 1 \}} | 0.17993 ^{\{ 13 \}} | 0.14516 ^{\{ 10 \}} | 0.1804 ^{\{ 14 \}} | |
D_{abs} | 0.01719 ^{\{ 2 \}} | 0.01841 ^{\{ 5 \}} | 0.01851 ^{\{ 6 \}} | 0.0174 ^{\{ 3 \}} | 0.01866 ^{\{ 7 \}} | 0.02028 ^{\{ 9 \}} | 0.01868 ^{\{ 8 \}} | 0.01825 ^{\{ 4 \}} | 0.02193 ^{\{ 12 \}} | 0.02128 ^{\{ 11 \}} | 0.02281 ^{\{ 13 \}} | 0.0171 ^{\{ 1 \}} | 0.02382 ^{\{ 15 \}} | 0.02076 ^{\{ 10 \}} | 0.02362 ^{\{ 14 \}} | |
D_{max} | 0.02788 ^{\{ 2 \}} | 0.03027 ^{\{ 5 \}} | 0.03093 ^{\{ 7 \}} | 0.02818 ^{\{ 3 \}} | 0.03111 ^{\{ 8 \}} | 0.03422 ^{\{ 10 \}} | 0.03083 ^{\{ 6 \}} | 0.02987 ^{\{ 4 \}} | 0.03496 ^{\{ 12 \}} | 0.03435 ^{\{ 11 \}} | 0.03748 ^{\{ 13 \}} | 0.02581 ^{\{ 1 \}} | 0.03887 ^{\{ 15 \}} | 0.03338 ^{\{ 9 \}} | 0.03869 ^{\{ 14 \}} | |
\sum Ranks | 16 ^{\{ 2 \}} | 42 ^{\{ 5 \}} | 63 ^{\{ 7 \}} | 24 ^{\{ 3 \}} | 72 ^{\{ 10 \}} | 92 ^{\{ 12 \}} | 57 ^{\{ 6 \}} | 34 ^{\{ 4 \}} | 64 ^{\{ 8 \}} | 82 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 8 ^{\{ 1 \}} | 107 ^{\{ 13 \}} | 71 ^{\{ 9 \}} | 112 ^{\{ 14 \}} | |
300 | BIAS( \hat{\delta} ) | 0.10138 ^{\{ 2 \}} | 0.12388 ^{\{ 6 \}} | 0.13609 ^{\{ 10 \}} | 0.11995 ^{\{ 4 \}} | 0.13703 ^{\{ 11 \}} | 0.15219 ^{\{ 12 \}} | 0.11897 ^{\{ 3 \}} | 0.12185 ^{\{ 5 \}} | 0.13147 ^{\{ 8 \}} | 0.13467 ^{\{ 9 \}} | 0.17695 ^{\{ 15 \}} | 0.04844 ^{\{ 1 \}} | 0.1656 ^{\{ 14 \}} | 0.12572 ^{\{ 7 \}} | 0.15695 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.03889 ^{\{ 2 \}} | 0.04166 ^{\{ 4 \}} | 0.04473 ^{\{ 8 \}} | 0.04246 ^{\{ 6 \}} | 0.04436 ^{\{ 7 \}} | 0.04875 ^{\{ 12 \}} | 0.04023 ^{\{ 3 \}} | 0.04196 ^{\{ 5 \}} | 0.04867 ^{\{ 11 \}} | 0.0481 ^{\{ 10 \}} | 0.06487 ^{\{ 15 \}} | 0.02362 ^{\{ 1 \}} | 0.05683 ^{\{ 14 \}} | 0.04507 ^{\{ 9 \}} | 0.05475 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01642 ^{\{ 2 \}} | 0.02414 ^{\{ 6 \}} | 0.02962 ^{\{ 10 \}} | 0.0216 ^{\{ 3 \}} | 0.02872 ^{\{ 9 \}} | 0.03784 ^{\{ 12 \}} | 0.02234 ^{\{ 4 \}} | 0.02409 ^{\{ 5 \}} | 0.03044 ^{\{ 11 \}} | 0.02838 ^{\{ 8 \}} | 0.04928 ^{\{ 15 \}} | 0.00716 ^{\{ 1 \}} | 0.04261 ^{\{ 14 \}} | 0.02478 ^{\{ 7 \}} | 0.03848 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.00251 ^{\{ 2 \}} | 0.00274 ^{\{ 4 \}} | 0.00319 ^{\{ 8 \}} | 0.00286 ^{\{ 6 \}} | 0.00308 ^{\{ 7 \}} | 0.0039 ^{\{ 11 \}} | 0.00252 ^{\{ 3 \}} | 0.00285 ^{\{ 5 \}} | 0.00411 ^{\{ 12 \}} | 0.00379 ^{\{ 10 \}} | 0.00722 ^{\{ 15 \}} | 0.00119 ^{\{ 1 \}} | 0.00549 ^{\{ 14 \}} | 0.00329 ^{\{ 9 \}} | 0.00507 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.04055 ^{\{ 2 \}} | 0.04955 ^{\{ 6 \}} | 0.05444 ^{\{ 10 \}} | 0.04798 ^{\{ 4 \}} | 0.05481 ^{\{ 11 \}} | 0.06088 ^{\{ 12 \}} | 0.04759 ^{\{ 3 \}} | 0.04874 ^{\{ 5 \}} | 0.05259 ^{\{ 8 \}} | 0.05387 ^{\{ 9 \}} | 0.07078 ^{\{ 15 \}} | 0.01938 ^{\{ 1 \}} | 0.06624 ^{\{ 14 \}} | 0.05029 ^{\{ 7 \}} | 0.06278 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.09723 ^{\{ 2 \}} | 0.10415 ^{\{ 4 \}} | 0.11183 ^{\{ 8 \}} | 0.10614 ^{\{ 6 \}} | 0.11091 ^{\{ 7 \}} | 0.12186 ^{\{ 12 \}} | 0.10057 ^{\{ 3 \}} | 0.10489 ^{\{ 5 \}} | 0.12168 ^{\{ 11 \}} | 0.12025 ^{\{ 10 \}} | 0.16218 ^{\{ 15 \}} | 0.05905 ^{\{ 1 \}} | 0.14207 ^{\{ 14 \}} | 0.11268 ^{\{ 9 \}} | 0.13687 ^{\{ 13 \}} | |
D_{abs} | 0.01417 ^{\{ 1 \}} | 0.01453 ^{\{ 2 \}} | 0.01543 ^{\{ 7 \}} | 0.01481 ^{\{ 5 \}} | 0.01544 ^{\{ 8 \}} | 0.01573 ^{\{ 9 \}} | 0.01455 ^{\{ 3 \}} | 0.0152 ^{\{ 6 \}} | 0.01852 ^{\{ 13 \}} | 0.01696 ^{\{ 11 \}} | 0.0176 ^{\{ 12 \}} | 0.0147 ^{\{ 4 \}} | 0.01974 ^{\{ 15 \}} | 0.01602 ^{\{ 10 \}} | 0.01923 ^{\{ 14 \}} | |
D_{max} | 0.02287 ^{\{ 2 \}} | 0.02379 ^{\{ 4 \}} | 0.02546 ^{\{ 7 \}} | 0.02403 ^{\{ 5 \}} | 0.02569 ^{\{ 8 \}} | 0.0263 ^{\{ 10 \}} | 0.02377 ^{\{ 3 \}} | 0.02487 ^{\{ 6 \}} | 0.02963 ^{\{ 13 \}} | 0.02753 ^{\{ 11 \}} | 0.02918 ^{\{ 12 \}} | 0.02235 ^{\{ 1 \}} | 0.03243 ^{\{ 15 \}} | 0.02583 ^{\{ 9 \}} | 0.03132 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 36 ^{\{ 4 \}} | 68 ^{\{ 8.5 \}} | 39 ^{\{ 5 \}} | 68 ^{\{ 8.5 \}} | 90 ^{\{ 12 \}} | 25 ^{\{ 3 \}} | 42 ^{\{ 6 \}} | 87 ^{\{ 11 \}} | 78 ^{\{ 10 \}} | 114 ^{\{ 14.5 \}} | 11 ^{\{ 1 \}} | 114 ^{\{ 14.5 \}} | 67 ^{\{ 7 \}} | 106 ^{\{ 13 \}} | |
400 | BIAS( \hat{\delta} ) | 0.08539 ^{\{ 2 \}} | 0.10758 ^{\{ 5 \}} | 0.12255 ^{\{ 11 \}} | 0.10162 ^{\{ 3 \}} | 0.12057 ^{\{ 10 \}} | 0.13935 ^{\{ 12 \}} | 0.10965 ^{\{ 6 \}} | 0.11168 ^{\{ 7 \}} | 0.10747 ^{\{ 4 \}} | 0.11634 ^{\{ 9 \}} | 0.15238 ^{\{ 15 \}} | 0.03778 ^{\{ 1 \}} | 0.14254 ^{\{ 13 \}} | 0.11229 ^{\{ 8 \}} | 0.14255 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.03288 ^{\{ 2 \}} | 0.03624 ^{\{ 4 \}} | 0.04047 ^{\{ 10 \}} | 0.03533 ^{\{ 3 \}} | 0.03953 ^{\{ 9 \}} | 0.04295 ^{\{ 12 \}} | 0.03632 ^{\{ 5 \}} | 0.03807 ^{\{ 6 \}} | 0.03897 ^{\{ 7 \}} | 0.04102 ^{\{ 11 \}} | 0.05635 ^{\{ 15 \}} | 0.01866 ^{\{ 1 \}} | 0.0513 ^{\{ 14 \}} | 0.03935 ^{\{ 8 \}} | 0.04946 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01164 ^{\{ 2 \}} | 0.01815 ^{\{ 4 \}} | 0.02321 ^{\{ 11 \}} | 0.01641 ^{\{ 3 \}} | 0.02278 ^{\{ 10 \}} | 0.02969 ^{\{ 12 \}} | 0.01878 ^{\{ 5 \}} | 0.01988 ^{\{ 7 \}} | 0.02066 ^{\{ 8 \}} | 0.02178 ^{\{ 9 \}} | 0.03737 ^{\{ 15 \}} | 0.00425 ^{\{ 1 \}} | 0.03212 ^{\{ 14 \}} | 0.01965 ^{\{ 6 \}} | 0.03198 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.00175 ^{\{ 2 \}} | 0.00205 ^{\{ 4 \}} | 0.00259 ^{\{ 9 \}} | 0.00201 ^{\{ 3 \}} | 0.0025 ^{\{ 8 \}} | 0.00289 ^{\{ 12 \}} | 0.00208 ^{\{ 5 \}} | 0.00229 ^{\{ 6 \}} | 0.00265 ^{\{ 10 \}} | 0.00283 ^{\{ 11 \}} | 0.0054 ^{\{ 15 \}} | 0.00073 ^{\{ 1 \}} | 0.00429 ^{\{ 14 \}} | 0.00241 ^{\{ 7 \}} | 0.00393 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.03416 ^{\{ 2 \}} | 0.04303 ^{\{ 5 \}} | 0.04902 ^{\{ 11 \}} | 0.04065 ^{\{ 3 \}} | 0.04823 ^{\{ 10 \}} | 0.05574 ^{\{ 12 \}} | 0.04386 ^{\{ 6 \}} | 0.04467 ^{\{ 7 \}} | 0.04299 ^{\{ 4 \}} | 0.04654 ^{\{ 9 \}} | 0.06095 ^{\{ 15 \}} | 0.01511 ^{\{ 1 \}} | 0.05701 ^{\{ 13 \}} | 0.04492 ^{\{ 8 \}} | 0.05702 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.08219 ^{\{ 2 \}} | 0.09059 ^{\{ 4 \}} | 0.10117 ^{\{ 10 \}} | 0.08833 ^{\{ 3 \}} | 0.09881 ^{\{ 9 \}} | 0.10736 ^{\{ 12 \}} | 0.0908 ^{\{ 5 \}} | 0.09518 ^{\{ 6 \}} | 0.09744 ^{\{ 7 \}} | 0.10255 ^{\{ 11 \}} | 0.14089 ^{\{ 15 \}} | 0.04664 ^{\{ 1 \}} | 0.12824 ^{\{ 14 \}} | 0.09839 ^{\{ 8 \}} | 0.12364 ^{\{ 13 \}} | |
D_{abs} | 0.01182 ^{\{ 1 \}} | 0.01268 ^{\{ 4 \}} | 0.01351 ^{\{ 7.5 \}} | 0.01208 ^{\{ 3 \}} | 0.01351 ^{\{ 7.5 \}} | 0.01356 ^{\{ 9 \}} | 0.01291 ^{\{ 5 \}} | 0.0134 ^{\{ 6 \}} | 0.0148 ^{\{ 11 \}} | 0.01494 ^{\{ 12 \}} | 0.016 ^{\{ 13 \}} | 0.01204 ^{\{ 2 \}} | 0.01733 ^{\{ 15 \}} | 0.01386 ^{\{ 10 \}} | 0.01687 ^{\{ 14 \}} | |
D_{max} | 0.01903 ^{\{ 2 \}} | 0.02082 ^{\{ 4 \}} | 0.02243 ^{\{ 8 \}} | 0.01969 ^{\{ 3 \}} | 0.02228 ^{\{ 7 \}} | 0.02295 ^{\{ 10 \}} | 0.02117 ^{\{ 5 \}} | 0.02188 ^{\{ 6 \}} | 0.02381 ^{\{ 11 \}} | 0.02425 ^{\{ 12 \}} | 0.02642 ^{\{ 13 \}} | 0.01826 ^{\{ 1 \}} | 0.02832 ^{\{ 15 \}} | 0.02247 ^{\{ 9 \}} | 0.02769 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 34 ^{\{ 4 \}} | 77.5 ^{\{ 10 \}} | 24 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 91 ^{\{ 12 \}} | 42 ^{\{ 5 \}} | 51 ^{\{ 6 \}} | 62 ^{\{ 7 \}} | 84 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 9 ^{\{ 1 \}} | 112 ^{\{ 14 \}} | 64 ^{\{ 8 \}} | 108 ^{\{ 13 \}} |
Parameter | n | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
\delta=0.7, \beta=2.5 | 30 | 4.0 | 7.0 | 11.0 | 6.0 | 12.5 | 15.0 | 12.5 | 10.0 | 2.0 | 3.0 | 14.0 | 1.0 | 8.5 | 8.5 | 5.0 |
60 | 3.0 | 11.0 | 14.0 | 4.0 | 10.0 | 15.0 | 12.0 | 9.0 | 2.0 | 5.0 | 8.0 | 1.0 | 6.0 | 13.0 | 7.0 | |
100 | 3.0 | 11.0 | 13.0 | 4.0 | 12.0 | 15.0 | 9.0 | 7.0 | 2.0 | 6.0 | 14.0 | 1.0 | 5.0 | 10.0 | 8.0 | |
200 | 2.0 | 7.0 | 14.0 | 5.0 | 13.0 | 15.0 | 10.0 | 4.0 | 3.0 | 8.0 | 12.0 | 1.0 | 6.0 | 11.0 | 9.0 | |
300 | 2.0 | 6.0 | 13.0 | 4.0 | 14.0 | 15.0 | 11.0 | 5.0 | 3.0 | 7.0 | 12.0 | 1.0 | 8.5 | 10.0 | 8.5 | |
400 | 2.0 | 9.0 | 14.0 | 4.0 | 12.5 | 15.0 | 7.0 | 5.0 | 3.0 | 11.0 | 12.5 | 1.0 | 6.0 | 8.0 | 10.0 | |
\delta=0.25, \beta=0.75 | 30 | 2.0 | 6.0 | 11.5 | 7.0 | 10.0 | 13.0 | 8.0 | 5.0 | 3.5 | 3.5 | 11.5 | 1.0 | 15.0 | 9.0 | 14.0 |
60 | 2.0 | 7.0 | 10.0 | 5.0 | 11.0 | 13.0 | 8.0 | 4.0 | 3.0 | 6.0 | 12.0 | 1.0 | 15.0 | 9.0 | 14.0 | |
100 | 2.0 | 3.5 | 8.0 | 6.0 | 9.0 | 14.0 | 3.5 | 7.0 | 5.0 | 11.0 | 13.0 | 1.0 | 12.0 | 10.0 | 15.0 | |
200 | 2.0 | 3.0 | 10.0 | 4.5 | 11.0 | 13.0 | 7.0 | 4.5 | 6.0 | 8.0 | 14.0 | 1.0 | 15.0 | 9.0 | 12.0 | |
300 | 1.0 | 7.0 | 12.0 | 3.0 | 11.0 | 14.0 | 5.0 | 6.0 | 4.0 | 9.0 | 15.0 | 2.0 | 13.0 | 8.0 | 10.0 | |
400 | 2.0 | 5.0 | 10.0 | 4.0 | 8.0 | 13.0 | 6.0 | 3.0 | 7.0 | 12.0 | 15.0 | 1.0 | 14.0 | 9.0 | 11.0 | |
\delta=1.5, \beta=1.5 | 30 | 2.0 | 8.0 | 12.0 | 5.0 | 11.0 | 13.0 | 10.0 | 4.0 | 3.0 | 7.0 | 14.0 | 1.0 | 15.0 | 6.0 | 9.0 |
60 | 1.0 | 6.0 | 10.0 | 4.0 | 11.5 | 14.0 | 7.0 | 3.0 | 5.0 | 8.5 | 11.5 | 2.0 | 15.0 | 8.5 | 13.0 | |
100 | 1.0 | 7.0 | 11.0 | 4.0 | 12.0 | 13.0 | 8.0 | 3.0 | 6.0 | 9.0 | 10.0 | 2.0 | 15.0 | 5.0 | 14.0 | |
200 | 1.5 | 5.0 | 10.0 | 3.0 | 11.0 | 14.0 | 6.5 | 4.0 | 6.5 | 9.0 | 12.0 | 1.5 | 15.0 | 8.0 | 13.0 | |
300 | 1.0 | 5.0 | 10.5 | 4.0 | 9.0 | 13.0 | 7.0 | 3.0 | 6.0 | 10.5 | 12.0 | 2.0 | 15.0 | 8.0 | 14.0 | |
400 | 1.0 | 7.0 | 10.0 | 3.0 | 11.0 | 14.0 | 6.0 | 4.0 | 9.0 | 8.0 | 12.0 | 2.0 | 15.0 | 5.0 | 13.0 | |
\delta=0.5, \beta=2.0 | 30 | 3.0 | 9.0 | 14.0 | 5.0 | 8.0 | 15.0 | 7.0 | 6.0 | 2.0 | 4.0 | 13.0 | 1.0 | 11.0 | 12.0 | 10.0 |
60 | 3.0 | 5.0 | 14.0 | 4.0 | 11.0 | 15.0 | 12.0 | 9.0 | 2.0 | 6.0 | 13.0 | 1.0 | 10.0 | 8.0 | 7.0 | |
100 | 3.0 | 9.0 | 13.0 | 4.0 | 14.0 | 15.0 | 10.0 | 5.0 | 2.0 | 11.0 | 12.0 | 1.0 | 7.5 | 7.5 | 6.0 | |
200 | 3.0 | 9.0 | 12.0 | 4.0 | 13.0 | 15.0 | 8.0 | 5.0 | 2.0 | 10.0 | 14.0 | 1.0 | 6.0 | 11.0 | 7.0 | |
300 | 2.0 | 8.0 | 14.0 | 4.0 | 13.0 | 15.0 | 9.0 | 5.0 | 3.0 | 11.0 | 12.0 | 1.0 | 6.0 | 10.0 | 7.0 | |
400 | 2.0 | 8.0 | 14.0 | 4.0 | 12.0 | 15.0 | 10.0 | 5.0 | 3.0 | 11.0 | 13.0 | 1.0 | 9.0 | 6.0 | 7.0 | |
\delta=2.5, \beta=0.4 | 30 | 2.0 | 6.0 | 11.0 | 4.5 | 10.0 | 15.0 | 4.5 | 8.0 | 3.0 | 9.0 | 13.0 | 1.0 | 14.0 | 7.0 | 12.0 |
60 | 2.0 | 3.0 | 7.0 | 4.0 | 10.0 | 12.0 | 8.0 | 6.0 | 5.0 | 11.0 | 13.0 | 1.0 | 14.5 | 9.0 | 14.5 | |
100 | 2.0 | 4.0 | 9.0 | 5.0 | 8.0 | 12.0 | 6.0 | 3.0 | 7.0 | 11.0 | 14.0 | 1.0 | 15.0 | 10.0 | 13.0 | |
200 | 2.0 | 5.0 | 7.0 | 3.0 | 10.0 | 12.0 | 6.0 | 4.0 | 8.0 | 11.0 | 15.0 | 1.0 | 13.0 | 9.0 | 14.0 | |
300 | 2.0 | 4.0 | 8.5 | 5.0 | 8.5 | 12.0 | 3.0 | 6.0 | 11.0 | 10.0 | 14.5 | 1.0 | 14.5 | 7.0 | 13.0 | |
400 | 2.0 | 4.0 | 10.0 | 3.0 | 9.0 | 12.0 | 5.0 | 6.0 | 7.0 | 11.0 | 15.0 | 1.0 | 14.0 | 8.0 | 13.0 | |
\sum Ranks | 62.5 | 194.5 | 337.5 | 129.0 | 326.0 | 416.0 | 232.0 | 158.5 | 134.0 | 257.5 | 386.0 | 35.5 | 348.5 | 259.5 | 323.0 | |
Overall Rank | 2 | 6 | 12 | 3 | 11 | 15 | 7 | 5 | 4 | 8 | 14 | 1 | 13 | 9 | 10 |
The simulation results show that all parameter estimation methods for the proposed model have high accuracy and are close to the true values. The computed measures generally decrease with increasing sample size ( n ). KE emerges as the most effective parameter estimation method, with an overall score of 35.5 (see Table 7).
In order to have graphical benchmark, the values from Table 2 are also visualized in Figures 3–11.
In these figures, we can see the fast decay of all curves with relatively small values for n . This confirms the efficiency of the estimation methods considered in the context of the PUILD.
In this subsection, randomly generated datasets have been used to find different values of the estimated entropy measures derived in Section 4. These numerical values are presented in Tables 8–11. A comparative analysis of the estimated entropy values using different measures, metrics such as BIAS, MSE, and MRE, and sample sizes ( n ) is performed. These measures are denoted by their respective abbreviations, such as RE for Rényi entropy, ExE for exponential entropy, HCE for Havrda and Charvat entropy, ArE for Arimoto entropy, TsE for Tsallis entropy, AA1E for Awad and Alawneh 1 entropy, AA2E for Awad and Alawneh 2 entropy, ShE for Shannon entropy, EX for extropy, and WEX for weighted extropy. These numerical values are obtained by the following procedure:
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.49641) | (0.60871) | (-0.53065) | (-0.39129) | (-0.43960) | (-0.29722) | (0.38680) | (-0.56988) | (-0.94070) | (-0.73958) | ||
20 | \hat{E} | -0.53095 | 0.59059 | -0.56088 | -0.40941 | -0.46465 | -0.34499 | 0.45684 | -0.60354 | -0.97991 | -0.76199 |
BIAS | 0.07837 | 0.04604 | 0.07246 | 0.04604 | 0.06003 | 0.05431 | 0.07916 | 0.08133 | 0.08408 | 0.08889 | |
MSE | 0.00993 | 0.00330 | 0.00831 | 0.00330 | 0.00570 | 0.00859 | 0.01919 | 0.01077 | 0.01236 | 0.01358 | |
MRE | 0.15788 | 0.07564 | 0.13656 | 0.11767 | 0.13656 | 0.18274 | 0.20465 | 0.14272 | 0.08938 | 0.12019 | |
60 | \hat{E} | -0.51151 | 0.60061 | -0.54401 | -0.39939 | -0.45067 | -0.33185 | 0.43748 | -0.58617 | -0.96029 | -0.75437 |
BIAS | 0.04763 | 0.02850 | 0.04447 | 0.02850 | 0.03684 | 0.04169 | 0.06052 | 0.04871 | 0.04965 | 0.05278 | |
MSE | 0.00366 | 0.00128 | 0.00315 | 0.00128 | 0.00216 | 0.00597 | 0.01312 | 0.00382 | 0.00409 | 0.00461 | |
MRE | 0.09595 | 0.04683 | 0.08379 | 0.07285 | 0.08379 | 0.14026 | 0.15646 | 0.08548 | 0.05278 | 0.07137 | |
100 | \hat{E} | -0.50365 | 0.60492 | -0.53699 | -0.39508 | -0.44486 | -0.32447 | 0.42653 | -0.57848 | -0.95201 | -0.74854 |
BIAS | 0.03577 | 0.02159 | 0.03354 | 0.02159 | 0.02778 | 0.03496 | 0.05049 | 0.03662 | 0.03749 | 0.04155 | |
MSE | 0.00203 | 0.00073 | 0.00178 | 0.00073 | 0.00122 | 0.00432 | 0.00937 | 0.00212 | 0.00225 | 0.00277 | |
MRE | 0.07206 | 0.03546 | 0.06320 | 0.05517 | 0.06320 | 0.11763 | 0.13054 | 0.06426 | 0.03985 | 0.05618 | |
150 | \hat{E} | -0.50062 | 0.60654 | -0.53431 | -0.39346 | -0.44263 | -0.31955 | 0.41918 | -0.57587 | -0.94935 | -0.74777 |
BIAS | 0.02875 | 0.01742 | 0.02701 | 0.01742 | 0.02237 | 0.02987 | 0.04291 | 0.02945 | 0.03022 | 0.03410 | |
MSE | 0.00131 | 0.00048 | 0.00115 | 0.00048 | 0.00079 | 0.00307 | 0.00658 | 0.00136 | 0.00147 | 0.00187 | |
MRE | 0.05791 | 0.02861 | 0.05090 | 0.04451 | 0.05090 | 0.10051 | 0.11094 | 0.05167 | 0.03213 | 0.04610 | |
200 | \hat{E} | -0.49922 | 0.60731 | -0.53305 | -0.39269 | -0.44159 | -0.31498 | 0.41241 | -0.57418 | -0.94720 | -0.74586 |
BIAS | 0.02534 | 0.01537 | 0.02382 | 0.01537 | 0.01973 | 0.02508 | 0.03583 | 0.02571 | 0.02619 | 0.02975 | |
MSE | 0.00100 | 0.00037 | 0.00088 | 0.00037 | 0.00061 | 0.00205 | 0.00433 | 0.00104 | 0.00110 | 0.00142 | |
MRE | 0.05105 | 0.02525 | 0.04489 | 0.03928 | 0.04489 | 0.08439 | 0.09264 | 0.04512 | 0.02784 | 0.04023 | |
250 | \hat{E} | -0.49872 | 0.60755 | -0.53263 | -0.39245 | -0.44125 | -0.31227 | 0.40843 | -0.57363 | -0.94642 | -0.74544 |
BIAS | 0.02237 | 0.01358 | 0.02104 | 0.01358 | 0.01743 | 0.02194 | 0.03125 | 0.02259 | 0.02298 | 0.02633 | |
MSE | 0.00078 | 0.00029 | 0.00069 | 0.00029 | 0.00047 | 0.00152 | 0.00317 | 0.00080 | 0.00085 | 0.00111 | |
MRE | 0.04507 | 0.02231 | 0.03965 | 0.03471 | 0.03965 | 0.07382 | 0.08078 | 0.03964 | 0.02443 | 0.03560 | |
300 | \hat{E} | -0.49867 | 0.60754 | -0.53262 | -0.39246 | -0.44124 | -0.30972 | 0.40472 | -0.57329 | -0.94570 | -0.74456 |
BIAS | 0.02057 | 0.01249 | 0.01934 | 0.01249 | 0.01602 | 0.01937 | 0.02751 | 0.02078 | 0.02112 | 0.02426 | |
MSE | 0.00066 | 0.00024 | 0.00059 | 0.00024 | 0.00040 | 0.00114 | 0.00237 | 0.00068 | 0.00071 | 0.00094 | |
MRE | 0.04143 | 0.02051 | 0.03645 | 0.03191 | 0.03645 | 0.06516 | 0.07111 | 0.03647 | 0.02246 | 0.03281 | |
400 | \hat{E} | -0.49783 | 0.60800 | -0.53187 | -0.39200 | -0.44061 | -0.30769 | 0.40175 | -0.57248 | -0.94480 | -0.74409 |
BIAS | 0.01785 | 0.01084 | 0.01679 | 0.01084 | 0.01391 | 0.01677 | 0.02374 | 0.01791 | 0.01813 | 0.02087 | |
MSE | 0.00050 | 0.00019 | 0.00045 | 0.00019 | 0.00031 | 0.00078 | 0.00159 | 0.00051 | 0.00053 | 0.00070 | |
MRE | 0.03595 | 0.01782 | 0.03165 | 0.02772 | 0.03165 | 0.05643 | 0.06138 | 0.03143 | 0.01927 | 0.02822 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.06783) | (0.93442) | (-0.08051) | (-0.06558) | (-0.06670) | (-0.55698) | (0.77529) | (-0.11870) | (-0.61333) | (-0.19500) | ||
20 | \hat{E} | -0.08600 | 0.91826 | -0.10119 | -0.08174 | -0.08383 | -0.57359 | 0.81240 | -0.14688 | -0.64206 | -0.20407 |
BIAS | 0.03295 | 0.03015 | 0.03804 | 0.03015 | 0.03152 | 0.13286 | 0.21409 | 0.05684 | 0.06063 | 0.02178 | |
MSE | 0.00181 | 0.00148 | 0.00238 | 0.00148 | 0.00163 | 0.02637 | 0.06957 | 0.00541 | 0.00640 | 0.00083 | |
MRE | 0.48578 | 0.03227 | 0.47257 | 0.45978 | 0.47257 | 0.23853 | 0.27615 | 0.47882 | 0.09886 | 0.11170 | |
60 | \hat{E} | -0.07058 | 0.93218 | -0.08350 | -0.06782 | -0.06918 | -0.54615 | 0.76306 | -0.12237 | -0.61731 | -0.19918 |
BIAS | 0.01987 | 0.01844 | 0.02311 | 0.01844 | 0.01914 | 0.08949 | 0.14217 | 0.03542 | 0.03742 | 0.01242 | |
MSE | 0.00071 | 0.00060 | 0.00095 | 0.00060 | 0.00065 | 0.01272 | 0.03216 | 0.00224 | 0.00257 | 0.00026 | |
MRE | 0.29296 | 0.01973 | 0.28701 | 0.28119 | 0.28701 | 0.16067 | 0.18337 | 0.29838 | 0.06101 | 0.06367 | |
100 | \hat{E} | -0.07079 | 0.93192 | -0.08380 | -0.06808 | -0.06942 | -0.55144 | 0.77021 | -0.12307 | -0.61799 | -0.19801 |
BIAS | 0.01830 | 0.01700 | 0.02129 | 0.01700 | 0.01764 | 0.07792 | 0.12400 | 0.03260 | 0.03419 | 0.00970 | |
MSE | 0.00057 | 0.00049 | 0.00077 | 0.00049 | 0.00053 | 0.00945 | 0.02392 | 0.00179 | 0.00197 | 0.00015 | |
MRE | 0.26984 | 0.01819 | 0.26448 | 0.25923 | 0.26448 | 0.13990 | 0.15994 | 0.27467 | 0.05574 | 0.04975 | |
150 | \hat{E} | -0.06964 | 0.93291 | -0.08251 | -0.06709 | -0.06835 | -0.55277 | 0.77125 | -0.12140 | -0.61630 | -0.19696 |
BIAS | 0.01545 | 0.01438 | 0.01799 | 0.01438 | 0.01490 | 0.06590 | 0.10492 | 0.02758 | 0.02884 | 0.00791 | |
MSE | 0.00039 | 0.00034 | 0.00053 | 0.00034 | 0.00036 | 0.00675 | 0.01708 | 0.00124 | 0.00136 | 0.00010 | |
MRE | 0.22776 | 0.01539 | 0.22349 | 0.21929 | 0.22349 | 0.11832 | 0.13533 | 0.23232 | 0.04703 | 0.04055 | |
200 | \hat{E} | -0.06936 | 0.93313 | -0.08220 | -0.06687 | -0.06810 | -0.55416 | 0.77283 | -0.12100 | -0.61582 | -0.19650 |
BIAS | 0.01350 | 0.01258 | 0.01573 | 0.01258 | 0.01303 | 0.05742 | 0.09143 | 0.02413 | 0.02521 | 0.00679 | |
MSE | 0.00029 | 0.00025 | 0.00039 | 0.00025 | 0.00027 | 0.00513 | 0.01298 | 0.00092 | 0.00101 | 0.00007 | |
MRE | 0.19902 | 0.01346 | 0.19539 | 0.19182 | 0.19539 | 0.10308 | 0.11793 | 0.20332 | 0.04110 | 0.03481 | |
250 | \hat{E} | -0.06880 | 0.93362 | -0.08157 | -0.06638 | -0.06758 | -0.55348 | 0.77135 | -0.12007 | -0.61481 | -0.19632 |
BIAS | 0.01209 | 0.01127 | 0.01409 | 0.01127 | 0.01167 | 0.05144 | 0.08187 | 0.02162 | 0.02254 | 0.00606 | |
MSE | 0.00023 | 0.00020 | 0.00031 | 0.00020 | 0.00021 | 0.00412 | 0.01041 | 0.00073 | 0.00079 | 0.00006 | |
MRE | 0.17817 | 0.01206 | 0.17501 | 0.17189 | 0.17501 | 0.09235 | 0.10559 | 0.18216 | 0.03675 | 0.03107 | |
300 | \hat{E} | -0.06849 | 0.93389 | -0.08122 | -0.06611 | -0.06728 | -0.55377 | 0.77148 | -0.11958 | -0.61429 | -0.19608 |
BIAS | 0.01080 | 0.01008 | 0.01259 | 0.01008 | 0.01043 | 0.04597 | 0.07317 | 0.01930 | 0.02008 | 0.00549 | |
MSE | 0.00018 | 0.00016 | 0.00024 | 0.00016 | 0.00017 | 0.00331 | 0.00836 | 0.00057 | 0.00062 | 0.00005 | |
MRE | 0.15919 | 0.01079 | 0.15641 | 0.15367 | 0.15641 | 0.08253 | 0.09437 | 0.16256 | 0.03273 | 0.02814 | |
400 | \hat{E} | -0.06779 | 0.93451 | -0.08043 | -0.06549 | -0.06663 | -0.55280 | 0.76960 | -0.11845 | -0.61313 | -0.19585 |
BIAS | 0.00922 | 0.00861 | 0.01075 | 0.00861 | 0.00891 | 0.03952 | 0.06287 | 0.01649 | 0.01714 | 0.00468 | |
MSE | 0.00013 | 0.00011 | 0.00018 | 0.00011 | 0.00012 | 0.00246 | 0.00620 | 0.00042 | 0.00045 | 0.00003 | |
MRE | 0.13587 | 0.00921 | 0.13356 | 0.13128 | 0.13356 | 0.07096 | 0.08109 | 0.13893 | 0.02795 | 0.02402 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.50778) | (0.60183) | (-0.54131) | (-0.39817) | (-0.44844) | (-0.41510) | (0.55685) | (-0.56943) | (-0.94546) | (-0.64094) | ||
20 | \hat{E} | -0.53963 | 0.58474 | -0.56950 | -0.41526 | -0.47179 | -0.42231 | 0.56992 | -0.60692 | -0.99220 | -0.68195 |
BIAS | 0.06486 | 0.03768 | 0.05965 | 0.03768 | 0.04941 | 0.06592 | 0.09848 | 0.07228 | 0.08395 | 0.06627 | |
MSE | 0.00716 | 0.00232 | 0.00593 | 0.00232 | 0.00407 | 0.00626 | 0.01412 | 0.00905 | 0.01324 | 0.00793 | |
MRE | 0.12773 | 0.06261 | 0.11019 | 0.09464 | 0.11019 | 0.15882 | 0.17685 | 0.12693 | 0.08879 | 0.10339 | |
60 | \hat{E} | -0.51075 | 0.60061 | -0.54366 | -0.39939 | -0.45038 | -0.39987 | 0.53536 | -0.57260 | -0.94830 | -0.65192 |
BIAS | 0.03416 | 0.02045 | 0.03190 | 0.02045 | 0.02642 | 0.04356 | 0.06430 | 0.03772 | 0.04263 | 0.03160 | |
MSE | 0.00191 | 0.00068 | 0.00165 | 0.00068 | 0.00114 | 0.00303 | 0.00661 | 0.00240 | 0.00323 | 0.00169 | |
MRE | 0.06728 | 0.03398 | 0.05893 | 0.05135 | 0.05893 | 0.10495 | 0.11547 | 0.06624 | 0.04509 | 0.04930 | |
100 | \hat{E} | -0.50817 | 0.60195 | -0.54141 | -0.39805 | -0.44852 | -0.40209 | 0.53840 | -0.56973 | -0.94512 | -0.64743 |
BIAS | 0.02656 | 0.01595 | 0.02484 | 0.01595 | 0.02058 | 0.03806 | 0.05627 | 0.02990 | 0.03474 | 0.02354 | |
MSE | 0.00118 | 0.00042 | 0.00102 | 0.00042 | 0.00070 | 0.00235 | 0.00513 | 0.00154 | 0.00218 | 0.00093 | |
MRE | 0.05231 | 0.02650 | 0.04589 | 0.04005 | 0.04589 | 0.09169 | 0.10105 | 0.05251 | 0.03675 | 0.03672 | |
150 | \hat{E} | -0.50866 | 0.60156 | -0.54195 | -0.39844 | -0.44896 | -0.40642 | 0.54463 | -0.57036 | -0.94611 | -0.64604 |
BIAS | 0.02290 | 0.01375 | 0.02142 | 0.01375 | 0.01774 | 0.03357 | 0.04972 | 0.02612 | 0.03078 | 0.01965 | |
MSE | 0.00085 | 0.00031 | 0.00074 | 0.00031 | 0.00051 | 0.00177 | 0.00388 | 0.00113 | 0.00162 | 0.00064 | |
MRE | 0.04510 | 0.02284 | 0.03956 | 0.03452 | 0.03956 | 0.08087 | 0.08930 | 0.04588 | 0.03256 | 0.03065 | |
200 | \hat{E} | -0.50805 | 0.60185 | -0.54143 | -0.39815 | -0.44853 | -0.40717 | 0.54558 | -0.56956 | -0.94505 | -0.64468 |
BIAS | 0.01964 | 0.01180 | 0.01838 | 0.01180 | 0.01523 | 0.02931 | 0.04342 | 0.02235 | 0.02632 | 0.01691 | |
MSE | 0.00061 | 0.00022 | 0.00054 | 0.00022 | 0.00037 | 0.00134 | 0.00294 | 0.00081 | 0.00115 | 0.00046 | |
MRE | 0.03868 | 0.01961 | 0.03395 | 0.02965 | 0.03395 | 0.07061 | 0.07797 | 0.03925 | 0.02783 | 0.02639 | |
250 | \hat{E} | -0.50768 | 0.60204 | -0.54111 | -0.39796 | -0.44827 | -0.40773 | 0.54631 | -0.56909 | -0.94448 | -0.64387 |
BIAS | 0.01792 | 0.01078 | 0.01678 | 0.01078 | 0.01390 | 0.02639 | 0.03910 | 0.02037 | 0.02394 | 0.01534 | |
MSE | 0.00051 | 0.00018 | 0.00044 | 0.00018 | 0.00030 | 0.00108 | 0.00237 | 0.00066 | 0.00093 | 0.00038 | |
MRE | 0.03529 | 0.01791 | 0.03099 | 0.02707 | 0.03099 | 0.06359 | 0.07022 | 0.03578 | 0.02532 | 0.02394 | |
300 | \hat{E} | -0.50732 | 0.60223 | -0.54079 | -0.39777 | -0.44801 | -0.40808 | 0.54678 | -0.56866 | -0.94398 | -0.64321 |
BIAS | 0.01623 | 0.00976 | 0.01520 | 0.00976 | 0.01259 | 0.02435 | 0.03607 | 0.01864 | 0.02221 | 0.01370 | |
MSE | 0.00042 | 0.00015 | 0.00037 | 0.00015 | 0.00025 | 0.00091 | 0.00200 | 0.00056 | 0.00078 | 0.00030 | |
MRE | 0.03196 | 0.01623 | 0.02807 | 0.02452 | 0.02807 | 0.05866 | 0.06477 | 0.03274 | 0.02349 | 0.02138 | |
400 | \hat{E} | -0.50687 | 0.60247 | -0.54040 | -0.39753 | -0.44768 | -0.40877 | 0.54769 | -0.56812 | -0.94337 | -0.64228 |
BIAS | 0.01396 | 0.00840 | 0.01307 | 0.00840 | 0.01083 | 0.02063 | 0.03056 | 0.01594 | 0.01885 | 0.01196 | |
MSE | 0.00031 | 0.00011 | 0.00027 | 0.00011 | 0.00019 | 0.00065 | 0.00142 | 0.00040 | 0.00055 | 0.00023 | |
MRE | 0.02748 | 0.01396 | 0.02415 | 0.02111 | 0.02415 | 0.04969 | 0.05487 | 0.02800 | 0.01994 | 0.01866 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.24470) | (0.78294) | (-0.27803) | (-0.21706) | (-0.23032) | (-0.93457) | (1.43805) | (-0.42447) | (-0.96243) | (-0.20000) | ||
20 | \hat{E} | -0.22848 | 0.79765 | -0.25933 | -0.20235 | -0.21483 | -0.89872 | 1.37489 | -0.39453 | -0.93073 | -0.20048 |
BIAS | 0.06023 | 0.04755 | 0.06458 | 0.04755 | 0.05350 | 0.08863 | 0.16768 | 0.10001 | 0.12505 | 0.01564 | |
MSE | 0.00513 | 0.00318 | 0.00587 | 0.00318 | 0.00403 | 0.01248 | 0.04391 | 0.01414 | 0.02254 | 0.00041 | |
MRE | 0.24614 | 0.06073 | 0.23228 | 0.21907 | 0.23228 | 0.09483 | 0.11660 | 0.23562 | 0.12993 | 0.07820 | |
60 | \hat{E} | -0.22444 | 0.79973 | -0.25575 | -0.20027 | -0.21187 | -0.90532 | 1.38427 | -0.39031 | -0.92289 | -0.19737 |
BIAS | 0.03918 | 0.03122 | 0.04221 | 0.03122 | 0.03497 | 0.05784 | 0.10973 | 0.06527 | 0.08070 | 0.00931 | |
MSE | 0.00233 | 0.00150 | 0.00272 | 0.00150 | 0.00187 | 0.00543 | 0.01921 | 0.00653 | 0.00994 | 0.00013 | |
MRE | 0.16010 | 0.03987 | 0.15182 | 0.14382 | 0.15182 | 0.06189 | 0.07631 | 0.15377 | 0.08385 | 0.04657 | |
100 | \hat{E} | -0.22910 | 0.79586 | -0.26088 | -0.20414 | -0.21612 | -0.91192 | 1.39608 | -0.39810 | -0.93166 | -0.19806 |
BIAS | 0.03308 | 0.02624 | 0.03556 | 0.02624 | 0.02946 | 0.04549 | 0.08654 | 0.05470 | 0.06766 | 0.00776 | |
MSE | 0.00179 | 0.00113 | 0.00207 | 0.00113 | 0.00142 | 0.00351 | 0.01253 | 0.00486 | 0.00735 | 0.00010 | |
MRE | 0.13519 | 0.03351 | 0.12789 | 0.12087 | 0.12789 | 0.04868 | 0.06018 | 0.12887 | 0.07030 | 0.03881 | |
150 | \hat{E} | -0.23449 | 0.79145 | -0.26676 | -0.20855 | -0.22099 | -0.91935 | 1.40990 | -0.40710 | -0.94232 | -0.19880 |
BIAS | 0.02881 | 0.02270 | 0.03087 | 0.02270 | 0.02557 | 0.03868 | 0.07394 | 0.04755 | 0.05926 | 0.00679 | |
MSE | 0.00132 | 0.00082 | 0.00151 | 0.00082 | 0.00104 | 0.00245 | 0.00889 | 0.00358 | 0.00556 | 0.00007 | |
MRE | 0.11772 | 0.02900 | 0.11102 | 0.10459 | 0.11102 | 0.04138 | 0.05141 | 0.11203 | 0.06158 | 0.03394 | |
200 | \hat{E} | -0.23524 | 0.79074 | -0.26765 | -0.20926 | -0.22173 | -0.92085 | 1.41249 | -0.40845 | -0.94356 | -0.19891 |
BIAS | 0.02538 | 0.02001 | 0.02721 | 0.02001 | 0.02254 | 0.03402 | 0.06510 | 0.04188 | 0.05223 | 0.00608 | |
MSE | 0.00098 | 0.00061 | 0.00113 | 0.00061 | 0.00078 | 0.00184 | 0.00669 | 0.00268 | 0.00415 | 0.00006 | |
MRE | 0.10374 | 0.02556 | 0.09785 | 0.09220 | 0.09785 | 0.03640 | 0.04527 | 0.09866 | 0.05426 | 0.03038 | |
250 | \hat{E} | -0.23646 | 0.78972 | -0.26900 | -0.21028 | -0.22285 | -0.92295 | 1.41637 | -0.41058 | -0.94608 | -0.19896 |
BIAS | 0.02336 | 0.01841 | 0.02504 | 0.01841 | 0.02074 | 0.03051 | 0.05844 | 0.03856 | 0.04798 | 0.00550 | |
MSE | 0.00082 | 0.00051 | 0.00095 | 0.00051 | 0.00065 | 0.00149 | 0.00544 | 0.00224 | 0.00346 | 0.00005 | |
MRE | 0.09548 | 0.02352 | 0.09005 | 0.08483 | 0.09005 | 0.03265 | 0.04064 | 0.09083 | 0.04986 | 0.02752 | |
300 | \hat{E} | -0.23566 | 0.79028 | -0.26819 | -0.20972 | -0.22217 | -0.92243 | 1.41526 | -0.40936 | -0.94439 | -0.19879 |
BIAS | 0.02066 | 0.01630 | 0.02215 | 0.01630 | 0.01835 | 0.02713 | 0.05194 | 0.03399 | 0.04211 | 0.00489 | |
MSE | 0.00066 | 0.00041 | 0.00076 | 0.00041 | 0.00052 | 0.00124 | 0.00450 | 0.00181 | 0.00278 | 0.00004 | |
MRE | 0.08442 | 0.02082 | 0.07967 | 0.07511 | 0.07967 | 0.02903 | 0.03612 | 0.08008 | 0.04375 | 0.02447 | |
400 | \hat{E} | -0.23764 | 0.78865 | -0.27036 | -0.21135 | -0.22397 | -0.92502 | 1.42006 | -0.41266 | -0.94823 | -0.19911 |
BIAS | 0.01779 | 0.01402 | 0.01906 | 0.01402 | 0.01579 | 0.02303 | 0.04416 | 0.02926 | 0.03636 | 0.00412 | |
MSE | 0.00048 | 0.00030 | 0.00055 | 0.00030 | 0.00038 | 0.00085 | 0.00310 | 0.00129 | 0.00200 | 0.00003 | |
MRE | 0.07271 | 0.01790 | 0.06856 | 0.06457 | 0.06856 | 0.02464 | 0.03071 | 0.06893 | 0.03777 | 0.02061 |
ⅰ) Initialize our proposed model parameters and use them to generate random datasets.
ⅱ) Initialize \kappa for the entropy measures that depend on it, then determine all their initial values, say E_0 .
ⅲ) Use the MLEs to determine the estimated entropy value, say \hat{E} , by the substitution method.
ⅳ) Finally, determine the corresponding BIAS, MSE, and MRE. Then, repeat the previous steps thousands of times.
The results in Tables 8–11 show that all the estimated entropy measures tend to their initial values as the sample size increases and the other error-type measures decrease.
In order to have graphical benchmark, the values from Table 8 are also visualized in Figures 12–14.
In these figures, we can observe the fast decay of all curves with relatively small values for n . This confirms the efficiency of the estimation of the entropy measures considered in the context of the PUILD.
In this section, we illustrate the importance and adaptability of the underlying PUILD model for fitting real-world unit data across different disciplines. In fact, two real datasets are considered and described below.
The first dataset contains failure rates for twenty mechanical parts. It was studied by Murthy et al. [53]. The corresponding values are 0.067, 0.068, 0.076, 0.081, 0.084, 0.085, 0.085, 0.086, 0.089, 0.098, 0.098, 0.114, 0.114, 0.115, 0.121, 0.125, 0.131, 0.149, 0.160, and 0.485.
The second dataset was studied by Krishna et al. [54]. It is about the highest flood level (measured in millions of cubic feet per second) that occurred at Harrisburg, Pennsylvania, on the Susquehanna River during twenty years spanning from 1890 to 1969. The corresponding values are 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.324, 0.296, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, and 0.265.
Some graphical representations of these two datasets are shown in Figures 15 and 16, respectively. These include the histograms, kernel density estimates, violin plots, box plots, total time on test plots, and quantile-quantile (QQ) plots.
These figures show that the first dataset is mainly right-skewed with some outliers and has an increasing HRF and that the second dataset is almost symmetrical also with an increasing HRF. These characteristics can be handled by the PUILD model as developed in the theoretical results.
The models compared with the PUILD model are derived from the UIL distribution, the exponentiated Topp-Leone (ETL) distribution [55], the Kumaraswamy (Km) distribution, the beta (Be) distribution, and the transformed gamma (TrG) distribution [56]. Preliminary tests show that our proposed model performs very well when compared to the others, which are known for their ability to fit the real datasets considered, using the maximum likelihood method. All the relevant parameter estimates and standard errors (SEs) for the two real datasets are presented in Tables 12 and 13, respectively.
Model | \hat{\delta} | SE( \hat{\delta} ) | \hat{\beta} | SE( \hat{\beta} ) |
PUIL | 2.4144 | 0.4321 | 0.0068 | 0.0072 |
UIL | 0.2045 | 0.0326 | ||
ETL | 1.7370 | 0.2896 | 9.7115 | 3.8780 |
Km | 1.5878 | 0.2444 | 21.8673 | 10.2082 |
Be | 3.1127 | 0.9368 | 21.8246 | 7.0422 |
TrG | 14.6813 | 2.3213 |
Model | \hat{\delta} | SE( \hat{\delta} ) | \hat{\beta} | SE( \hat{\beta} ) |
PUIL | 2.9709 | 0.5340 | 0.1091 | 0.0645 |
UIL | 0.9867 | 0.1666 | ||
ETL | 4.6858 | 0.9595 | 4.1306 | 1.5083 |
Km | 3.4039 | 0.6073 | 12.0731 | 5.4978 |
Be | 6.9757 | 2.1638 | 9.3522 | 2.9276 |
TrG | 3.4438 | 0.54452 |
To support this claim, we use a variety of information criteria (ICs), including Akaike IC (Aic), corrected Akaike IC (Caic), Bayesian IC (Bic), and Hannan-Quinn IC (Hqic), to determine which model is most appropriate for fitting the two datasets. We also consider the goodness of fit metrics, including Anderson-Darling (A), Cramér-von Mises (W), and Kolmogorov-Smirnov (KS) with its p-value (KSp). The main novelty of this part is that we use new measures of uncertainty to compare the models, namely, ShE, DEX, and WEX. It is known that the model with less uncertainty information is the best.
All compared measures for the two datasets are presented in Tables 14 and 15, respectively.
Model | Aic | Caic | Bic | Hqic | A | W | KS | KSp | ShE | DEX | WEX |
PUIL | -71.5426 | -70.8367 | -69.5511 | -71.1538 | 0.4162 | 0.0500 | 0.1259 | 0.9092 | -1.9713 | -4.6321 | -0.4527 |
UIL | -57.9514 | -57.7292 | -56.9557 | -57.7570 | 2.6972 | 0.5309 | 0.3036 | 0.05012 | -0.9807 | -1.9026 | -0.1944 |
ETL | 48.2272 | -47.5213 | -46.2358 | -47.8385 | 2.6147 | 0.4524 | 0.2641 | 0.1229 | -1.2521 | -2.0067 | -0.2121 |
Km | -47.2969 | -46.5910 | -45.3054 | -46.9081 | 2.6889 | 0.4681 | 0.2627 | 0.1265 | -1.2290 | -1.9560 | -0.2031 |
Be | -51.7626 | -51.0567 | -49.7711 | -51.3738 | 2.2611 | 0.3727 | 0.2538 | 0.1521 | -1.3941 | -2.3467 | -0.2561 |
TrG | -51.8497 | -51.6275 | -50.8540 | -51.6553 | 2.5040 | 0.4327 | 0.2709 | 0.1062 | -1.2456 | -2.0362 | -0.2010 |
Model | Aic | Caic | Bic | Hqic | A | W | KS | KSp | ShE | DEX | WEX |
PUIL | -30.0341 | -29.3282 | -28.0427 | -29.6454 | 0.2522 | 0.0425 | 0.1226 | 0.9247 | -0.8583 | -1.4576 | -0.5632 |
UIL | -16.4854 | -16.2631 | -15.4896 | -16.2910 | 2.4384 | 0.4656 | 0.3009 | 0.0535 | -0.2021 | -0.6580 | -0.2946 |
ETL | -23.6156 | -22.9097 | -21.6241 | -23.2268 | 0.8845 | 0.1553 | 0.2110 | 0.3353 | -0.6532 | -1.0934 | -0.4659 |
Km | -22.0935 | -21.3876 | -20.1020 | -21.7047 | 1.0040 | 0.17602 | 0.2151 | 0.3132 | -0.6031 | -1.0336 | -0.4439 |
Be | -24.6329 | -23.9270 | -22.6414 | -24.2441 | 0.7991 | 0.1345 | 0.2038 | 0.3771 | -0.7158 | -1.1654 | -0.4923 |
TrG | -15.3907 | -15.1684 | -14.3949 | -15.1963 | 2.1343 | 0.3959 | 0.2905 | 0.0684 | -0.2401 | -0.6892 | -0.2587 |
Plots of the estimated CDFs and histograms with the estimated PDFs of all compared models for the two datasets are shown in Figures 17 and 18, respectively.
Clearly, the estimated curves fit the empirical objects very well.
The behavior of the L-LF with our proposed model estimates is also shown in Figures 19 and 20, respectively.
As expected, the uniqueness of these estimates is confirmed, as shown by the unique red point.
This article focuses on the PUILD, which is presented as a new valuable generalization of the UILD. An analysis showed that it has a PDF that can be unimodal, decreasing, increasing, or right-skewed. On the other hand, the HRF can be U-shaped, N-shaped, or increasing. The mode, quantiles, median, skewness, moments, variance, coefficient of variation, index of dispersion, harmonic mean, incomplete moments, inverse moments, and Lorenz and Bonferroni curves are among the many measures calculated in closed form. ShE, RE, ExE, HCE, ArE, TsE, AA1E, AA2E, Ex and WEX are the uncertainty measures computed. The incomplete gamma function was a key mathematical tool in this context. Methods such as maximum likelihood, Anderson-Darling, Cramér-von-Mises, least squares, right-tail Anderson-Darling, weighted least squares, left-tail Anderson-Darling, minimum spacing absolute distance, minimum spacing absolute-logarithmic distance, Anderson-Darling left-tail second order, Kolmogorov, minimum spacing square distance, minimum spacing square-logarithmic distance, and minimum spacing Linex distance were used. The invariance property of the MLEs was also used to estimate the different uncertainty measures. A simulation study validates these methods. The significance of the model associated with the PUILD compared to various current statistical models, including the UIL, exponentiated Topp-Leone, Kumaraswamy, and beta and transformed gamma models, is illustrated by two applications using real datasets.
Ahmed M. Gemeay, Najwan Alsadat, Christophe Chesneau, Mohammed Elgarhy: Writing – original draft, Formal analysis, Validation, Writing – review & editing. The authors contributed equally to this work. All the authors have read and approved the final version of the manuscript for publication.
The authors declare that they have not used Artificial Intelligence tools in the creation of this article.
This research is supported by researchers Supporting Project number (RSPD2024R548), King Saud University, Riyadh, Saudi Arabia.
The authors declare no conflict of interest.
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Name of the entropy | Reference | Expression |
Shannon | [44] | S(\beta, \delta)=-\displaystyle {\int}_{0}^{1}g(z; \beta, \delta) \log \left[ g(z; \beta, \delta)\right] dz |
Rényi | [45] | R_{\kappa}(\beta, \delta)=\frac{1}{1-\kappa} \log\left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right] |
Exponential | [46] | E_{\kappa}(\beta, \delta)=\left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right]^{\frac{1}{1-\kappa}} |
Havrda and Charvat | [47] | HC_{\kappa}(\beta, \delta)=\frac{1}{2^{1-\kappa}-1} \left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz-1\right] |
Arimoto | [48] | A_{\kappa}(\beta, \delta)=\frac{\kappa}{1-\kappa} \left\lbrace \left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right]^{\frac{1}{\kappa}}-1\right\rbrace |
Tsallis | [49] | T_{\kappa}(\beta, \delta)=\frac{1}{\kappa-1} \left[1-\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right] |
Awad and Alawneh 1 | [50] | AA1_{\kappa}(\beta, \delta)=\frac{1}{\kappa-1} \log \left\lbrace\left[\sup\limits_{z\in\mathbb{R}}g(z; \beta, \delta)\right]^{1-\kappa}\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right\rbrace |
Awad and Alawneh 2 | [50] | AA2_{\kappa}(\beta, \delta)=\frac{1}{2^{1-\kappa}-1} \left[\left\lbrace\left[\sup\limits_{z\in\mathbb{R}}g(z; \beta, \delta)\right]^{1-\kappa}\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right\rbrace -1\right] |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.22373 ^{\{ 7 \}} | 0.22577 ^{\{ 8 \}} | 0.25312 ^{\{ 14 \}} | 0.20754 ^{\{ 5 \}} | 0.25032 ^{\{ 13 \}} | 0.2571 ^{\{ 15 \}} | 0.2331 ^{\{ 11 \}} | 0.23345 ^{\{ 12 \}} | 0.13758 ^{\{ 2 \}} | 0.18677 ^{\{ 3 \}} | 0.22978 ^{\{ 10 \}} | 0.08802 ^{\{ 1 \}} | 0.22914 ^{\{ 9 \}} | 0.2176 ^{\{ 6 \}} | 0.20036 ^{\{ 4 \}} |
BIAS( \hat{\beta} ) | 0.7744 ^{\{ 7 \}} | 0.80526 ^{\{ 9 \}} | 0.75617 ^{\{ 6 \}} | 0.82699 ^{\{ 11 \}} | 0.77464 ^{\{ 8 \}} | 0.80846 ^{\{ 10 \}} | 0.84149 ^{\{ 13 \}} | 0.83473 ^{\{ 12 \}} | 0.29217 ^{\{ 2 \}} | 0.7215 ^{\{ 4 \}} | 0.85207 ^{\{ 15 \}} | 0.04434 ^{\{ 1 \}} | 0.72321 ^{\{ 5 \}} | 0.84278 ^{\{ 14 \}} | 0.64726 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.07932 ^{\{ 8 \}} | 0.07826 ^{\{ 7 \}} | 0.09935 ^{\{ 14 \}} | 0.06438 ^{\{ 4 \}} | 0.09656 ^{\{ 13 \}} | 0.09952 ^{\{ 15 \}} | 0.08196 ^{\{ 9 \}} | 0.08382 ^{\{ 11 \}} | 0.03691 ^{\{ 2 \}} | 0.05693 ^{\{ 3 \}} | 0.08258 ^{\{ 10 \}} | 0.01303 ^{\{ 1 \}} | 0.08619 ^{\{ 12 \}} | 0.07251 ^{\{ 6 \}} | 0.06923 ^{\{ 5 \}} | |
MSE( \hat{\beta} ) | 0.85507 ^{\{ 7 \}} | 0.93048 ^{\{ 10 \}} | 0.79375 ^{\{ 5 \}} | 1.00678 ^{\{ 13 \}} | 0.83857 ^{\{ 6 \}} | 0.87698 ^{\{ 8 \}} | 0.99586 ^{\{ 12 \}} | 0.98746 ^{\{ 11 \}} | 0.29807 ^{\{ 2 \}} | 0.88098 ^{\{ 9 \}} | 1.04622 ^{\{ 15 \}} | 0.00824 ^{\{ 1 \}} | 0.74237 ^{\{ 4 \}} | 1.02031 ^{\{ 14 \}} | 0.62886 ^{\{ 3 \}} | |
MRE( \hat{\delta} ) | 0.31961 ^{\{ 7 \}} | 0.32252 ^{\{ 8 \}} | 0.36159 ^{\{ 14 \}} | 0.29648 ^{\{ 5 \}} | 0.3576 ^{\{ 13 \}} | 0.36728 ^{\{ 15 \}} | 0.333 ^{\{ 11 \}} | 0.3335 ^{\{ 12 \}} | 0.19654 ^{\{ 2 \}} | 0.26682 ^{\{ 3 \}} | 0.32825 ^{\{ 10 \}} | 0.12574 ^{\{ 1 \}} | 0.32734 ^{\{ 9 \}} | 0.31086 ^{\{ 6 \}} | 0.28622 ^{\{ 4 \}} | |
MRE( \hat{\beta} ) | 0.30976 ^{\{ 7 \}} | 0.3221 ^{\{ 9 \}} | 0.30247 ^{\{ 6 \}} | 0.33079 ^{\{ 11 \}} | 0.30985 ^{\{ 8 \}} | 0.32339 ^{\{ 10 \}} | 0.33659 ^{\{ 13 \}} | 0.33389 ^{\{ 12 \}} | 0.11687 ^{\{ 2 \}} | 0.2886 ^{\{ 4 \}} | 0.34083 ^{\{ 15 \}} | 0.01774 ^{\{ 1 \}} | 0.28928 ^{\{ 5 \}} | 0.33711 ^{\{ 14 \}} | 0.2589 ^{\{ 3 \}} | |
D_{abs} | 0.04005 ^{\{ 1 \}} | 0.04126 ^{\{ 4 \}} | 0.04356 ^{\{ 10 \}} | 0.04111 ^{\{ 2 \}} | 0.04425 ^{\{ 11 \}} | 0.04479 ^{\{ 12 \}} | 0.04172 ^{\{ 6 \}} | 0.0414 ^{\{ 5 \}} | 0.04519 ^{\{ 13 \}} | 0.04312 ^{\{ 9 \}} | 0.04124 ^{\{ 3 \}} | 0.04269 ^{\{ 8 \}} | 0.06087 ^{\{ 15 \}} | 0.04232 ^{\{ 7 \}} | 0.05767 ^{\{ 14 \}} | |
D_{max} | 0.06512 ^{\{ 3 \}} | 0.06645 ^{\{ 4 \}} | 0.07113 ^{\{ 12 \}} | 0.06499 ^{\{ 2 \}} | 0.0708 ^{\{ 11 \}} | 0.07279 ^{\{ 13 \}} | 0.06712 ^{\{ 8 \}} | 0.06667 ^{\{ 5 \}} | 0.06964 ^{\{ 10 \}} | 0.06785 ^{\{ 9 \}} | 0.06693 ^{\{ 6 \}} | 0.06424 ^{\{ 1 \}} | 0.09209 ^{\{ 15 \}} | 0.06696 ^{\{ 7 \}} | 0.08699 ^{\{ 14 \}} | |
\sum Ranks | 47 ^{\{ 4 \}} | 59 ^{\{ 7 \}} | 81 ^{\{ 11 \}} | 53 ^{\{ 6 \}} | 83 ^{\{ 12.5 \}} | 98 ^{\{ 15 \}} | 83 ^{\{ 12.5 \}} | 80 ^{\{ 10 \}} | 35 ^{\{ 2 \}} | 44 ^{\{ 3 \}} | 84 ^{\{ 14 \}} | 15 ^{\{ 1 \}} | 74 ^{\{ 8.5 \}} | 74 ^{\{ 8.5 \}} | 50 ^{\{ 5 \}} | |
60 | BIAS( \hat{\delta} ) | 0.17259 ^{\{ 5 \}} | 0.19678 ^{\{ 11 \}} | 0.22989 ^{\{ 15 \}} | 0.17065 ^{\{ 4 \}} | 0.20041 ^{\{ 13 \}} | 0.22807 ^{\{ 14 \}} | 0.19883 ^{\{ 12 \}} | 0.19121 ^{\{ 9 \}} | 0.1085 ^{\{ 2 \}} | 0.16308 ^{\{ 3 \}} | 0.19184 ^{\{ 10 \}} | 0.05999 ^{\{ 1 \}} | 0.18004 ^{\{ 7 \}} | 0.18312 ^{\{ 8 \}} | 0.17361 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.68127 ^{\{ 5 \}} | 0.78854 ^{\{ 14 \}} | 0.7418 ^{\{ 8 \}} | 0.74548 ^{\{ 9 \}} | 0.71544 ^{\{ 7 \}} | 0.77753 ^{\{ 13 \}} | 0.7739 ^{\{ 12 \}} | 0.74844 ^{\{ 10 \}} | 0.27546 ^{\{ 2 \}} | 0.69382 ^{\{ 6 \}} | 0.75274 ^{\{ 11 \}} | 0.03594 ^{\{ 1 \}} | 0.58942 ^{\{ 3 \}} | 0.80753 ^{\{ 15 \}} | 0.59255 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.04735 ^{\{ 5 \}} | 0.05737 ^{\{ 9 \}} | 0.0815 ^{\{ 15 \}} | 0.04443 ^{\{ 4 \}} | 0.06477 ^{\{ 13 \}} | 0.08028 ^{\{ 14 \}} | 0.06309 ^{\{ 12 \}} | 0.05805 ^{\{ 10 \}} | 0.02402 ^{\{ 2 \}} | 0.04275 ^{\{ 3 \}} | 0.05944 ^{\{ 11 \}} | 0.00602 ^{\{ 1 \}} | 0.05663 ^{\{ 8 \}} | 0.05151 ^{\{ 6 \}} | 0.05354 ^{\{ 7 \}} | |
MSE( \hat{\beta} ) | 0.71243 ^{\{ 5 \}} | 0.92878 ^{\{ 14 \}} | 0.76945 ^{\{ 7 \}} | 0.87146 ^{\{ 12 \}} | 0.74299 ^{\{ 6 \}} | 0.85999 ^{\{ 10 \}} | 0.91218 ^{\{ 13 \}} | 0.84322 ^{\{ 9 \}} | 0.26265 ^{\{ 2 \}} | 0.81005 ^{\{ 8 \}} | 0.86264 ^{\{ 11 \}} | 0.00554 ^{\{ 1 \}} | 0.50974 ^{\{ 3 \}} | 0.99557 ^{\{ 15 \}} | 0.5388 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.24656 ^{\{ 5 \}} | 0.28111 ^{\{ 11 \}} | 0.32842 ^{\{ 15 \}} | 0.24378 ^{\{ 4 \}} | 0.2863 ^{\{ 13 \}} | 0.32581 ^{\{ 14 \}} | 0.28404 ^{\{ 12 \}} | 0.27316 ^{\{ 9 \}} | 0.155 ^{\{ 2 \}} | 0.23297 ^{\{ 3 \}} | 0.27405 ^{\{ 10 \}} | 0.08571 ^{\{ 1 \}} | 0.25719 ^{\{ 7 \}} | 0.26159 ^{\{ 8 \}} | 0.24802 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.27251 ^{\{ 5 \}} | 0.31542 ^{\{ 14 \}} | 0.29672 ^{\{ 8 \}} | 0.29819 ^{\{ 9 \}} | 0.28618 ^{\{ 7 \}} | 0.31101 ^{\{ 13 \}} | 0.30956 ^{\{ 12 \}} | 0.29938 ^{\{ 10 \}} | 0.11018 ^{\{ 2 \}} | 0.27753 ^{\{ 6 \}} | 0.3011 ^{\{ 11 \}} | 0.01437 ^{\{ 1 \}} | 0.23577 ^{\{ 3 \}} | 0.32301 ^{\{ 15 \}} | 0.23702 ^{\{ 4 \}} | |
D_{abs} | 0.0295 ^{\{ 2 \}} | 0.03057 ^{\{ 5 \}} | 0.03189 ^{\{ 13 \}} | 0.02811 ^{\{ 1 \}} | 0.03141 ^{\{ 11 \}} | 0.03108 ^{\{ 9 \}} | 0.03081 ^{\{ 6 \}} | 0.03091 ^{\{ 8 \}} | 0.03083 ^{\{ 7 \}} | 0.0314 ^{\{ 10 \}} | 0.02956 ^{\{ 3 \}} | 0.02965 ^{\{ 4 \}} | 0.04047 ^{\{ 14 \}} | 0.03166 ^{\{ 12 \}} | 0.04048 ^{\{ 15 \}} | |
D_{max} | 0.04787 ^{\{ 3 \}} | 0.04999 ^{\{ 6 \}} | 0.05351 ^{\{ 13 \}} | 0.04551 ^{\{ 2 \}} | 0.05135 ^{\{ 11 \}} | 0.05238 ^{\{ 12 \}} | 0.05037 ^{\{ 7 \}} | 0.05048 ^{\{ 9 \}} | 0.04829 ^{\{ 4 \}} | 0.05038 ^{\{ 8 \}} | 0.04885 ^{\{ 5 \}} | 0.04457 ^{\{ 1 \}} | 0.06312 ^{\{ 14 \}} | 0.05093 ^{\{ 10 \}} | 0.06349 ^{\{ 15 \}} | |
\sum Ranks | 35 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 94 ^{\{ 14 \}} | 45 ^{\{ 4 \}} | 81 ^{\{ 10 \}} | 99 ^{\{ 15 \}} | 86 ^{\{ 12 \}} | 74 ^{\{ 9 \}} | 23 ^{\{ 2 \}} | 47 ^{\{ 5 \}} | 72 ^{\{ 8 \}} | 11 ^{\{ 1 \}} | 59 ^{\{ 6 \}} | 89 ^{\{ 13 \}} | 61 ^{\{ 7 \}} | |
100 | BIAS( \hat{\delta} ) | 0.14625 ^{\{ 7 \}} | 0.16744 ^{\{ 12 \}} | 0.19457 ^{\{ 14 \}} | 0.14162 ^{\{ 5 \}} | 0.185 ^{\{ 13 \}} | 0.21524 ^{\{ 15 \}} | 0.16618 ^{\{ 11 \}} | 0.15543 ^{\{ 9 \}} | 0.0977 ^{\{ 2 \}} | 0.13859 ^{\{ 4 \}} | 0.15888 ^{\{ 10 \}} | 0.0488 ^{\{ 1 \}} | 0.13599 ^{\{ 3 \}} | 0.15167 ^{\{ 8 \}} | 0.14603 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.59238 ^{\{ 5 \}} | 0.68527 ^{\{ 13 \}} | 0.68361 ^{\{ 12 \}} | 0.67297 ^{\{ 9 \}} | 0.67575 ^{\{ 10 \}} | 0.75015 ^{\{ 15 \}} | 0.66326 ^{\{ 8 \}} | 0.62369 ^{\{ 6 \}} | 0.27316 ^{\{ 2 \}} | 0.64513 ^{\{ 7 \}} | 0.70079 ^{\{ 14 \}} | 0.03289 ^{\{ 1 \}} | 0.48933 ^{\{ 3 \}} | 0.67978 ^{\{ 11 \}} | 0.52629 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.03434 ^{\{ 6 \}} | 0.04318 ^{\{ 12 \}} | 0.0602 ^{\{ 14 \}} | 0.02976 ^{\{ 3 \}} | 0.05233 ^{\{ 13 \}} | 0.07238 ^{\{ 15 \}} | 0.04301 ^{\{ 11 \}} | 0.0378 ^{\{ 8 \}} | 0.02044 ^{\{ 2 \}} | 0.03082 ^{\{ 4 \}} | 0.03962 ^{\{ 10 \}} | 0.0038 ^{\{ 1 \}} | 0.03378 ^{\{ 5 \}} | 0.03602 ^{\{ 7 \}} | 0.03945 ^{\{ 9 \}} | |
MSE( \hat{\beta} ) | 0.56392 ^{\{ 5 \}} | 0.7543 ^{\{ 13 \}} | 0.69846 ^{\{ 9 \}} | 0.74207 ^{\{ 12 \}} | 0.66928 ^{\{ 7 \}} | 0.84187 ^{\{ 15 \}} | 0.68905 ^{\{ 8 \}} | 0.62999 ^{\{ 6 \}} | 0.25309 ^{\{ 2 \}} | 0.73522 ^{\{ 10 \}} | 0.78489 ^{\{ 14 \}} | 0.00524 ^{\{ 1 \}} | 0.37739 ^{\{ 3 \}} | 0.74173 ^{\{ 11 \}} | 0.48398 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.20892 ^{\{ 7 \}} | 0.2392 ^{\{ 12 \}} | 0.27796 ^{\{ 14 \}} | 0.20231 ^{\{ 5 \}} | 0.26429 ^{\{ 13 \}} | 0.30748 ^{\{ 15 \}} | 0.2374 ^{\{ 11 \}} | 0.22204 ^{\{ 9 \}} | 0.13957 ^{\{ 2 \}} | 0.19798 ^{\{ 4 \}} | 0.22697 ^{\{ 10 \}} | 0.06972 ^{\{ 1 \}} | 0.19427 ^{\{ 3 \}} | 0.21668 ^{\{ 8 \}} | 0.20861 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.23695 ^{\{ 5 \}} | 0.27411 ^{\{ 13 \}} | 0.27344 ^{\{ 12 \}} | 0.26919 ^{\{ 9 \}} | 0.2703 ^{\{ 10 \}} | 0.30006 ^{\{ 15 \}} | 0.26531 ^{\{ 8 \}} | 0.24948 ^{\{ 6 \}} | 0.10926 ^{\{ 2 \}} | 0.25805 ^{\{ 7 \}} | 0.28032 ^{\{ 14 \}} | 0.01315 ^{\{ 1 \}} | 0.19573 ^{\{ 3 \}} | 0.27191 ^{\{ 11 \}} | 0.21052 ^{\{ 4 \}} | |
D_{abs} | 0.02341 ^{\{ 2 \}} | 0.02377 ^{\{ 4 \}} | 0.02389 ^{\{ 5 \}} | 0.02292 ^{\{ 1 \}} | 0.02502 ^{\{ 8 \}} | 0.02535 ^{\{ 10 \}} | 0.02347 ^{\{ 3 \}} | 0.024 ^{\{ 6 \}} | 0.02561 ^{\{ 12 \}} | 0.02526 ^{\{ 9 \}} | 0.02537 ^{\{ 11 \}} | 0.02447 ^{\{ 7 \}} | 0.03101 ^{\{ 15 \}} | 0.02653 ^{\{ 13 \}} | 0.03041 ^{\{ 14 \}} | |
D_{max} | 0.03833 ^{\{ 3 \}} | 0.03924 ^{\{ 5 \}} | 0.04063 ^{\{ 9 \}} | 0.03709 ^{\{ 2 \}} | 0.0422 ^{\{ 11 \}} | 0.04366 ^{\{ 13 \}} | 0.03885 ^{\{ 4 \}} | 0.03946 ^{\{ 6 \}} | 0.04025 ^{\{ 7 \}} | 0.04062 ^{\{ 8 \}} | 0.04131 ^{\{ 10 \}} | 0.03704 ^{\{ 1 \}} | 0.0492 ^{\{ 15 \}} | 0.04265 ^{\{ 12 \}} | 0.04866 ^{\{ 14 \}} | |
\sum Ranks | 40 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 89 ^{\{ 13 \}} | 46 ^{\{ 4 \}} | 85 ^{\{ 12 \}} | 113 ^{\{ 15 \}} | 64 ^{\{ 9 \}} | 56 ^{\{ 7 \}} | 31 ^{\{ 2 \}} | 53 ^{\{ 6 \}} | 93 ^{\{ 14 \}} | 14 ^{\{ 1 \}} | 50 ^{\{ 5 \}} | 81 ^{\{ 10 \}} | 61 ^{\{ 8 \}} | |
200 | BIAS( \hat{\delta} ) | 0.09648 ^{\{ 3 \}} | 0.12482 ^{\{ 10 \}} | 0.15103 ^{\{ 14 \}} | 0.11384 ^{\{ 6 \}} | 0.1425 ^{\{ 13 \}} | 0.17269 ^{\{ 15 \}} | 0.13156 ^{\{ 12 \}} | 0.11484 ^{\{ 7 \}} | 0.07786 ^{\{ 2 \}} | 0.11322 ^{\{ 5 \}} | 0.12492 ^{\{ 11 \}} | 0.03359 ^{\{ 1 \}} | 0.11157 ^{\{ 4 \}} | 0.11946 ^{\{ 8 \}} | 0.12256 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.42875 ^{\{ 3 \}} | 0.5295 ^{\{ 7 \}} | 0.61287 ^{\{ 14 \}} | 0.54979 ^{\{ 8 \}} | 0.59034 ^{\{ 13 \}} | 0.63661 ^{\{ 15 \}} | 0.55697 ^{\{ 10 \}} | 0.50632 ^{\{ 6 \}} | 0.25968 ^{\{ 2 \}} | 0.55546 ^{\{ 9 \}} | 0.58785 ^{\{ 12 \}} | 0.02817 ^{\{ 1 \}} | 0.437 ^{\{ 4 \}} | 0.5818 ^{\{ 11 \}} | 0.45147 ^{\{ 5 \}} | |
MSE( \hat{\delta} ) | 0.01468 ^{\{ 3 \}} | 0.02419 ^{\{ 10 \}} | 0.03546 ^{\{ 14 \}} | 0.01995 ^{\{ 5 \}} | 0.03181 ^{\{ 13 \}} | 0.04731 ^{\{ 15 \}} | 0.02608 ^{\{ 11 \}} | 0.02091 ^{\{ 6 \}} | 0.01268 ^{\{ 2 \}} | 0.01992 ^{\{ 4 \}} | 0.02357 ^{\{ 9 \}} | 0.00198 ^{\{ 1 \}} | 0.023 ^{\{ 8 \}} | 0.02177 ^{\{ 7 \}} | 0.02768 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.31499 ^{\{ 3 \}} | 0.46709 ^{\{ 7 \}} | 0.61924 ^{\{ 14 \}} | 0.53484 ^{\{ 9 \}} | 0.58357 ^{\{ 12 \}} | 0.65048 ^{\{ 15 \}} | 0.5114 ^{\{ 8 \}} | 0.44203 ^{\{ 6 \}} | 0.2079 ^{\{ 2 \}} | 0.55213 ^{\{ 10 \}} | 0.56088 ^{\{ 11 \}} | 0.00473 ^{\{ 1 \}} | 0.34757 ^{\{ 4 \}} | 0.58871 ^{\{ 13 \}} | 0.37455 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.13782 ^{\{ 3 \}} | 0.17831 ^{\{ 10 \}} | 0.21575 ^{\{ 14 \}} | 0.16263 ^{\{ 6 \}} | 0.20357 ^{\{ 13 \}} | 0.2467 ^{\{ 15 \}} | 0.18794 ^{\{ 12 \}} | 0.16405 ^{\{ 7 \}} | 0.11122 ^{\{ 2 \}} | 0.16174 ^{\{ 5 \}} | 0.17846 ^{\{ 11 \}} | 0.04798 ^{\{ 1 \}} | 0.15939 ^{\{ 4 \}} | 0.17065 ^{\{ 8 \}} | 0.17509 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.1715 ^{\{ 3 \}} | 0.2118 ^{\{ 7 \}} | 0.24515 ^{\{ 14 \}} | 0.21992 ^{\{ 8 \}} | 0.23613 ^{\{ 13 \}} | 0.25465 ^{\{ 15 \}} | 0.22279 ^{\{ 10 \}} | 0.20253 ^{\{ 6 \}} | 0.10387 ^{\{ 2 \}} | 0.22218 ^{\{ 9 \}} | 0.23514 ^{\{ 12 \}} | 0.01127 ^{\{ 1 \}} | 0.1748 ^{\{ 4 \}} | 0.23272 ^{\{ 11 \}} | 0.18059 ^{\{ 5 \}} | |
D_{abs} | 0.01667 ^{\{ 2 \}} | 0.0172 ^{\{ 5 \}} | 0.01824 ^{\{ 9 \}} | 0.01714 ^{\{ 4 \}} | 0.01785 ^{\{ 8 \}} | 0.01886 ^{\{ 11 \}} | 0.01762 ^{\{ 6 \}} | 0.01696 ^{\{ 3 \}} | 0.01783 ^{\{ 7 \}} | 0.02031 ^{\{ 13 \}} | 0.01851 ^{\{ 10 \}} | 0.01629 ^{\{ 1 \}} | 0.02275 ^{\{ 15 \}} | 0.01899 ^{\{ 12 \}} | 0.02256 ^{\{ 14 \}} | |
D_{max} | 0.0269 ^{\{ 2 \}} | 0.02854 ^{\{ 6 \}} | 0.03108 ^{\{ 11 \}} | 0.02791 ^{\{ 4 \}} | 0.03005 ^{\{ 8 \}} | 0.03272 ^{\{ 13 \}} | 0.0292 ^{\{ 7 \}} | 0.02788 ^{\{ 3 \}} | 0.02821 ^{\{ 5 \}} | 0.03259 ^{\{ 12 \}} | 0.03021 ^{\{ 9 \}} | 0.02461 ^{\{ 1 \}} | 0.03658 ^{\{ 15 \}} | 0.03087 ^{\{ 10 \}} | 0.03649 ^{\{ 14 \}} | |
\sum Ranks | 22 ^{\{ 2 \}} | 62 ^{\{ 7 \}} | 104 ^{\{ 14 \}} | 50 ^{\{ 5 \}} | 93 ^{\{ 13 \}} | 114 ^{\{ 15 \}} | 76 ^{\{ 10 \}} | 44 ^{\{ 4 \}} | 24 ^{\{ 3 \}} | 67 ^{\{ 8 \}} | 85 ^{\{ 12 \}} | 8 ^{\{ 1 \}} | 58 ^{\{ 6 \}} | 80 ^{\{ 11 \}} | 73 ^{\{ 9 \}} | |
300 | BIAS( \hat{\delta} ) | 0.08469 ^{\{ 3 \}} | 0.10079 ^{\{ 10 \}} | 0.1252 ^{\{ 14 \}} | 0.09076 ^{\{ 4 \}} | 0.12392 ^{\{ 13 \}} | 0.14567 ^{\{ 15 \}} | 0.10608 ^{\{ 12 \}} | 0.09495 ^{\{ 6 \}} | 0.06793 ^{\{ 2 \}} | 0.09395 ^{\{ 5 \}} | 0.10567 ^{\{ 11 \}} | 0.02746 ^{\{ 1 \}} | 0.09748 ^{\{ 7 \}} | 0.10026 ^{\{ 8 \}} | 0.10057 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.36972 ^{\{ 3 \}} | 0.43786 ^{\{ 8 \}} | 0.50879 ^{\{ 13 \}} | 0.43499 ^{\{ 7 \}} | 0.53725 ^{\{ 14 \}} | 0.57736 ^{\{ 15 \}} | 0.45602 ^{\{ 10 \}} | 0.42746 ^{\{ 6 \}} | 0.24286 ^{\{ 2 \}} | 0.44069 ^{\{ 9 \}} | 0.5055 ^{\{ 12 \}} | 0.0244 ^{\{ 1 \}} | 0.38919 ^{\{ 5 \}} | 0.46693 ^{\{ 11 \}} | 0.3866 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.01148 ^{\{ 3 \}} | 0.01591 ^{\{ 8 \}} | 0.02484 ^{\{ 14 \}} | 0.01253 ^{\{ 4 \}} | 0.02397 ^{\{ 13 \}} | 0.03315 ^{\{ 15 \}} | 0.01778 ^{\{ 10 \}} | 0.01434 ^{\{ 6 \}} | 0.00976 ^{\{ 2 \}} | 0.01432 ^{\{ 5 \}} | 0.01768 ^{\{ 9 \}} | 0.00126 ^{\{ 1 \}} | 0.01805 ^{\{ 11 \}} | 0.01511 ^{\{ 7 \}} | 0.01817 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.24338 ^{\{ 3 \}} | 0.33897 ^{\{ 7 \}} | 0.44168 ^{\{ 12 \}} | 0.32176 ^{\{ 6 \}} | 0.51632 ^{\{ 14 \}} | 0.54731 ^{\{ 15 \}} | 0.35494 ^{\{ 9 \}} | 0.34003 ^{\{ 8 \}} | 0.17342 ^{\{ 2 \}} | 0.36794 ^{\{ 10 \}} | 0.44555 ^{\{ 13 \}} | 0.00288 ^{\{ 1 \}} | 0.29013 ^{\{ 5 \}} | 0.37214 ^{\{ 11 \}} | 0.26368 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.12099 ^{\{ 3 \}} | 0.14399 ^{\{ 10 \}} | 0.17886 ^{\{ 14 \}} | 0.12966 ^{\{ 4 \}} | 0.17703 ^{\{ 13 \}} | 0.2081 ^{\{ 15 \}} | 0.15154 ^{\{ 12 \}} | 0.13565 ^{\{ 6 \}} | 0.09705 ^{\{ 2 \}} | 0.13422 ^{\{ 5 \}} | 0.15096 ^{\{ 11 \}} | 0.03922 ^{\{ 1 \}} | 0.13925 ^{\{ 7 \}} | 0.14323 ^{\{ 8 \}} | 0.14368 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.14789 ^{\{ 3 \}} | 0.17514 ^{\{ 8 \}} | 0.20352 ^{\{ 13 \}} | 0.174 ^{\{ 7 \}} | 0.2149 ^{\{ 14 \}} | 0.23094 ^{\{ 15 \}} | 0.18241 ^{\{ 10 \}} | 0.17099 ^{\{ 6 \}} | 0.09715 ^{\{ 2 \}} | 0.17628 ^{\{ 9 \}} | 0.2022 ^{\{ 12 \}} | 0.00976 ^{\{ 1 \}} | 0.15568 ^{\{ 5 \}} | 0.18677 ^{\{ 11 \}} | 0.15464 ^{\{ 4 \}} | |
D_{abs} | 0.01342 ^{\{ 1 \}} | 0.0141 ^{\{ 5 \}} | 0.01471 ^{\{ 7 \}} | 0.01402 ^{\{ 4 \}} | 0.01492 ^{\{ 8 \}} | 0.01569 ^{\{ 12 \}} | 0.015 ^{\{ 9 \}} | 0.01378 ^{\{ 3 \}} | 0.01434 ^{\{ 6 \}} | 0.01559 ^{\{ 11 \}} | 0.01595 ^{\{ 13 \}} | 0.01376 ^{\{ 2 \}} | 0.01845 ^{\{ 15 \}} | 0.01547 ^{\{ 10 \}} | 0.0179 ^{\{ 14 \}} | |
D_{max} | 0.02182 ^{\{ 2 \}} | 0.02339 ^{\{ 6 \}} | 0.025 ^{\{ 8 \}} | 0.0228 ^{\{ 5 \}} | 0.02526 ^{\{ 10 \}} | 0.02734 ^{\{ 13 \}} | 0.02465 ^{\{ 7 \}} | 0.02256 ^{\{ 3 \}} | 0.02277 ^{\{ 4 \}} | 0.02521 ^{\{ 9 \}} | 0.02581 ^{\{ 12 \}} | 0.02079 ^{\{ 1 \}} | 0.03004 ^{\{ 15 \}} | 0.02528 ^{\{ 11 \}} | 0.029 ^{\{ 14 \}} | |
\sum Ranks | 21 ^{\{ 2 \}} | 62 ^{\{ 6 \}} | 95 ^{\{ 13 \}} | 41 ^{\{ 4 \}} | 99 ^{\{ 14 \}} | 115 ^{\{ 15 \}} | 79 ^{\{ 11 \}} | 44 ^{\{ 5 \}} | 22 ^{\{ 3 \}} | 63 ^{\{ 7 \}} | 93 ^{\{ 12 \}} | 9 ^{\{ 1 \}} | 70 ^{\{ 8.5 \}} | 77 ^{\{ 10 \}} | 70 ^{\{ 8.5 \}} | |
400 | BIAS( \hat{\delta} ) | 0.07174 ^{\{ 3 \}} | 0.09293 ^{\{ 11 \}} | 0.10943 ^{\{ 13 \}} | 0.07801 ^{\{ 4 \}} | 0.11145 ^{\{ 14 \}} | 0.13305 ^{\{ 15 \}} | 0.09175 ^{\{ 10 \}} | 0.08463 ^{\{ 6 \}} | 0.0625 ^{\{ 2 \}} | 0.08834 ^{\{ 8 \}} | 0.09467 ^{\{ 12 \}} | 0.02457 ^{\{ 1 \}} | 0.08264 ^{\{ 5 \}} | 0.08637 ^{\{ 7 \}} | 0.09147 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.31908 ^{\{ 3 \}} | 0.40392 ^{\{ 9 \}} | 0.47059 ^{\{ 14 \}} | 0.35775 ^{\{ 5 \}} | 0.46071 ^{\{ 13 \}} | 0.53833 ^{\{ 15 \}} | 0.40286 ^{\{ 8 \}} | 0.38381 ^{\{ 7 \}} | 0.2324 ^{\{ 2 \}} | 0.4171 ^{\{ 11 \}} | 0.45466 ^{\{ 12 \}} | 0.02287 ^{\{ 1 \}} | 0.33301 ^{\{ 4 \}} | 0.40844 ^{\{ 10 \}} | 0.38044 ^{\{ 6 \}} | |
MSE( \hat{\delta} ) | 0.00819 ^{\{ 3 \}} | 0.01379 ^{\{ 10 \}} | 0.01888 ^{\{ 13 \}} | 0.00948 ^{\{ 4 \}} | 0.01913 ^{\{ 14 \}} | 0.02688 ^{\{ 15 \}} | 0.01303 ^{\{ 9 \}} | 0.01121 ^{\{ 5 \}} | 0.00806 ^{\{ 2 \}} | 0.01219 ^{\{ 7 \}} | 0.01427 ^{\{ 12 \}} | 0.00096 ^{\{ 1 \}} | 0.01264 ^{\{ 8 \}} | 0.01176 ^{\{ 6 \}} | 0.01408 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.17003 ^{\{ 3 \}} | 0.27561 ^{\{ 8 \}} | 0.39787 ^{\{ 14 \}} | 0.22918 ^{\{ 5 \}} | 0.3543 ^{\{ 12 \}} | 0.49644 ^{\{ 15 \}} | 0.28197 ^{\{ 9 \}} | 0.25617 ^{\{ 7 \}} | 0.15577 ^{\{ 2 \}} | 0.30793 ^{\{ 11 \}} | 0.38113 ^{\{ 13 \}} | 0.00249 ^{\{ 1 \}} | 0.21587 ^{\{ 4 \}} | 0.30226 ^{\{ 10 \}} | 0.24672 ^{\{ 6 \}} | |
MRE( \hat{\delta} ) | 0.10249 ^{\{ 3 \}} | 0.13276 ^{\{ 11 \}} | 0.15633 ^{\{ 13 \}} | 0.11145 ^{\{ 4 \}} | 0.15921 ^{\{ 14 \}} | 0.19008 ^{\{ 15 \}} | 0.13108 ^{\{ 10 \}} | 0.1209 ^{\{ 6 \}} | 0.08929 ^{\{ 2 \}} | 0.1262 ^{\{ 8 \}} | 0.13524 ^{\{ 12 \}} | 0.0351 ^{\{ 1 \}} | 0.11806 ^{\{ 5 \}} | 0.12339 ^{\{ 7 \}} | 0.13067 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.12763 ^{\{ 3 \}} | 0.16157 ^{\{ 9 \}} | 0.18824 ^{\{ 14 \}} | 0.1431 ^{\{ 5 \}} | 0.18428 ^{\{ 13 \}} | 0.21533 ^{\{ 15 \}} | 0.16114 ^{\{ 8 \}} | 0.15352 ^{\{ 7 \}} | 0.09296 ^{\{ 2 \}} | 0.16684 ^{\{ 11 \}} | 0.18186 ^{\{ 12 \}} | 0.00915 ^{\{ 1 \}} | 0.1332 ^{\{ 4 \}} | 0.16337 ^{\{ 10 \}} | 0.15218 ^{\{ 6 \}} | |
D_{abs} | 0.01202 ^{\{ 1 \}} | 0.01289 ^{\{ 5 \}} | 0.01357 ^{\{ 9 \}} | 0.01216 ^{\{ 3 \}} | 0.01333 ^{\{ 8 \}} | 0.01381 ^{\{ 11 \}} | 0.01248 ^{\{ 4 \}} | 0.01302 ^{\{ 7 \}} | 0.01293 ^{\{ 6 \}} | 0.01404 ^{\{ 12 \}} | 0.01407 ^{\{ 13 \}} | 0.01203 ^{\{ 2 \}} | 0.01585 ^{\{ 15 \}} | 0.01372 ^{\{ 10 \}} | 0.01554 ^{\{ 14 \}} | |
D_{max} | 0.01947 ^{\{ 2 \}} | 0.02129 ^{\{ 7 \}} | 0.02291 ^{\{ 12 \}} | 0.01985 ^{\{ 3 \}} | 0.0227 ^{\{ 9 \}} | 0.02414 ^{\{ 13 \}} | 0.02063 ^{\{ 5 \}} | 0.02113 ^{\{ 6 \}} | 0.02061 ^{\{ 4 \}} | 0.02288 ^{\{ 10 \}} | 0.02289 ^{\{ 11 \}} | 0.01819 ^{\{ 1 \}} | 0.02568 ^{\{ 15 \}} | 0.02227 ^{\{ 8 \}} | 0.02537 ^{\{ 14 \}} | |
\sum Ranks | 21 ^{\{ 2 \}} | 70 ^{\{ 9 \}} | 102 ^{\{ 14 \}} | 33 ^{\{ 4 \}} | 97 ^{\{ 12.5 \}} | 114 ^{\{ 15 \}} | 63 ^{\{ 7 \}} | 51 ^{\{ 5 \}} | 22 ^{\{ 3 \}} | 78 ^{\{ 11 \}} | 97 ^{\{ 12.5 \}} | 9 ^{\{ 1 \}} | 60 ^{\{ 6 \}} | 68 ^{\{ 8 \}} | 75 ^{\{ 10 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.03792 ^{\{ 2 \}} | 0.04631 ^{\{ 7 \}} | 0.05819 ^{\{ 14 \}} | 0.04377 ^{\{ 5 \}} | 0.05299 ^{\{ 10 \}} | 0.06162 ^{\{ 15 \}} | 0.04975 ^{\{ 9 \}} | 0.0476 ^{\{ 8 \}} | 0.03992 ^{\{ 3 \}} | 0.04343 ^{\{ 4 \}} | 0.05571 ^{\{ 11.5 \}} | 0.0272 ^{\{ 1 \}} | 0.05571 ^{\{ 11.5 \}} | 0.04408 ^{\{ 6 \}} | 0.05653 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.20078 ^{\{ 3 \}} | 0.22619 ^{\{ 6 \}} | 0.24363 ^{\{ 12 \}} | 0.23866 ^{\{ 10 \}} | 0.23765 ^{\{ 9 \}} | 0.23982 ^{\{ 11 \}} | 0.23231 ^{\{ 7 \}} | 0.21986 ^{\{ 5 \}} | 0.16668 ^{\{ 2 \}} | 0.20972 ^{\{ 4 \}} | 0.25183 ^{\{ 14 \}} | 0.09301 ^{\{ 1 \}} | 0.26527 ^{\{ 15 \}} | 0.23274 ^{\{ 8 \}} | 0.24674 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00219 ^{\{ 2 \}} | 0.00339 ^{\{ 7 \}} | 0.00555 ^{\{ 14 \}} | 0.00291 ^{\{ 4 \}} | 0.00433 ^{\{ 10 \}} | 0.00612 ^{\{ 15 \}} | 0.00397 ^{\{ 9 \}} | 0.00366 ^{\{ 8 \}} | 0.00287 ^{\{ 3 \}} | 0.00308 ^{\{ 5 \}} | 0.00528 ^{\{ 13 \}} | 0.0014 ^{\{ 1 \}} | 0.00479 ^{\{ 11 \}} | 0.00312 ^{\{ 6 \}} | 0.00498 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.0644 ^{\{ 3 \}} | 0.07884 ^{\{ 6 \}} | 0.08607 ^{\{ 10 \}} | 0.08968 ^{\{ 12 \}} | 0.08623 ^{\{ 11 \}} | 0.08533 ^{\{ 8 \}} | 0.08135 ^{\{ 7 \}} | 0.07377 ^{\{ 4 \}} | 0.05803 ^{\{ 2 \}} | 0.07457 ^{\{ 5 \}} | 0.09151 ^{\{ 14 \}} | 0.02381 ^{\{ 1 \}} | 0.10724 ^{\{ 15 \}} | 0.08546 ^{\{ 9 \}} | 0.08994 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.15167 ^{\{ 2 \}} | 0.18523 ^{\{ 7 \}} | 0.23277 ^{\{ 14 \}} | 0.17509 ^{\{ 5 \}} | 0.21197 ^{\{ 10 \}} | 0.24649 ^{\{ 15 \}} | 0.199 ^{\{ 9 \}} | 0.19039 ^{\{ 8 \}} | 0.15967 ^{\{ 3 \}} | 0.1737 ^{\{ 4 \}} | 0.22284 ^{\{ 11.5 \}} | 0.10879 ^{\{ 1 \}} | 0.22284 ^{\{ 11.5 \}} | 0.17631 ^{\{ 6 \}} | 0.22614 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.26771 ^{\{ 3 \}} | 0.30159 ^{\{ 6 \}} | 0.32484 ^{\{ 12 \}} | 0.31822 ^{\{ 10 \}} | 0.31687 ^{\{ 9 \}} | 0.31975 ^{\{ 11 \}} | 0.30975 ^{\{ 7 \}} | 0.29315 ^{\{ 5 \}} | 0.22223 ^{\{ 2 \}} | 0.27962 ^{\{ 4 \}} | 0.33577 ^{\{ 14 \}} | 0.12402 ^{\{ 1 \}} | 0.35369 ^{\{ 15 \}} | 0.31032 ^{\{ 8 \}} | 0.32898 ^{\{ 13 \}} | |
D_{abs} | 0.03978 ^{\{ 1 \}} | 0.04375 ^{\{ 5 \}} | 0.04853 ^{\{ 12 \}} | 0.04421 ^{\{ 7 \}} | 0.04553 ^{\{ 8 \}} | 0.04857 ^{\{ 13 \}} | 0.04381 ^{\{ 6 \}} | 0.04289 ^{\{ 2 \}} | 0.04619 ^{\{ 9 \}} | 0.04303 ^{\{ 3 \}} | 0.04796 ^{\{ 11 \}} | 0.0432 ^{\{ 4 \}} | 0.05475 ^{\{ 15 \}} | 0.04735 ^{\{ 10 \}} | 0.05436 ^{\{ 14 \}} | |
D_{max} | 0.06553 ^{\{ 1 \}} | 0.07131 ^{\{ 6 \}} | 0.08238 ^{\{ 12 \}} | 0.07085 ^{\{ 4 \}} | 0.07553 ^{\{ 10 \}} | 0.08325 ^{\{ 13 \}} | 0.0722 ^{\{ 7 \}} | 0.07117 ^{\{ 5 \}} | 0.07504 ^{\{ 8 \}} | 0.07035 ^{\{ 3 \}} | 0.07991 ^{\{ 11 \}} | 0.06891 ^{\{ 2 \}} | 0.08996 ^{\{ 15 \}} | 0.07547 ^{\{ 9 \}} | 0.0896 ^{\{ 14 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 50 ^{\{ 6 \}} | 100 ^{\{ 11.5 \}} | 57 ^{\{ 7 \}} | 77 ^{\{ 10 \}} | 101 ^{\{ 13 \}} | 61 ^{\{ 8 \}} | 45 ^{\{ 5 \}} | 32 ^{\{ 3.5 \}} | 32 ^{\{ 3.5 \}} | 100 ^{\{ 11.5 \}} | 12 ^{\{ 1 \}} | 109 ^{\{ 15 \}} | 62 ^{\{ 9 \}} | 105 ^{\{ 14 \}} | |
60 | BIAS( \hat{\delta} ) | 0.02673 ^{\{ 2 \}} | 0.03656 ^{\{ 8 \}} | 0.04072 ^{\{ 12 \}} | 0.03364 ^{\{ 4 \}} | 0.04024 ^{\{ 11 \}} | 0.04471 ^{\{ 14 \}} | 0.03721 ^{\{ 9 \}} | 0.03459 ^{\{ 6 \}} | 0.02974 ^{\{ 3 \}} | 0.03401 ^{\{ 5 \}} | 0.03937 ^{\{ 10 \}} | 0.02163 ^{\{ 1 \}} | 0.04522 ^{\{ 15 \}} | 0.03572 ^{\{ 7 \}} | 0.04219 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.15368 ^{\{ 3 \}} | 0.1865 ^{\{ 8 \}} | 0.19022 ^{\{ 9 \}} | 0.18644 ^{\{ 7 \}} | 0.20018 ^{\{ 11 \}} | 0.20036 ^{\{ 12 \}} | 0.18376 ^{\{ 6 \}} | 0.17252 ^{\{ 4 \}} | 0.13065 ^{\{ 2 \}} | 0.17919 ^{\{ 5 \}} | 0.20222 ^{\{ 14 \}} | 0.08068 ^{\{ 1 \}} | 0.22493 ^{\{ 15 \}} | 0.19677 ^{\{ 10 \}} | 0.20108 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00113 ^{\{ 2 \}} | 0.00214 ^{\{ 8 \}} | 0.00273 ^{\{ 12 \}} | 0.00169 ^{\{ 4 \}} | 0.00252 ^{\{ 10 \}} | 0.00318 ^{\{ 15 \}} | 0.00217 ^{\{ 9 \}} | 0.00193 ^{\{ 7 \}} | 0.0016 ^{\{ 3 \}} | 0.00189 ^{\{ 6 \}} | 0.00253 ^{\{ 11 \}} | 0.00092 ^{\{ 1 \}} | 0.00314 ^{\{ 14 \}} | 0.00187 ^{\{ 5 \}} | 0.00278 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.04175 ^{\{ 3 \}} | 0.05644 ^{\{ 6 \}} | 0.05686 ^{\{ 7 \}} | 0.05842 ^{\{ 9 \}} | 0.06424 ^{\{ 14 \}} | 0.06282 ^{\{ 13 \}} | 0.05305 ^{\{ 5 \}} | 0.04927 ^{\{ 4 \}} | 0.03699 ^{\{ 2 \}} | 0.05786 ^{\{ 8 \}} | 0.06276 ^{\{ 12 \}} | 0.01743 ^{\{ 1 \}} | 0.07899 ^{\{ 15 \}} | 0.06166 ^{\{ 10 \}} | 0.06264 ^{\{ 11 \}} | |
MRE( \hat{\delta} ) | 0.10693 ^{\{ 2 \}} | 0.14623 ^{\{ 8 \}} | 0.16287 ^{\{ 12 \}} | 0.13457 ^{\{ 4 \}} | 0.16097 ^{\{ 11 \}} | 0.17885 ^{\{ 14 \}} | 0.14882 ^{\{ 9 \}} | 0.13837 ^{\{ 6 \}} | 0.11896 ^{\{ 3 \}} | 0.13604 ^{\{ 5 \}} | 0.15748 ^{\{ 10 \}} | 0.08653 ^{\{ 1 \}} | 0.18086 ^{\{ 15 \}} | 0.14289 ^{\{ 7 \}} | 0.16877 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.20491 ^{\{ 3 \}} | 0.24867 ^{\{ 8 \}} | 0.25362 ^{\{ 9 \}} | 0.24859 ^{\{ 7 \}} | 0.2669 ^{\{ 11 \}} | 0.26715 ^{\{ 12 \}} | 0.24501 ^{\{ 6 \}} | 0.23003 ^{\{ 4 \}} | 0.17419 ^{\{ 2 \}} | 0.23892 ^{\{ 5 \}} | 0.26963 ^{\{ 14 \}} | 0.10757 ^{\{ 1 \}} | 0.2999 ^{\{ 15 \}} | 0.26236 ^{\{ 10 \}} | 0.2681 ^{\{ 13 \}} | |
D_{abs} | 0.02865 ^{\{ 1 \}} | 0.03235 ^{\{ 3 \}} | 0.03403 ^{\{ 10 \}} | 0.03266 ^{\{ 6 \}} | 0.0338 ^{\{ 8 \}} | 0.03443 ^{\{ 11 \}} | 0.03265 ^{\{ 5 \}} | 0.03189 ^{\{ 2 \}} | 0.03385 ^{\{ 9 \}} | 0.03286 ^{\{ 7 \}} | 0.03487 ^{\{ 12 \}} | 0.03252 ^{\{ 4 \}} | 0.04035 ^{\{ 14 \}} | 0.03501 ^{\{ 13 \}} | 0.04078 ^{\{ 15 \}} | |
D_{max} | 0.04675 ^{\{ 1 \}} | 0.05334 ^{\{ 5 \}} | 0.0571 ^{\{ 11 \}} | 0.05293 ^{\{ 4 \}} | 0.05628 ^{\{ 9 \}} | 0.05832 ^{\{ 13 \}} | 0.05371 ^{\{ 7 \}} | 0.05256 ^{\{ 3 \}} | 0.05479 ^{\{ 8 \}} | 0.05366 ^{\{ 6 \}} | 0.05734 ^{\{ 12 \}} | 0.05239 ^{\{ 2 \}} | 0.06677 ^{\{ 14 \}} | 0.05649 ^{\{ 10 \}} | 0.06719 ^{\{ 15 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 54 ^{\{ 7 \}} | 82 ^{\{ 10 \}} | 45 ^{\{ 5 \}} | 85 ^{\{ 11 \}} | 104 ^{\{ 13 \}} | 56 ^{\{ 8 \}} | 36 ^{\{ 4 \}} | 32 ^{\{ 3 \}} | 47 ^{\{ 6 \}} | 95 ^{\{ 12 \}} | 12 ^{\{ 1 \}} | 117 ^{\{ 15 \}} | 72 ^{\{ 9 \}} | 106 ^{\{ 14 \}} | |
100 | BIAS( \hat{\delta} ) | 0.0231 ^{\{ 2 \}} | 0.02843 ^{\{ 6 \}} | 0.03137 ^{\{ 10 \}} | 0.02728 ^{\{ 4 \}} | 0.0316 ^{\{ 11 \}} | 0.03625 ^{\{ 15 \}} | 0.02813 ^{\{ 5 \}} | 0.02848 ^{\{ 7 \}} | 0.02475 ^{\{ 3 \}} | 0.02932 ^{\{ 9 \}} | 0.03253 ^{\{ 12 \}} | 0.01602 ^{\{ 1 \}} | 0.03265 ^{\{ 13 \}} | 0.029 ^{\{ 8 \}} | 0.03311 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.1315 ^{\{ 3 \}} | 0.14273 ^{\{ 5 \}} | 0.14892 ^{\{ 7 \}} | 0.15193 ^{\{ 8 \}} | 0.15411 ^{\{ 9 \}} | 0.16605 ^{\{ 14 \}} | 0.14096 ^{\{ 4 \}} | 0.14515 ^{\{ 6 \}} | 0.1185 ^{\{ 2 \}} | 0.15562 ^{\{ 10 \}} | 0.1731 ^{\{ 15 \}} | 0.06434 ^{\{ 1 \}} | 0.16197 ^{\{ 12 \}} | 0.16008 ^{\{ 11 \}} | 0.16502 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00086 ^{\{ 2 \}} | 0.00126 ^{\{ 5 \}} | 0.00157 ^{\{ 11 \}} | 0.00112 ^{\{ 4 \}} | 0.00156 ^{\{ 10 \}} | 0.00208 ^{\{ 15 \}} | 0.00127 ^{\{ 6 \}} | 0.00128 ^{\{ 7 \}} | 0.00107 ^{\{ 3 \}} | 0.00136 ^{\{ 9 \}} | 0.00166 ^{\{ 12.5 \}} | 0.00051 ^{\{ 1 \}} | 0.00166 ^{\{ 12.5 \}} | 0.00129 ^{\{ 8 \}} | 0.00177 ^{\{ 14 \}} | |
MSE( \hat{\beta} ) | 0.03144 ^{\{ 3 \}} | 0.03254 ^{\{ 4 \}} | 0.03644 ^{\{ 7 \}} | 0.03847 ^{\{ 8 \}} | 0.03959 ^{\{ 9 \}} | 0.04527 ^{\{ 14 \}} | 0.03344 ^{\{ 5 \}} | 0.03458 ^{\{ 6 \}} | 0.0301 ^{\{ 2 \}} | 0.04335 ^{\{ 12 \}} | 0.04773 ^{\{ 15 \}} | 0.01114 ^{\{ 1 \}} | 0.04165 ^{\{ 10 \}} | 0.04281 ^{\{ 11 \}} | 0.04409 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.09238 ^{\{ 2 \}} | 0.11374 ^{\{ 6 \}} | 0.12547 ^{\{ 10 \}} | 0.10911 ^{\{ 4 \}} | 0.12639 ^{\{ 11 \}} | 0.14502 ^{\{ 15 \}} | 0.1125 ^{\{ 5 \}} | 0.11391 ^{\{ 7 \}} | 0.09899 ^{\{ 3 \}} | 0.11729 ^{\{ 9 \}} | 0.13012 ^{\{ 12 \}} | 0.0641 ^{\{ 1 \}} | 0.13062 ^{\{ 13 \}} | 0.11599 ^{\{ 8 \}} | 0.13244 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.17533 ^{\{ 3 \}} | 0.1903 ^{\{ 5 \}} | 0.19856 ^{\{ 7 \}} | 0.20257 ^{\{ 8 \}} | 0.20548 ^{\{ 9 \}} | 0.2214 ^{\{ 14 \}} | 0.18795 ^{\{ 4 \}} | 0.19354 ^{\{ 6 \}} | 0.158 ^{\{ 2 \}} | 0.20749 ^{\{ 10 \}} | 0.2308 ^{\{ 15 \}} | 0.08578 ^{\{ 1 \}} | 0.21596 ^{\{ 12 \}} | 0.21345 ^{\{ 11 \}} | 0.22003 ^{\{ 13 \}} | |
D_{abs} | 0.02498 ^{\{ 2 \}} | 0.02503 ^{\{ 3 \}} | 0.02656 ^{\{ 7 \}} | 0.02588 ^{\{ 5 \}} | 0.02596 ^{\{ 6 \}} | 0.02722 ^{\{ 11 \}} | 0.02554 ^{\{ 4 \}} | 0.02663 ^{\{ 8 \}} | 0.02832 ^{\{ 12 \}} | 0.02699 ^{\{ 9 \}} | 0.02868 ^{\{ 13 \}} | 0.02409 ^{\{ 1 \}} | 0.03054 ^{\{ 14 \}} | 0.02703 ^{\{ 10 \}} | 0.03077 ^{\{ 15 \}} | |
D_{max} | 0.04055 ^{\{ 2 \}} | 0.04137 ^{\{ 3 \}} | 0.04456 ^{\{ 10 \}} | 0.04216 ^{\{ 5 \}} | 0.04346 ^{\{ 6 \}} | 0.04638 ^{\{ 12 \}} | 0.04211 ^{\{ 4 \}} | 0.04374 ^{\{ 7 \}} | 0.04578 ^{\{ 11 \}} | 0.04428 ^{\{ 9 \}} | 0.04702 ^{\{ 13 \}} | 0.03867 ^{\{ 1 \}} | 0.0502 ^{\{ 14 \}} | 0.04408 ^{\{ 8 \}} | 0.05067 ^{\{ 15 \}} | |
\sum Ranks | 19 ^{\{ 2 \}} | 37 ^{\{ 3.5 \}} | 69 ^{\{ 8 \}} | 46 ^{\{ 6 \}} | 71 ^{\{ 9 \}} | 110 ^{\{ 14 \}} | 37 ^{\{ 3.5 \}} | 54 ^{\{ 7 \}} | 38 ^{\{ 5 \}} | 77 ^{\{ 11 \}} | 107.5 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 100.5 ^{\{ 12 \}} | 75 ^{\{ 10 \}} | 111 ^{\{ 15 \}} | |
200 | BIAS( \hat{\delta} ) | 0.0173 ^{\{ 2 \}} | 0.01938 ^{\{ 7 \}} | 0.02284 ^{\{ 11 \}} | 0.019 ^{\{ 4 \}} | 0.02223 ^{\{ 10 \}} | 0.02627 ^{\{ 15 \}} | 0.02055 ^{\{ 9 \}} | 0.01934 ^{\{ 6 \}} | 0.01838 ^{\{ 3 \}} | 0.01911 ^{\{ 5 \}} | 0.02522 ^{\{ 13 \}} | 0.01175 ^{\{ 1 \}} | 0.02527 ^{\{ 14 \}} | 0.01943 ^{\{ 8 \}} | 0.02293 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.09413 ^{\{ 3 \}} | 0.09871 ^{\{ 5 \}} | 0.10837 ^{\{ 11 \}} | 0.1018 ^{\{ 7 \}} | 0.10854 ^{\{ 12 \}} | 0.11908 ^{\{ 13 \}} | 0.10176 ^{\{ 6 \}} | 0.0986 ^{\{ 4 \}} | 0.08773 ^{\{ 2 \}} | 0.1022 ^{\{ 8 \}} | 0.14097 ^{\{ 15 \}} | 0.04869 ^{\{ 1 \}} | 0.13008 ^{\{ 14 \}} | 0.10478 ^{\{ 9 \}} | 0.10707 ^{\{ 10 \}} | |
MSE( \hat{\delta} ) | 0.00047 ^{\{ 2 \}} | 0.00059 ^{\{ 5.5 \}} | 0.00082 ^{\{ 11 \}} | 0.00057 ^{\{ 3 \}} | 0.00078 ^{\{ 10 \}} | 0.00108 ^{\{ 15 \}} | 0.00065 ^{\{ 9 \}} | 0.00058 ^{\{ 4 \}} | 0.00061 ^{\{ 7 \}} | 0.00062 ^{\{ 8 \}} | 0.00098 ^{\{ 13.5 \}} | 0.00029 ^{\{ 1 \}} | 0.00098 ^{\{ 13.5 \}} | 0.00059 ^{\{ 5.5 \}} | 0.00091 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.01471 ^{\{ 2 \}} | 0.01633 ^{\{ 5 \}} | 0.01863 ^{\{ 9 \}} | 0.01774 ^{\{ 7 \}} | 0.01979 ^{\{ 11 \}} | 0.0231 ^{\{ 13 \}} | 0.01716 ^{\{ 6 \}} | 0.01534 ^{\{ 3 \}} | 0.01628 ^{\{ 4 \}} | 0.0195 ^{\{ 10 \}} | 0.03311 ^{\{ 15 \}} | 0.00672 ^{\{ 1 \}} | 0.02761 ^{\{ 14 \}} | 0.01843 ^{\{ 8 \}} | 0.02024 ^{\{ 12 \}} | |
MRE( \hat{\delta} ) | 0.06921 ^{\{ 2 \}} | 0.07753 ^{\{ 7 \}} | 0.09134 ^{\{ 11 \}} | 0.07601 ^{\{ 4 \}} | 0.08891 ^{\{ 10 \}} | 0.10508 ^{\{ 15 \}} | 0.0822 ^{\{ 9 \}} | 0.07736 ^{\{ 6 \}} | 0.07351 ^{\{ 3 \}} | 0.07643 ^{\{ 5 \}} | 0.10088 ^{\{ 13 \}} | 0.04701 ^{\{ 1 \}} | 0.10108 ^{\{ 14 \}} | 0.07774 ^{\{ 8 \}} | 0.09172 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.1255 ^{\{ 3 \}} | 0.13161 ^{\{ 5 \}} | 0.1445 ^{\{ 11 \}} | 0.13574 ^{\{ 7 \}} | 0.14471 ^{\{ 12 \}} | 0.15878 ^{\{ 13 \}} | 0.13568 ^{\{ 6 \}} | 0.13146 ^{\{ 4 \}} | 0.11697 ^{\{ 2 \}} | 0.13626 ^{\{ 8 \}} | 0.18796 ^{\{ 15 \}} | 0.06491 ^{\{ 1 \}} | 0.17344 ^{\{ 14 \}} | 0.1397 ^{\{ 9 \}} | 0.14275 ^{\{ 10 \}} | |
D_{abs} | 0.01803 ^{\{ 3 \}} | 0.01795 ^{\{ 2 \}} | 0.01849 ^{\{ 6 \}} | 0.01827 ^{\{ 4 \}} | 0.01852 ^{\{ 7 \}} | 0.01943 ^{\{ 9 \}} | 0.01842 ^{\{ 5 \}} | 0.01853 ^{\{ 8 \}} | 0.01976 ^{\{ 12 \}} | 0.01957 ^{\{ 10 \}} | 0.02149 ^{\{ 13 \}} | 0.0176 ^{\{ 1 \}} | 0.0237 ^{\{ 15 \}} | 0.01974 ^{\{ 11 \}} | 0.02287 ^{\{ 14 \}} | |
D_{max} | 0.02919 ^{\{ 2 \}} | 0.02952 ^{\{ 3 \}} | 0.0311 ^{\{ 8 \}} | 0.02977 ^{\{ 4 \}} | 0.03109 ^{\{ 7 \}} | 0.03335 ^{\{ 12 \}} | 0.0306 ^{\{ 6 \}} | 0.03031 ^{\{ 5 \}} | 0.03239 ^{\{ 11 \}} | 0.03179 ^{\{ 9 \}} | 0.03524 ^{\{ 13 \}} | 0.02807 ^{\{ 1 \}} | 0.03898 ^{\{ 15 \}} | 0.03199 ^{\{ 10 \}} | 0.03733 ^{\{ 14 \}} | |
\sum Ranks | 19 ^{\{ 2 \}} | 39.5 ^{\{ 3 \}} | 78 ^{\{ 10 \}} | 40 ^{\{ 4.5 \}} | 79 ^{\{ 11 \}} | 105 ^{\{ 13 \}} | 56 ^{\{ 7 \}} | 40 ^{\{ 4.5 \}} | 44 ^{\{ 6 \}} | 63 ^{\{ 8 \}} | 110.5 ^{\{ 14 \}} | 8 ^{\{ 1 \}} | 113.5 ^{\{ 15 \}} | 68.5 ^{\{ 9 \}} | 96 ^{\{ 12 \}} | |
300 | BIAS( \hat{\delta} ) | 0.01419 ^{\{ 2 \}} | 0.01736 ^{\{ 9 \}} | 0.01883 ^{\{ 13 \}} | 0.01549 ^{\{ 4 \}} | 0.01842 ^{\{ 12 \}} | 0.02076 ^{\{ 15 \}} | 0.01633 ^{\{ 7 \}} | 0.01636 ^{\{ 8 \}} | 0.01534 ^{\{ 3 \}} | 0.01629 ^{\{ 6 \}} | 0.01982 ^{\{ 14 \}} | 0.00983 ^{\{ 1 \}} | 0.01821 ^{\{ 11 \}} | 0.01577 ^{\{ 5 \}} | 0.01778 ^{\{ 10 \}} |
BIAS( \hat{\beta} ) | 0.0731 ^{\{ 2 \}} | 0.08699 ^{\{ 9 \}} | 0.08814 ^{\{ 11 \}} | 0.08227 ^{\{ 4 \}} | 0.09008 ^{\{ 12 \}} | 0.09593 ^{\{ 14 \}} | 0.08242 ^{\{ 5 \}} | 0.08312 ^{\{ 6 \}} | 0.07557 ^{\{ 3 \}} | 0.08758 ^{\{ 10 \}} | 0.10944 ^{\{ 15 \}} | 0.03688 ^{\{ 1 \}} | 0.09063 ^{\{ 13 \}} | 0.08574 ^{\{ 8 \}} | 0.08526 ^{\{ 7 \}} | |
MSE( \hat{\delta} ) | 0.00031 ^{\{ 2 \}} | 0.00048 ^{\{ 9 \}} | 0.00056 ^{\{ 12 \}} | 0.00038 ^{\{ 3 \}} | 0.00054 ^{\{ 10.5 \}} | 0.00067 ^{\{ 15 \}} | 0.00044 ^{\{ 8 \}} | 0.00041 ^{\{ 5.5 \}} | 0.00041 ^{\{ 5.5 \}} | 0.00043 ^{\{ 7 \}} | 0.00062 ^{\{ 14 \}} | 2e-04 ^{\{ 1 \}} | 0.00057 ^{\{ 13 \}} | 4e-04 ^{\{ 4 \}} | 0.00054 ^{\{ 10.5 \}} | |
MSE( \hat{\beta} ) | 0.00848 ^{\{ 2 \}} | 0.01219 ^{\{ 8 \}} | 0.01264 ^{\{ 10 \}} | 0.01114 ^{\{ 4 \}} | 0.01321 ^{\{ 11 \}} | 0.01472 ^{\{ 14 \}} | 0.01129 ^{\{ 5 \}} | 0.0111 ^{\{ 3 \}} | 0.01187 ^{\{ 6 \}} | 0.01322 ^{\{ 12 \}} | 0.02008 ^{\{ 15 \}} | 0.00412 ^{\{ 1 \}} | 0.01441 ^{\{ 13 \}} | 0.01218 ^{\{ 7 \}} | 0.01225 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.05677 ^{\{ 2 \}} | 0.06944 ^{\{ 9 \}} | 0.07531 ^{\{ 13 \}} | 0.06196 ^{\{ 4 \}} | 0.07367 ^{\{ 12 \}} | 0.08303 ^{\{ 15 \}} | 0.06531 ^{\{ 7 \}} | 0.06545 ^{\{ 8 \}} | 0.06137 ^{\{ 3 \}} | 0.06515 ^{\{ 6 \}} | 0.07927 ^{\{ 14 \}} | 0.03933 ^{\{ 1 \}} | 0.07285 ^{\{ 11 \}} | 0.06308 ^{\{ 5 \}} | 0.07112 ^{\{ 10 \}} | |
MRE( \hat{\beta} ) | 0.09747 ^{\{ 2 \}} | 0.11599 ^{\{ 9 \}} | 0.11753 ^{\{ 11 \}} | 0.10969 ^{\{ 4 \}} | 0.12011 ^{\{ 12 \}} | 0.12791 ^{\{ 14 \}} | 0.1099 ^{\{ 5 \}} | 0.11082 ^{\{ 6 \}} | 0.10076 ^{\{ 3 \}} | 0.11678 ^{\{ 10 \}} | 0.14592 ^{\{ 15 \}} | 0.04918 ^{\{ 1 \}} | 0.12084 ^{\{ 13 \}} | 0.11432 ^{\{ 8 \}} | 0.11368 ^{\{ 7 \}} | |
D_{abs} | 0.01447 ^{\{ 2 \}} | 0.01529 ^{\{ 7 \}} | 0.01535 ^{\{ 8 \}} | 0.01484 ^{\{ 3 \}} | 0.01519 ^{\{ 6 \}} | 0.01601 ^{\{ 9 \}} | 0.01493 ^{\{ 4 \}} | 0.01504 ^{\{ 5 \}} | 0.01637 ^{\{ 11 \}} | 0.01677 ^{\{ 12 \}} | 0.01742 ^{\{ 13 \}} | 0.01433 ^{\{ 1 \}} | 0.01906 ^{\{ 15 \}} | 0.0163 ^{\{ 10 \}} | 0.0188 ^{\{ 14 \}} | |
D_{max} | 0.02349 ^{\{ 2 \}} | 0.02522 ^{\{ 6 \}} | 0.02584 ^{\{ 8 \}} | 0.02421 ^{\{ 3 \}} | 0.02533 ^{\{ 7 \}} | 0.02724 ^{\{ 12 \}} | 0.02471 ^{\{ 4 \}} | 0.02483 ^{\{ 5 \}} | 0.02669 ^{\{ 10 \}} | 0.02712 ^{\{ 11 \}} | 0.02847 ^{\{ 13 \}} | 0.02302 ^{\{ 1 \}} | 0.03095 ^{\{ 15 \}} | 0.02641 ^{\{ 9 \}} | 0.0306 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 1 \}} | 64 ^{\{ 7 \}} | 84 ^{\{ 12 \}} | 28 ^{\{ 3 \}} | 80.5 ^{\{ 11 \}} | 106 ^{\{ 14 \}} | 43 ^{\{ 5 \}} | 44.5 ^{\{ 6 \}} | 42.5 ^{\{ 4 \}} | 72 ^{\{ 9 \}} | 111 ^{\{ 15 \}} | 21 ^{\{ 2 \}} | 102 ^{\{ 13 \}} | 67 ^{\{ 8 \}} | 79.5 ^{\{ 10 \}} | |
400 | BIAS( \hat{\delta} ) | 0.0123 ^{\{ 2 \}} | 0.01445 ^{\{ 7 \}} | 0.01582 ^{\{ 12 \}} | 0.01337 ^{\{ 4 \}} | 0.01565 ^{\{ 11 \}} | 0.01778 ^{\{ 14 \}} | 0.01429 ^{\{ 6 \}} | 0.01336 ^{\{ 3 \}} | 0.01397 ^{\{ 5 \}} | 0.01482 ^{\{ 9 \}} | 0.01814 ^{\{ 15 \}} | 0.0086 ^{\{ 1 \}} | 0.01606 ^{\{ 13 \}} | 0.01477 ^{\{ 8 \}} | 0.01543 ^{\{ 10 \}} |
BIAS( \hat{\beta} ) | 0.06611 ^{\{ 2 \}} | 0.07122 ^{\{ 6 \}} | 0.07645 ^{\{ 9 \}} | 0.07072 ^{\{ 5 \}} | 0.07554 ^{\{ 8 \}} | 0.08205 ^{\{ 14 \}} | 0.07194 ^{\{ 7 \}} | 0.06821 ^{\{ 3 \}} | 0.07008 ^{\{ 4 \}} | 0.07952 ^{\{ 13 \}} | 0.10105 ^{\{ 15 \}} | 0.03501 ^{\{ 1 \}} | 0.07876 ^{\{ 10 \}} | 0.07925 ^{\{ 12 \}} | 0.07879 ^{\{ 11 \}} | |
MSE( \hat{\delta} ) | 0.00023 ^{\{ 2 \}} | 0.00032 ^{\{ 6 \}} | 4e-04 ^{\{ 12 \}} | 0.00027 ^{\{ 3.5 \}} | 0.00039 ^{\{ 11 \}} | 0.00049 ^{\{ 14 \}} | 0.00032 ^{\{ 6 \}} | 0.00027 ^{\{ 3.5 \}} | 0.00035 ^{\{ 8 \}} | 0.00036 ^{\{ 9 \}} | 5e-04 ^{\{ 15 \}} | 0.00016 ^{\{ 1 \}} | 0.00044 ^{\{ 13 \}} | 0.00032 ^{\{ 6 \}} | 0.00037 ^{\{ 10 \}} | |
MSE( \hat{\beta} ) | 0.00715 ^{\{ 2 \}} | 0.00794 ^{\{ 5 \}} | 0.00928 ^{\{ 8 \}} | 0.00783 ^{\{ 4 \}} | 0.00904 ^{\{ 7 \}} | 0.01058 ^{\{ 12 \}} | 0.00843 ^{\{ 6 \}} | 0.00727 ^{\{ 3 \}} | 0.00986 ^{\{ 11 \}} | 0.011 ^{\{ 14 \}} | 0.0167 ^{\{ 15 \}} | 0.00343 ^{\{ 1 \}} | 0.01096 ^{\{ 13 \}} | 0.00975 ^{\{ 10 \}} | 0.00967 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.0492 ^{\{ 2 \}} | 0.05779 ^{\{ 7 \}} | 0.06329 ^{\{ 12 \}} | 0.05346 ^{\{ 4 \}} | 0.06258 ^{\{ 11 \}} | 0.07111 ^{\{ 14 \}} | 0.05715 ^{\{ 6 \}} | 0.05345 ^{\{ 3 \}} | 0.05587 ^{\{ 5 \}} | 0.05929 ^{\{ 9 \}} | 0.07256 ^{\{ 15 \}} | 0.0344 ^{\{ 1 \}} | 0.06425 ^{\{ 13 \}} | 0.0591 ^{\{ 8 \}} | 0.06173 ^{\{ 10 \}} | |
MRE( \hat{\beta} ) | 0.08815 ^{\{ 2 \}} | 0.09495 ^{\{ 6 \}} | 0.10194 ^{\{ 9 \}} | 0.0943 ^{\{ 5 \}} | 0.10072 ^{\{ 8 \}} | 0.1094 ^{\{ 14 \}} | 0.09591 ^{\{ 7 \}} | 0.09095 ^{\{ 3 \}} | 0.09344 ^{\{ 4 \}} | 0.10603 ^{\{ 13 \}} | 0.13473 ^{\{ 15 \}} | 0.04668 ^{\{ 1 \}} | 0.10501 ^{\{ 10 \}} | 0.10567 ^{\{ 12 \}} | 0.10505 ^{\{ 11 \}} | |
D_{abs} | 0.01235 ^{\{ 2 \}} | 0.01243 ^{\{ 3 \}} | 0.0136 ^{\{ 8 \}} | 0.01295 ^{\{ 6 \}} | 0.01324 ^{\{ 7 \}} | 0.0137 ^{\{ 9 \}} | 0.01286 ^{\{ 5 \}} | 0.01258 ^{\{ 4 \}} | 0.01459 ^{\{ 11 \}} | 0.01484 ^{\{ 12 \}} | 0.01561 ^{\{ 13 \}} | 0.0121 ^{\{ 1 \}} | 0.01697 ^{\{ 15 \}} | 0.01421 ^{\{ 10 \}} | 0.01637 ^{\{ 14 \}} | |
D_{max} | 0.02003 ^{\{ 2 \}} | 0.02073 ^{\{ 4 \}} | 0.02267 ^{\{ 8 \}} | 0.0211 ^{\{ 5 \}} | 0.02216 ^{\{ 7 \}} | 0.02335 ^{\{ 10 \}} | 0.02123 ^{\{ 6 \}} | 0.02064 ^{\{ 3 \}} | 0.02367 ^{\{ 11 \}} | 0.02415 ^{\{ 12 \}} | 0.02563 ^{\{ 13 \}} | 0.01949 ^{\{ 1 \}} | 0.02748 ^{\{ 15 \}} | 0.02321 ^{\{ 9 \}} | 0.02659 ^{\{ 14 \}} | |
\sum Ranks | 16 ^{\{ 2 \}} | 44 ^{\{ 5 \}} | 80 ^{\{ 10 \}} | 36.5 ^{\{ 4 \}} | 70 ^{\{ 8 \}} | 100 ^{\{ 13 \}} | 49 ^{\{ 6 \}} | 25.5 ^{\{ 3 \}} | 59 ^{\{ 7 \}} | 91 ^{\{ 12 \}} | 116 ^{\{ 15 \}} | 8 ^{\{ 1 \}} | 101 ^{\{ 14 \}} | 75 ^{\{ 9 \}} | 89 ^{\{ 11 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.36362 ^{\{ 4 \}} | 0.41591 ^{\{ 10 \}} | 0.46651 ^{\{ 14 \}} | 0.3611 ^{\{ 3 \}} | 0.43973 ^{\{ 13 \}} | 0.47679 ^{\{ 15 \}} | 0.42017 ^{\{ 11 \}} | 0.3771 ^{\{ 7 \}} | 0.31372 ^{\{ 2 \}} | 0.37161 ^{\{ 6 \}} | 0.42351 ^{\{ 12 \}} | 0.18844 ^{\{ 1 \}} | 0.41564 ^{\{ 9 \}} | 0.3691 ^{\{ 5 \}} | 0.40653 ^{\{ 8 \}} |
BIAS( \hat{\beta} ) | 0.44791 ^{\{ 3 \}} | 0.47888 ^{\{ 6 \}} | 0.50494 ^{\{ 11 \}} | 0.48893 ^{\{ 8 \}} | 0.50806 ^{\{ 12 \}} | 0.49815 ^{\{ 10 \}} | 0.50822 ^{\{ 13 \}} | 0.46815 ^{\{ 4 \}} | 0.37554 ^{\{ 2 \}} | 0.47535 ^{\{ 5 \}} | 0.51634 ^{\{ 15 \}} | 0.26964 ^{\{ 1 \}} | 0.51083 ^{\{ 14 \}} | 0.48372 ^{\{ 7 \}} | 0.49761 ^{\{ 9 \}} | |
MSE( \hat{\delta} ) | 0.21037 ^{\{ 4 \}} | 0.26991 ^{\{ 10 \}} | 0.33182 ^{\{ 14 \}} | 0.20202 ^{\{ 3 \}} | 0.29159 ^{\{ 13 \}} | 0.35861 ^{\{ 15 \}} | 0.27624 ^{\{ 11 \}} | 0.22664 ^{\{ 7 \}} | 0.17428 ^{\{ 2 \}} | 0.21641 ^{\{ 5 \}} | 0.2846 ^{\{ 12 \}} | 0.05936 ^{\{ 1 \}} | 0.26835 ^{\{ 9 \}} | 0.21794 ^{\{ 6 \}} | 0.25878 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.29849 ^{\{ 3 \}} | 0.33823 ^{\{ 5 \}} | 0.36564 ^{\{ 11 \}} | 0.36137 ^{\{ 10 \}} | 0.37711 ^{\{ 13 \}} | 0.3567 ^{\{ 8 \}} | 0.3741 ^{\{ 12 \}} | 0.32343 ^{\{ 4 \}} | 0.25591 ^{\{ 2 \}} | 0.35154 ^{\{ 7 \}} | 0.38344 ^{\{ 15 \}} | 0.12377 ^{\{ 1 \}} | 0.38059 ^{\{ 14 \}} | 0.35106 ^{\{ 6 \}} | 0.35683 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.24241 ^{\{ 4 \}} | 0.27727 ^{\{ 10 \}} | 0.31101 ^{\{ 14 \}} | 0.24073 ^{\{ 3 \}} | 0.29315 ^{\{ 13 \}} | 0.31786 ^{\{ 15 \}} | 0.28011 ^{\{ 11 \}} | 0.2514 ^{\{ 7 \}} | 0.20915 ^{\{ 2 \}} | 0.24774 ^{\{ 6 \}} | 0.28234 ^{\{ 12 \}} | 0.12563 ^{\{ 1 \}} | 0.27709 ^{\{ 9 \}} | 0.24606 ^{\{ 5 \}} | 0.27102 ^{\{ 8 \}} | |
MRE( \hat{\beta} ) | 0.29861 ^{\{ 3 \}} | 0.31926 ^{\{ 6 \}} | 0.33662 ^{\{ 11 \}} | 0.32595 ^{\{ 8 \}} | 0.33871 ^{\{ 12 \}} | 0.3321 ^{\{ 10 \}} | 0.33882 ^{\{ 13 \}} | 0.3121 ^{\{ 4 \}} | 0.25036 ^{\{ 2 \}} | 0.3169 ^{\{ 5 \}} | 0.34423 ^{\{ 15 \}} | 0.17976 ^{\{ 1 \}} | 0.34056 ^{\{ 14 \}} | 0.32248 ^{\{ 7 \}} | 0.33174 ^{\{ 9 \}} | |
D_{abs} | 0.03944 ^{\{ 1 \}} | 0.04453 ^{\{ 8 \}} | 0.04527 ^{\{ 10 \}} | 0.04151 ^{\{ 3 \}} | 0.04361 ^{\{ 5 \}} | 0.04561 ^{\{ 12 \}} | 0.04232 ^{\{ 4 \}} | 0.04094 ^{\{ 2 \}} | 0.04552 ^{\{ 11 \}} | 0.04852 ^{\{ 13 \}} | 0.04426 ^{\{ 7 \}} | 0.04386 ^{\{ 6 \}} | 0.05483 ^{\{ 15 \}} | 0.04475 ^{\{ 9 \}} | 0.05458 ^{\{ 14 \}} | |
D_{max} | 0.06565 ^{\{ 1 \}} | 0.07283 ^{\{ 8 \}} | 0.07644 ^{\{ 11 \}} | 0.06681 ^{\{ 2 \}} | 0.0725 ^{\{ 7 \}} | 0.07674 ^{\{ 12 \}} | 0.06966 ^{\{ 5 \}} | 0.06724 ^{\{ 3 \}} | 0.07288 ^{\{ 9 \}} | 0.07755 ^{\{ 13 \}} | 0.07318 ^{\{ 10 \}} | 0.06774 ^{\{ 4 \}} | 0.0875 ^{\{ 15 \}} | 0.07206 ^{\{ 6 \}} | 0.08689 ^{\{ 14 \}} | |
\sum Ranks | 23 ^{\{ 2 \}} | 63 ^{\{ 8 \}} | 96 ^{\{ 12 \}} | 40 ^{\{ 5 \}} | 88 ^{\{ 11 \}} | 97 ^{\{ 13 \}} | 80 ^{\{ 10 \}} | 38 ^{\{ 4 \}} | 32 ^{\{ 3 \}} | 60 ^{\{ 7 \}} | 98 ^{\{ 14 \}} | 16 ^{\{ 1 \}} | 99 ^{\{ 15 \}} | 51 ^{\{ 6 \}} | 79 ^{\{ 9 \}} | |
60 | BIAS( \hat{\delta} ) | 0.25193 ^{\{ 2 \}} | 0.31471 ^{\{ 8 \}} | 0.35735 ^{\{ 14 \}} | 0.27976 ^{\{ 4 \}} | 0.34812 ^{\{ 12 \}} | 0.40958 ^{\{ 15 \}} | 0.32601 ^{\{ 10 \}} | 0.28841 ^{\{ 5 \}} | 0.26064 ^{\{ 3 \}} | 0.29235 ^{\{ 7 \}} | 0.32535 ^{\{ 9 \}} | 0.17636 ^{\{ 1 \}} | 0.35405 ^{\{ 13 \}} | 0.28887 ^{\{ 6 \}} | 0.34765 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.34938 ^{\{ 2 \}} | 0.40316 ^{\{ 5 \}} | 0.41962 ^{\{ 7 \}} | 0.412 ^{\{ 6 \}} | 0.42457 ^{\{ 9 \}} | 0.45051 ^{\{ 14 \}} | 0.42184 ^{\{ 8 \}} | 0.38627 ^{\{ 4 \}} | 0.34945 ^{\{ 3 \}} | 0.42834 ^{\{ 10 \}} | 0.44339 ^{\{ 12 \}} | 0.24504 ^{\{ 1 \}} | 0.46935 ^{\{ 15 \}} | 0.43185 ^{\{ 11 \}} | 0.44997 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.10131 ^{\{ 2 \}} | 0.14878 ^{\{ 8 \}} | 0.20227 ^{\{ 14 \}} | 0.12128 ^{\{ 4 \}} | 0.19801 ^{\{ 13 \}} | 0.25959 ^{\{ 15 \}} | 0.16855 ^{\{ 10 \}} | 0.12991 ^{\{ 5 \}} | 0.12121 ^{\{ 3 \}} | 0.13012 ^{\{ 6 \}} | 0.1646 ^{\{ 9 \}} | 0.05318 ^{\{ 1 \}} | 0.19114 ^{\{ 11 \}} | 0.13018 ^{\{ 7 \}} | 0.1939 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.19887 ^{\{ 2 \}} | 0.24543 ^{\{ 5 \}} | 0.2617 ^{\{ 6 \}} | 0.28333 ^{\{ 9 \}} | 0.28214 ^{\{ 8 \}} | 0.30154 ^{\{ 13 \}} | 0.27339 ^{\{ 7 \}} | 0.23646 ^{\{ 4 \}} | 0.23071 ^{\{ 3 \}} | 0.29922 ^{\{ 11 \}} | 0.30099 ^{\{ 12 \}} | 0.1013 ^{\{ 1 \}} | 0.3262 ^{\{ 15 \}} | 0.29724 ^{\{ 10 \}} | 0.31345 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.16795 ^{\{ 2 \}} | 0.20981 ^{\{ 8 \}} | 0.23823 ^{\{ 14 \}} | 0.18651 ^{\{ 4 \}} | 0.23208 ^{\{ 12 \}} | 0.27306 ^{\{ 15 \}} | 0.21734 ^{\{ 10 \}} | 0.19227 ^{\{ 5 \}} | 0.17376 ^{\{ 3 \}} | 0.1949 ^{\{ 7 \}} | 0.2169 ^{\{ 9 \}} | 0.11757 ^{\{ 1 \}} | 0.23603 ^{\{ 13 \}} | 0.19258 ^{\{ 6 \}} | 0.23177 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.23292 ^{\{ 2 \}} | 0.26877 ^{\{ 5 \}} | 0.27975 ^{\{ 7 \}} | 0.27467 ^{\{ 6 \}} | 0.28305 ^{\{ 9 \}} | 0.30034 ^{\{ 14 \}} | 0.28122 ^{\{ 8 \}} | 0.25751 ^{\{ 4 \}} | 0.23297 ^{\{ 3 \}} | 0.28556 ^{\{ 10 \}} | 0.2956 ^{\{ 12 \}} | 0.16336 ^{\{ 1 \}} | 0.3129 ^{\{ 15 \}} | 0.2879 ^{\{ 11 \}} | 0.29998 ^{\{ 13 \}} | |
D_{abs} | 0.02961 ^{\{ 1 \}} | 0.03051 ^{\{ 2 \}} | 0.03132 ^{\{ 6 \}} | 0.03103 ^{\{ 5 \}} | 0.03319 ^{\{ 9 \}} | 0.03268 ^{\{ 7 \}} | 0.03101 ^{\{ 4 \}} | 0.03063 ^{\{ 3 \}} | 0.03576 ^{\{ 13 \}} | 0.03372 ^{\{ 11 \}} | 0.03366 ^{\{ 10 \}} | 0.03275 ^{\{ 8 \}} | 0.03914 ^{\{ 14 \}} | 0.03378 ^{\{ 12 \}} | 0.03955 ^{\{ 15 \}} | |
D_{max} | 0.04842 ^{\{ 1 \}} | 0.05047 ^{\{ 4 \}} | 0.05302 ^{\{ 7 \}} | 0.04997 ^{\{ 2 \}} | 0.05541 ^{\{ 11 \}} | 0.05606 ^{\{ 12 \}} | 0.05152 ^{\{ 6 \}} | 0.05036 ^{\{ 3 \}} | 0.0573 ^{\{ 13 \}} | 0.05469 ^{\{ 9 \}} | 0.05517 ^{\{ 10 \}} | 0.0514 ^{\{ 5 \}} | 0.06451 ^{\{ 14 \}} | 0.05445 ^{\{ 8 \}} | 0.06483 ^{\{ 15 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 45 ^{\{ 6 \}} | 75 ^{\{ 10 \}} | 40 ^{\{ 4 \}} | 83 ^{\{ 11.5 \}} | 105 ^{\{ 14 \}} | 63 ^{\{ 7 \}} | 33 ^{\{ 3 \}} | 44 ^{\{ 5 \}} | 71 ^{\{ 8.5 \}} | 83 ^{\{ 11.5 \}} | 19 ^{\{ 2 \}} | 110 ^{\{ 15 \}} | 71 ^{\{ 8.5 \}} | 104 ^{\{ 13 \}} | |
100 | BIAS( \hat{\delta} ) | 0.20656 ^{\{ 2 \}} | 0.25391 ^{\{ 8 \}} | 0.30172 ^{\{ 13 \}} | 0.23433 ^{\{ 5 \}} | 0.29934 ^{\{ 12 \}} | 0.327 ^{\{ 15 \}} | 0.25672 ^{\{ 9 \}} | 0.23765 ^{\{ 6 \}} | 0.2224 ^{\{ 3 \}} | 0.24672 ^{\{ 7 \}} | 0.26161 ^{\{ 10 \}} | 0.16346 ^{\{ 1 \}} | 0.30254 ^{\{ 14 \}} | 0.23314 ^{\{ 4 \}} | 0.29396 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.28605 ^{\{ 2 \}} | 0.34848 ^{\{ 6 \}} | 0.37222 ^{\{ 10 \}} | 0.35822 ^{\{ 8 \}} | 0.38959 ^{\{ 13 \}} | 0.38082 ^{\{ 12 \}} | 0.34956 ^{\{ 7 \}} | 0.32677 ^{\{ 4 \}} | 0.30141 ^{\{ 3 \}} | 0.36475 ^{\{ 9 \}} | 0.37444 ^{\{ 11 \}} | 0.22382 ^{\{ 1 \}} | 0.40691 ^{\{ 15 \}} | 0.34202 ^{\{ 5 \}} | 0.40473 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.06736 ^{\{ 2 \}} | 0.10079 ^{\{ 8 \}} | 0.14079 ^{\{ 14 \}} | 0.08235 ^{\{ 3 \}} | 0.13993 ^{\{ 13 \}} | 0.17078 ^{\{ 15 \}} | 0.10453 ^{\{ 9 \}} | 0.09235 ^{\{ 6 \}} | 0.08912 ^{\{ 5 \}} | 0.09267 ^{\{ 7 \}} | 0.10813 ^{\{ 10 \}} | 0.0469 ^{\{ 1 \}} | 0.13908 ^{\{ 12 \}} | 0.08462 ^{\{ 4 \}} | 0.13209 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.14544 ^{\{ 2 \}} | 0.19976 ^{\{ 6 \}} | 0.22305 ^{\{ 9 \}} | 0.21696 ^{\{ 8 \}} | 0.23768 ^{\{ 13 \}} | 0.23128 ^{\{ 12 \}} | 0.20607 ^{\{ 7 \}} | 0.18067 ^{\{ 4 \}} | 0.1778 ^{\{ 3 \}} | 0.22367 ^{\{ 10 \}} | 0.22523 ^{\{ 11 \}} | 0.08729 ^{\{ 1 \}} | 0.26214 ^{\{ 15 \}} | 0.1935 ^{\{ 5 \}} | 0.25864 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.13771 ^{\{ 2 \}} | 0.16927 ^{\{ 8 \}} | 0.20115 ^{\{ 13 \}} | 0.15622 ^{\{ 5 \}} | 0.19956 ^{\{ 12 \}} | 0.218 ^{\{ 15 \}} | 0.17115 ^{\{ 9 \}} | 0.15844 ^{\{ 6 \}} | 0.14827 ^{\{ 3 \}} | 0.16448 ^{\{ 7 \}} | 0.17441 ^{\{ 10 \}} | 0.10897 ^{\{ 1 \}} | 0.2017 ^{\{ 14 \}} | 0.15543 ^{\{ 4 \}} | 0.19598 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.1907 ^{\{ 2 \}} | 0.23232 ^{\{ 6 \}} | 0.24815 ^{\{ 10 \}} | 0.23882 ^{\{ 8 \}} | 0.25972 ^{\{ 13 \}} | 0.25388 ^{\{ 12 \}} | 0.23304 ^{\{ 7 \}} | 0.21785 ^{\{ 4 \}} | 0.20094 ^{\{ 3 \}} | 0.24317 ^{\{ 9 \}} | 0.24963 ^{\{ 11 \}} | 0.14921 ^{\{ 1 \}} | 0.27127 ^{\{ 15 \}} | 0.22801 ^{\{ 5 \}} | 0.26982 ^{\{ 14 \}} | |
D_{abs} | 0.02235 ^{\{ 1 \}} | 0.02482 ^{\{ 3 \}} | 0.02569 ^{\{ 6 \}} | 0.02462 ^{\{ 2 \}} | 0.02622 ^{\{ 10 \}} | 0.02597 ^{\{ 8 \}} | 0.0254 ^{\{ 5 \}} | 0.02484 ^{\{ 4 \}} | 0.02732 ^{\{ 13 \}} | 0.02672 ^{\{ 12 \}} | 0.02663 ^{\{ 11 \}} | 0.02587 ^{\{ 7 \}} | 0.03105 ^{\{ 14 \}} | 0.02607 ^{\{ 9 \}} | 0.03134 ^{\{ 15 \}} | |
D_{max} | 0.03675 ^{\{ 1 \}} | 0.04106 ^{\{ 4 \}} | 0.04372 ^{\{ 10 \}} | 0.04 ^{\{ 2 \}} | 0.04399 ^{\{ 12 \}} | 0.04501 ^{\{ 13 \}} | 0.04203 ^{\{ 6 \}} | 0.04065 ^{\{ 3 \}} | 0.04393 ^{\{ 11 \}} | 0.04357 ^{\{ 8 \}} | 0.04363 ^{\{ 9 \}} | 0.0414 ^{\{ 5 \}} | 0.05119 ^{\{ 14 \}} | 0.04208 ^{\{ 7 \}} | 0.05143 ^{\{ 15 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 49 ^{\{ 7 \}} | 85 ^{\{ 11 \}} | 41 ^{\{ 4 \}} | 98 ^{\{ 12 \}} | 102 ^{\{ 13 \}} | 59 ^{\{ 8 \}} | 37 ^{\{ 3 \}} | 44 ^{\{ 6 \}} | 69 ^{\{ 9 \}} | 83 ^{\{ 10 \}} | 18 ^{\{ 2 \}} | 113 ^{\{ 15 \}} | 43 ^{\{ 5 \}} | 105 ^{\{ 14 \}} | |
200 | BIAS( \hat{\delta} ) | 0.15288 ^{\{ 2 \}} | 0.18956 ^{\{ 8 \}} | 0.21534 ^{\{ 11 \}} | 0.16331 ^{\{ 3 \}} | 0.21778 ^{\{ 12 \}} | 0.26177 ^{\{ 15 \}} | 0.1897 ^{\{ 9 \}} | 0.1732 ^{\{ 4 \}} | 0.17595 ^{\{ 5 \}} | 0.18315 ^{\{ 7 \}} | 0.20027 ^{\{ 10 \}} | 0.13041 ^{\{ 1 \}} | 0.2255 ^{\{ 14 \}} | 0.18271 ^{\{ 6 \}} | 0.22008 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.2161 ^{\{ 2 \}} | 0.25397 ^{\{ 6 \}} | 0.27965 ^{\{ 10 \}} | 0.24415 ^{\{ 4 \}} | 0.28824 ^{\{ 11 \}} | 0.32319 ^{\{ 15 \}} | 0.25894 ^{\{ 7 \}} | 0.23866 ^{\{ 3 \}} | 0.24499 ^{\{ 5 \}} | 0.27244 ^{\{ 9 \}} | 0.29863 ^{\{ 12 \}} | 0.1761 ^{\{ 1 \}} | 0.32229 ^{\{ 14 \}} | 0.26751 ^{\{ 8 \}} | 0.31447 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.03871 ^{\{ 2 \}} | 0.05626 ^{\{ 8 \}} | 0.07407 ^{\{ 12 \}} | 0.0409 ^{\{ 3 \}} | 0.07383 ^{\{ 11 \}} | 0.10743 ^{\{ 15 \}} | 0.05719 ^{\{ 9 \}} | 0.04689 ^{\{ 4 \}} | 0.0535 ^{\{ 7 \}} | 0.05235 ^{\{ 6 \}} | 0.06256 ^{\{ 10 \}} | 0.02939 ^{\{ 1 \}} | 0.07872 ^{\{ 14 \}} | 0.05169 ^{\{ 5 \}} | 0.07812 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.08188 ^{\{ 2 \}} | 0.11144 ^{\{ 5 \}} | 0.13504 ^{\{ 10 \}} | 0.10336 ^{\{ 4 \}} | 0.14447 ^{\{ 11 \}} | 0.17092 ^{\{ 14 \}} | 0.11355 ^{\{ 6 \}} | 0.09586 ^{\{ 3 \}} | 0.11854 ^{\{ 7 \}} | 0.13004 ^{\{ 9 \}} | 0.15599 ^{\{ 12 \}} | 0.0546 ^{\{ 1 \}} | 0.17622 ^{\{ 15 \}} | 0.12213 ^{\{ 8 \}} | 0.16875 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.10192 ^{\{ 2 \}} | 0.12638 ^{\{ 8 \}} | 0.14356 ^{\{ 11 \}} | 0.10888 ^{\{ 3 \}} | 0.14519 ^{\{ 12 \}} | 0.17452 ^{\{ 15 \}} | 0.12646 ^{\{ 9 \}} | 0.11547 ^{\{ 4 \}} | 0.1173 ^{\{ 5 \}} | 0.1221 ^{\{ 7 \}} | 0.13351 ^{\{ 10 \}} | 0.08694 ^{\{ 1 \}} | 0.15034 ^{\{ 14 \}} | 0.1218 ^{\{ 6 \}} | 0.14672 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.14407 ^{\{ 2 \}} | 0.16931 ^{\{ 6 \}} | 0.18643 ^{\{ 10 \}} | 0.16277 ^{\{ 4 \}} | 0.19216 ^{\{ 11 \}} | 0.21546 ^{\{ 15 \}} | 0.17263 ^{\{ 7 \}} | 0.1591 ^{\{ 3 \}} | 0.16333 ^{\{ 5 \}} | 0.18163 ^{\{ 9 \}} | 0.19909 ^{\{ 12 \}} | 0.1174 ^{\{ 1 \}} | 0.21486 ^{\{ 14 \}} | 0.17834 ^{\{ 8 \}} | 0.20965 ^{\{ 13 \}} | |
D_{abs} | 0.01708 ^{\{ 1 \}} | 0.01778 ^{\{ 4 \}} | 0.01841 ^{\{ 7 \}} | 0.01749 ^{\{ 2.5 \}} | 0.01868 ^{\{ 8 \}} | 0.01953 ^{\{ 9 \}} | 0.01788 ^{\{ 6 \}} | 0.01749 ^{\{ 2.5 \}} | 0.02059 ^{\{ 13 \}} | 0.02042 ^{\{ 12 \}} | 0.02035 ^{\{ 11 \}} | 0.01781 ^{\{ 5 \}} | 0.02273 ^{\{ 14 \}} | 0.01969 ^{\{ 10 \}} | 0.02289 ^{\{ 15 \}} | |
D_{max} | 0.02783 ^{\{ 1 \}} | 0.02966 ^{\{ 5 \}} | 0.03114 ^{\{ 7 \}} | 0.02835 ^{\{ 2 \}} | 0.03144 ^{\{ 8 \}} | 0.03378 ^{\{ 13 \}} | 0.02968 ^{\{ 6 \}} | 0.02876 ^{\{ 4 \}} | 0.03344 ^{\{ 12 \}} | 0.03311 ^{\{ 10 \}} | 0.03321 ^{\{ 11 \}} | 0.02872 ^{\{ 3 \}} | 0.03777 ^{\{ 15 \}} | 0.03208 ^{\{ 9 \}} | 0.03745 ^{\{ 14 \}} | |
\sum Ranks | 14 ^{\{ 1.5 \}} | 50 ^{\{ 5 \}} | 78 ^{\{ 10 \}} | 25.5 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 111 ^{\{ 14 \}} | 59 ^{\{ 6.5 \}} | 27.5 ^{\{ 4 \}} | 59 ^{\{ 6.5 \}} | 69 ^{\{ 9 \}} | 88 ^{\{ 12 \}} | 14 ^{\{ 1.5 \}} | 114 ^{\{ 15 \}} | 60 ^{\{ 8 \}} | 107 ^{\{ 13 \}} | |
300 | BIAS( \hat{\delta} ) | 0.12756 ^{\{ 2 \}} | 0.14717 ^{\{ 6 \}} | 0.17874 ^{\{ 12 \}} | 0.13713 ^{\{ 3 \}} | 0.17341 ^{\{ 11 \}} | 0.21053 ^{\{ 15 \}} | 0.16111 ^{\{ 9 \}} | 0.13772 ^{\{ 4 \}} | 0.1435 ^{\{ 5 \}} | 0.15587 ^{\{ 8 \}} | 0.16547 ^{\{ 10 \}} | 0.11423 ^{\{ 1 \}} | 0.19156 ^{\{ 14 \}} | 0.14743 ^{\{ 7 \}} | 0.18039 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.17764 ^{\{ 2 \}} | 0.19923 ^{\{ 4 \}} | 0.22686 ^{\{ 9 \}} | 0.20548 ^{\{ 6 \}} | 0.22944 ^{\{ 10 \}} | 0.26697 ^{\{ 13 \}} | 0.21258 ^{\{ 7 \}} | 0.18859 ^{\{ 3 \}} | 0.20381 ^{\{ 5 \}} | 0.23225 ^{\{ 11 \}} | 0.25014 ^{\{ 12 \}} | 0.14872 ^{\{ 1 \}} | 0.27192 ^{\{ 15 \}} | 0.21783 ^{\{ 8 \}} | 0.2676 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.02609 ^{\{ 2 \}} | 0.03343 ^{\{ 5 \}} | 0.04979 ^{\{ 12 \}} | 0.0295 ^{\{ 3 \}} | 0.0464 ^{\{ 11 \}} | 0.06938 ^{\{ 15 \}} | 0.04051 ^{\{ 9 \}} | 0.03057 ^{\{ 4 \}} | 0.03482 ^{\{ 7 \}} | 0.03819 ^{\{ 8 \}} | 0.0425 ^{\{ 10 \}} | 0.02264 ^{\{ 1 \}} | 0.05705 ^{\{ 14 \}} | 0.03431 ^{\{ 6 \}} | 0.05061 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.05098 ^{\{ 2 \}} | 0.06578 ^{\{ 4 \}} | 0.09046 ^{\{ 10 \}} | 0.0717 ^{\{ 5 \}} | 0.08987 ^{\{ 9 \}} | 0.12315 ^{\{ 13 \}} | 0.07547 ^{\{ 6 \}} | 0.06013 ^{\{ 3 \}} | 0.07747 ^{\{ 7 \}} | 0.09348 ^{\{ 11 \}} | 0.10805 ^{\{ 12 \}} | 0.04003 ^{\{ 1 \}} | 0.12352 ^{\{ 14 \}} | 0.08226 ^{\{ 8 \}} | 0.1243 ^{\{ 15 \}} | |
MRE( \hat{\delta} ) | 0.08504 ^{\{ 2 \}} | 0.09811 ^{\{ 6 \}} | 0.11916 ^{\{ 12 \}} | 0.09142 ^{\{ 3 \}} | 0.1156 ^{\{ 11 \}} | 0.14035 ^{\{ 15 \}} | 0.10741 ^{\{ 9 \}} | 0.09181 ^{\{ 4 \}} | 0.09567 ^{\{ 5 \}} | 0.10392 ^{\{ 8 \}} | 0.11031 ^{\{ 10 \}} | 0.07615 ^{\{ 1 \}} | 0.1277 ^{\{ 14 \}} | 0.09829 ^{\{ 7 \}} | 0.12026 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.11842 ^{\{ 2 \}} | 0.13282 ^{\{ 4 \}} | 0.15124 ^{\{ 9 \}} | 0.13699 ^{\{ 6 \}} | 0.15296 ^{\{ 10 \}} | 0.17798 ^{\{ 13 \}} | 0.14172 ^{\{ 7 \}} | 0.12573 ^{\{ 3 \}} | 0.13587 ^{\{ 5 \}} | 0.15484 ^{\{ 11 \}} | 0.16676 ^{\{ 12 \}} | 0.09915 ^{\{ 1 \}} | 0.18128 ^{\{ 15 \}} | 0.14522 ^{\{ 8 \}} | 0.1784 ^{\{ 14 \}} | |
D_{abs} | 0.01393 ^{\{ 1 \}} | 0.01439 ^{\{ 3 \}} | 0.01544 ^{\{ 8 \}} | 0.01449 ^{\{ 4 \}} | 0.01487 ^{\{ 6 \}} | 0.01613 ^{\{ 9 \}} | 0.0152 ^{\{ 7 \}} | 0.01395 ^{\{ 2 \}} | 0.0174 ^{\{ 13 \}} | 0.01678 ^{\{ 12 \}} | 0.01636 ^{\{ 11 \}} | 0.01461 ^{\{ 5 \}} | 0.01959 ^{\{ 15 \}} | 0.0163 ^{\{ 10 \}} | 0.01954 ^{\{ 14 \}} | |
D_{max} | 0.02275 ^{\{ 1 \}} | 0.02384 ^{\{ 5 \}} | 0.02607 ^{\{ 8 \}} | 0.02357 ^{\{ 3 \}} | 0.02515 ^{\{ 6 \}} | 0.0278 ^{\{ 12 \}} | 0.02531 ^{\{ 7 \}} | 0.02298 ^{\{ 2 \}} | 0.02812 ^{\{ 13 \}} | 0.02723 ^{\{ 11 \}} | 0.02671 ^{\{ 10 \}} | 0.02372 ^{\{ 4 \}} | 0.03222 ^{\{ 15 \}} | 0.02651 ^{\{ 9 \}} | 0.03184 ^{\{ 14 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 37 ^{\{ 5 \}} | 80 ^{\{ 10.5 \}} | 33 ^{\{ 4 \}} | 74 ^{\{ 9 \}} | 105 ^{\{ 13 \}} | 61 ^{\{ 7 \}} | 25 ^{\{ 3 \}} | 60 ^{\{ 6 \}} | 80 ^{\{ 10.5 \}} | 87 ^{\{ 12 \}} | 15 ^{\{ 2 \}} | 116 ^{\{ 15 \}} | 63 ^{\{ 8 \}} | 110 ^{\{ 14 \}} | |
400 | BIAS( \hat{\delta} ) | 0.10866 ^{\{ 2 \}} | 0.1345 ^{\{ 9 \}} | 0.15044 ^{\{ 11 \}} | 0.11647 ^{\{ 3 \}} | 0.15668 ^{\{ 13 \}} | 0.18648 ^{\{ 15 \}} | 0.13412 ^{\{ 8 \}} | 0.11906 ^{\{ 4 \}} | 0.13407 ^{\{ 7 \}} | 0.13152 ^{\{ 6 \}} | 0.1481 ^{\{ 10 \}} | 0.10545 ^{\{ 1 \}} | 0.16093 ^{\{ 14 \}} | 0.12007 ^{\{ 5 \}} | 0.15558 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.15352 ^{\{ 2 \}} | 0.18616 ^{\{ 8 \}} | 0.19179 ^{\{ 9 \}} | 0.16571 ^{\{ 4 \}} | 0.20261 ^{\{ 11 \}} | 0.23735 ^{\{ 15 \}} | 0.18354 ^{\{ 6 \}} | 0.16043 ^{\{ 3 \}} | 0.18538 ^{\{ 7 \}} | 0.19245 ^{\{ 10 \}} | 0.22086 ^{\{ 12 \}} | 0.1359 ^{\{ 1 \}} | 0.2328 ^{\{ 14 \}} | 0.17147 ^{\{ 5 \}} | 0.22366 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01904 ^{\{ 1 \}} | 0.02909 ^{\{ 8 \}} | 0.0361 ^{\{ 11 \}} | 0.02075 ^{\{ 3 \}} | 0.03897 ^{\{ 12 \}} | 0.05496 ^{\{ 15 \}} | 0.02796 ^{\{ 7 \}} | 0.02201 ^{\{ 4 \}} | 0.03078 ^{\{ 9 \}} | 0.02632 ^{\{ 6 \}} | 0.03496 ^{\{ 10 \}} | 0.01912 ^{\{ 2 \}} | 0.0414 ^{\{ 14 \}} | 0.02291 ^{\{ 5 \}} | 0.03957 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.03755 ^{\{ 2 \}} | 0.05973 ^{\{ 7 \}} | 0.06171 ^{\{ 9 \}} | 0.04387 ^{\{ 4 \}} | 0.07077 ^{\{ 11 \}} | 0.09576 ^{\{ 15 \}} | 0.05488 ^{\{ 6 \}} | 0.04111 ^{\{ 3 \}} | 0.06503 ^{\{ 10 \}} | 0.06059 ^{\{ 8 \}} | 0.08705 ^{\{ 13 \}} | 0.03291 ^{\{ 1 \}} | 0.09212 ^{\{ 14 \}} | 0.04832 ^{\{ 5 \}} | 0.08653 ^{\{ 12 \}} | |
MRE( \hat{\delta} ) | 0.07244 ^{\{ 2 \}} | 0.08967 ^{\{ 9 \}} | 0.10029 ^{\{ 11 \}} | 0.07765 ^{\{ 3 \}} | 0.10445 ^{\{ 13 \}} | 0.12432 ^{\{ 15 \}} | 0.08942 ^{\{ 8 \}} | 0.07938 ^{\{ 4 \}} | 0.08938 ^{\{ 7 \}} | 0.08768 ^{\{ 6 \}} | 0.09874 ^{\{ 10 \}} | 0.0703 ^{\{ 1 \}} | 0.10729 ^{\{ 14 \}} | 0.08004 ^{\{ 5 \}} | 0.10372 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.10235 ^{\{ 2 \}} | 0.1241 ^{\{ 8 \}} | 0.12786 ^{\{ 9 \}} | 0.11048 ^{\{ 4 \}} | 0.13507 ^{\{ 11 \}} | 0.15823 ^{\{ 15 \}} | 0.12236 ^{\{ 6 \}} | 0.10695 ^{\{ 3 \}} | 0.12359 ^{\{ 7 \}} | 0.1283 ^{\{ 10 \}} | 0.14724 ^{\{ 12 \}} | 0.0906 ^{\{ 1 \}} | 0.1552 ^{\{ 14 \}} | 0.11431 ^{\{ 5 \}} | 0.1491 ^{\{ 13 \}} | |
D_{abs} | 0.01246 ^{\{ 3 \}} | 0.01304 ^{\{ 6 \}} | 0.0132 ^{\{ 7 \}} | 0.01242 ^{\{ 1.5 \}} | 0.0134 ^{\{ 8 \}} | 0.0142 ^{\{ 10 \}} | 0.01274 ^{\{ 4 \}} | 0.01242 ^{\{ 1.5 \}} | 0.01486 ^{\{ 13 \}} | 0.01465 ^{\{ 11 \}} | 0.01477 ^{\{ 12 \}} | 0.01284 ^{\{ 5 \}} | 0.01692 ^{\{ 15 \}} | 0.01356 ^{\{ 9 \}} | 0.01666 ^{\{ 14 \}} | |
D_{max} | 0.0202 ^{\{ 1 \}} | 0.02154 ^{\{ 6 \}} | 0.02231 ^{\{ 8 \}} | 0.02028 ^{\{ 2 \}} | 0.02261 ^{\{ 9 \}} | 0.02465 ^{\{ 13 \}} | 0.02112 ^{\{ 5 \}} | 0.02043 ^{\{ 3 \}} | 0.02406 ^{\{ 11 \}} | 0.02377 ^{\{ 10 \}} | 0.02409 ^{\{ 12 \}} | 0.02086 ^{\{ 4 \}} | 0.02768 ^{\{ 15 \}} | 0.02207 ^{\{ 7 \}} | 0.02724 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 1 \}} | 61 ^{\{ 7 \}} | 75 ^{\{ 10 \}} | 24.5 ^{\{ 3 \}} | 88 ^{\{ 11 \}} | 113 ^{\{ 14 \}} | 50 ^{\{ 6 \}} | 25.5 ^{\{ 4 \}} | 71 ^{\{ 9 \}} | 67 ^{\{ 8 \}} | 91 ^{\{ 12 \}} | 16 ^{\{ 2 \}} | 114 ^{\{ 15 \}} | 46 ^{\{ 5 \}} | 103 ^{\{ 13 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.13624 ^{\{ 4 \}} | 0.15298 ^{\{ 10 \}} | 0.16819 ^{\{ 14 \}} | 0.13657 ^{\{ 5 \}} | 0.15175 ^{\{ 8 \}} | 0.18055 ^{\{ 15 \}} | 0.15251 ^{\{ 9 \}} | 0.14595 ^{\{ 6 \}} | 0.09288 ^{\{ 2 \}} | 0.12666 ^{\{ 3 \}} | 0.15422 ^{\{ 11 \}} | 0.05648 ^{\{ 1 \}} | 0.1601 ^{\{ 12 \}} | 0.14753 ^{\{ 7 \}} | 0.16209 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.57765 ^{\{ 4 \}} | 0.65842 ^{\{ 12 \}} | 0.63719 ^{\{ 9 \}} | 0.64979 ^{\{ 10 \}} | 0.60867 ^{\{ 7 \}} | 0.68047 ^{\{ 14 \}} | 0.65627 ^{\{ 11 \}} | 0.62937 ^{\{ 8 \}} | 0.25245 ^{\{ 2 \}} | 0.57701 ^{\{ 3 \}} | 0.66039 ^{\{ 13 \}} | 0.04775 ^{\{ 1 \}} | 0.5962 ^{\{ 6 \}} | 0.69113 ^{\{ 15 \}} | 0.58347 ^{\{ 5 \}} | |
MSE( \hat{\delta} ) | 0.02921 ^{\{ 5 \}} | 0.03635 ^{\{ 10 \}} | 0.04402 ^{\{ 14 \}} | 0.02809 ^{\{ 4 \}} | 0.03633 ^{\{ 9 \}} | 0.04928 ^{\{ 15 \}} | 0.03627 ^{\{ 8 \}} | 0.0338 ^{\{ 6 \}} | 0.01634 ^{\{ 2 \}} | 0.02598 ^{\{ 3 \}} | 0.03818 ^{\{ 11 \}} | 0.00526 ^{\{ 1 \}} | 0.04053 ^{\{ 12 \}} | 0.03407 ^{\{ 7 \}} | 0.0423 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.49086 ^{\{ 3 \}} | 0.62358 ^{\{ 11 \}} | 0.56339 ^{\{ 7 \}} | 0.62516 ^{\{ 12 \}} | 0.53029 ^{\{ 6 \}} | 0.6451 ^{\{ 14 \}} | 0.61704 ^{\{ 10 \}} | 0.58229 ^{\{ 9 \}} | 0.2075 ^{\{ 2 \}} | 0.56556 ^{\{ 8 \}} | 0.6369 ^{\{ 13 \}} | 0.01233 ^{\{ 1 \}} | 0.51078 ^{\{ 5 \}} | 0.68808 ^{\{ 15 \}} | 0.49454 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.27249 ^{\{ 4 \}} | 0.30596 ^{\{ 10 \}} | 0.33638 ^{\{ 14 \}} | 0.27314 ^{\{ 5 \}} | 0.3035 ^{\{ 8 \}} | 0.36111 ^{\{ 15 \}} | 0.30502 ^{\{ 9 \}} | 0.29189 ^{\{ 6 \}} | 0.18576 ^{\{ 2 \}} | 0.25331 ^{\{ 3 \}} | 0.30843 ^{\{ 11 \}} | 0.11296 ^{\{ 1 \}} | 0.32019 ^{\{ 12 \}} | 0.29507 ^{\{ 7 \}} | 0.32419 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.28882 ^{\{ 4 \}} | 0.32921 ^{\{ 12 \}} | 0.31859 ^{\{ 9 \}} | 0.3249 ^{\{ 10 \}} | 0.30434 ^{\{ 7 \}} | 0.34024 ^{\{ 14 \}} | 0.32814 ^{\{ 11 \}} | 0.31469 ^{\{ 8 \}} | 0.12623 ^{\{ 2 \}} | 0.28851 ^{\{ 3 \}} | 0.33019 ^{\{ 13 \}} | 0.02388 ^{\{ 1 \}} | 0.2981 ^{\{ 6 \}} | 0.34556 ^{\{ 15 \}} | 0.29173 ^{\{ 5 \}} | |
D_{abs} | 0.03768 ^{\{ 1 \}} | 0.04098 ^{\{ 5 \}} | 0.04468 ^{\{ 12 \}} | 0.04059 ^{\{ 2 \}} | 0.04585 ^{\{ 13 \}} | 0.04352 ^{\{ 10 \}} | 0.04069 ^{\{ 3 \}} | 0.04313 ^{\{ 9 \}} | 0.0425 ^{\{ 7 \}} | 0.04216 ^{\{ 6 \}} | 0.04287 ^{\{ 8 \}} | 0.04096 ^{\{ 4 \}} | 0.06514 ^{\{ 15 \}} | 0.04419 ^{\{ 11 \}} | 0.06029 ^{\{ 14 \}} | |
D_{max} | 0.06199 ^{\{ 1 \}} | 0.06671 ^{\{ 5 \}} | 0.07388 ^{\{ 13 \}} | 0.06517 ^{\{ 3 \}} | 0.07369 ^{\{ 12 \}} | 0.07286 ^{\{ 11 \}} | 0.06621 ^{\{ 4 \}} | 0.07003 ^{\{ 8 \}} | 0.06696 ^{\{ 6 \}} | 0.06706 ^{\{ 7 \}} | 0.07019 ^{\{ 9 \}} | 0.0623 ^{\{ 2 \}} | 0.10106 ^{\{ 15 \}} | 0.07066 ^{\{ 10 \}} | 0.09345 ^{\{ 14 \}} | |
\sum Ranks | 26 ^{\{ 3 \}} | 75 ^{\{ 9 \}} | 92 ^{\{ 14 \}} | 51 ^{\{ 5 \}} | 70 ^{\{ 8 \}} | 108 ^{\{ 15 \}} | 65 ^{\{ 7 \}} | 60 ^{\{ 6 \}} | 25 ^{\{ 2 \}} | 36 ^{\{ 4 \}} | 89 ^{\{ 13 \}} | 12 ^{\{ 1 \}} | 83 ^{\{ 11 \}} | 87 ^{\{ 12 \}} | 81 ^{\{ 10 \}} | |
60 | BIAS( \hat{\delta} ) | 0.10659 ^{\{ 3 \}} | 0.12172 ^{\{ 9 \}} | 0.14371 ^{\{ 14 \}} | 0.10716 ^{\{ 4 \}} | 0.13619 ^{\{ 13 \}} | 0.16304 ^{\{ 15 \}} | 0.13084 ^{\{ 11 \}} | 0.11699 ^{\{ 8 \}} | 0.07293 ^{\{ 2 \}} | 0.10752 ^{\{ 5 \}} | 0.12531 ^{\{ 10 \}} | 0.04248 ^{\{ 1 \}} | 0.13535 ^{\{ 12 \}} | 0.11105 ^{\{ 6 \}} | 0.11259 ^{\{ 7 \}} |
BIAS( \hat{\beta} ) | 0.50725 ^{\{ 5 \}} | 0.55301 ^{\{ 7 \}} | 0.6035 ^{\{ 12 \}} | 0.58515 ^{\{ 9 \}} | 0.5899 ^{\{ 11 \}} | 0.64435 ^{\{ 15 \}} | 0.60784 ^{\{ 14 \}} | 0.58714 ^{\{ 10 \}} | 0.23162 ^{\{ 2 \}} | 0.5467 ^{\{ 6 \}} | 0.60699 ^{\{ 13 \}} | 0.03925 ^{\{ 1 \}} | 0.49202 ^{\{ 4 \}} | 0.56928 ^{\{ 8 \}} | 0.4335 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.01774 ^{\{ 4 \}} | 0.02332 ^{\{ 9 \}} | 0.03276 ^{\{ 14 \}} | 0.01729 ^{\{ 3 \}} | 0.0283 ^{\{ 12 \}} | 0.04047 ^{\{ 15 \}} | 0.02684 ^{\{ 11 \}} | 0.0214 ^{\{ 7 \}} | 0.01089 ^{\{ 2 \}} | 0.01838 ^{\{ 5 \}} | 0.02516 ^{\{ 10 \}} | 0.00304 ^{\{ 1 \}} | 0.03231 ^{\{ 13 \}} | 0.02003 ^{\{ 6 \}} | 0.02258 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.41751 ^{\{ 5 \}} | 0.47226 ^{\{ 6 \}} | 0.53147 ^{\{ 10 \}} | 0.56899 ^{\{ 14 \}} | 0.50961 ^{\{ 8 \}} | 0.58852 ^{\{ 15 \}} | 0.55385 ^{\{ 12 \}} | 0.54235 ^{\{ 11 \}} | 0.18176 ^{\{ 2 \}} | 0.50391 ^{\{ 7 \}} | 0.56276 ^{\{ 13 \}} | 0.00761 ^{\{ 1 \}} | 0.36579 ^{\{ 4 \}} | 0.51548 ^{\{ 9 \}} | 0.28438 ^{\{ 3 \}} | |
MRE( \hat{\delta} ) | 0.21318 ^{\{ 3 \}} | 0.24344 ^{\{ 9 \}} | 0.28743 ^{\{ 14 \}} | 0.21432 ^{\{ 4 \}} | 0.27239 ^{\{ 13 \}} | 0.32609 ^{\{ 15 \}} | 0.26168 ^{\{ 11 \}} | 0.23398 ^{\{ 8 \}} | 0.14585 ^{\{ 2 \}} | 0.21505 ^{\{ 5 \}} | 0.25063 ^{\{ 10 \}} | 0.08496 ^{\{ 1 \}} | 0.27071 ^{\{ 12 \}} | 0.22211 ^{\{ 6 \}} | 0.22518 ^{\{ 7 \}} | |
MRE( \hat{\beta} ) | 0.25363 ^{\{ 5 \}} | 0.27651 ^{\{ 7 \}} | 0.30175 ^{\{ 12 \}} | 0.29257 ^{\{ 9 \}} | 0.29495 ^{\{ 11 \}} | 0.32217 ^{\{ 15 \}} | 0.30392 ^{\{ 14 \}} | 0.29357 ^{\{ 10 \}} | 0.11581 ^{\{ 2 \}} | 0.27335 ^{\{ 6 \}} | 0.30349 ^{\{ 13 \}} | 0.01962 ^{\{ 1 \}} | 0.24601 ^{\{ 4 \}} | 0.28464 ^{\{ 8 \}} | 0.21675 ^{\{ 3 \}} | |
D_{abs} | 0.02879 ^{\{ 1 \}} | 0.02974 ^{\{ 2 \}} | 0.03169 ^{\{ 9 \}} | 0.03041 ^{\{ 4 \}} | 0.03128 ^{\{ 7 \}} | 0.03293 ^{\{ 13 \}} | 0.03119 ^{\{ 6 \}} | 0.03093 ^{\{ 5 \}} | 0.03153 ^{\{ 8 \}} | 0.03238 ^{\{ 12 \}} | 0.03207 ^{\{ 10 \}} | 0.03029 ^{\{ 3 \}} | 0.04427 ^{\{ 15 \}} | 0.03231 ^{\{ 11 \}} | 0.04117 ^{\{ 14 \}} | |
D_{max} | 0.04743 ^{\{ 2 \}} | 0.04934 ^{\{ 4 \}} | 0.05375 ^{\{ 12 \}} | 0.0493 ^{\{ 3 \}} | 0.05256 ^{\{ 10 \}} | 0.0566 ^{\{ 13 \}} | 0.05167 ^{\{ 7 \}} | 0.05037 ^{\{ 6 \}} | 0.04949 ^{\{ 5 \}} | 0.05255 ^{\{ 9 \}} | 0.05272 ^{\{ 11 \}} | 0.04604 ^{\{ 1 \}} | 0.06992 ^{\{ 15 \}} | 0.05183 ^{\{ 8 \}} | 0.06489 ^{\{ 14 \}} | |
\sum Ranks | 28 ^{\{ 3 \}} | 53 ^{\{ 5 \}} | 97 ^{\{ 14 \}} | 50 ^{\{ 4 \}} | 85 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 86 ^{\{ 12 \}} | 65 ^{\{ 9 \}} | 25 ^{\{ 2 \}} | 55 ^{\{ 6 \}} | 90 ^{\{ 13 \}} | 10 ^{\{ 1 \}} | 79 ^{\{ 10 \}} | 62 ^{\{ 8 \}} | 59 ^{\{ 7 \}} | |
100 | BIAS( \hat{\delta} ) | 0.08286 ^{\{ 3 \}} | 0.10295 ^{\{ 11 \}} | 0.11725 ^{\{ 14 \}} | 0.08874 ^{\{ 4 \}} | 0.11484 ^{\{ 13 \}} | 0.14099 ^{\{ 15 \}} | 0.10506 ^{\{ 12 \}} | 0.09668 ^{\{ 9 \}} | 0.06209 ^{\{ 2 \}} | 0.09521 ^{\{ 8 \}} | 0.10061 ^{\{ 10 \}} | 0.0325 ^{\{ 1 \}} | 0.09485 ^{\{ 7 \}} | 0.09062 ^{\{ 6 \}} | 0.09056 ^{\{ 5 \}} |
BIAS( \hat{\beta} ) | 0.41544 ^{\{ 5 \}} | 0.51195 ^{\{ 9 \}} | 0.51848 ^{\{ 11 \}} | 0.50035 ^{\{ 7 \}} | 0.55605 ^{\{ 14 \}} | 0.57985 ^{\{ 15 \}} | 0.51445 ^{\{ 10 \}} | 0.48798 ^{\{ 6 \}} | 0.22274 ^{\{ 2 \}} | 0.52223 ^{\{ 12 \}} | 0.52272 ^{\{ 13 \}} | 0.03493 ^{\{ 1 \}} | 0.36852 ^{\{ 3 \}} | 0.5066 ^{\{ 8 \}} | 0.392 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.01073 ^{\{ 3 \}} | 0.01683 ^{\{ 10 \}} | 0.02226 ^{\{ 14 \}} | 0.01217 ^{\{ 4 \}} | 0.02079 ^{\{ 13 \}} | 0.03092 ^{\{ 15 \}} | 0.01737 ^{\{ 12 \}} | 0.01473 ^{\{ 7 \}} | 0.00763 ^{\{ 2 \}} | 0.01427 ^{\{ 6 \}} | 0.01587 ^{\{ 9 \}} | 0.00182 ^{\{ 1 \}} | 0.0169 ^{\{ 11 \}} | 0.0124 ^{\{ 5 \}} | 0.01525 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.29348 ^{\{ 5 \}} | 0.42649 ^{\{ 9 \}} | 0.43143 ^{\{ 11 \}} | 0.42826 ^{\{ 10 \}} | 0.49967 ^{\{ 14 \}} | 0.51432 ^{\{ 15 \}} | 0.42161 ^{\{ 8 \}} | 0.38818 ^{\{ 6 \}} | 0.15605 ^{\{ 2 \}} | 0.47725 ^{\{ 13 \}} | 0.43798 ^{\{ 12 \}} | 0.0066 ^{\{ 1 \}} | 0.21342 ^{\{ 3 \}} | 0.41474 ^{\{ 7 \}} | 0.23478 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.16573 ^{\{ 3 \}} | 0.2059 ^{\{ 11 \}} | 0.23451 ^{\{ 14 \}} | 0.17748 ^{\{ 4 \}} | 0.22968 ^{\{ 13 \}} | 0.28198 ^{\{ 15 \}} | 0.21011 ^{\{ 12 \}} | 0.19336 ^{\{ 9 \}} | 0.12419 ^{\{ 2 \}} | 0.19042 ^{\{ 8 \}} | 0.20121 ^{\{ 10 \}} | 0.065 ^{\{ 1 \}} | 0.18971 ^{\{ 7 \}} | 0.18125 ^{\{ 6 \}} | 0.18112 ^{\{ 5 \}} | |
MRE( \hat{\beta} ) | 0.20772 ^{\{ 5 \}} | 0.25598 ^{\{ 9 \}} | 0.25924 ^{\{ 11 \}} | 0.25018 ^{\{ 7 \}} | 0.27802 ^{\{ 14 \}} | 0.28993 ^{\{ 15 \}} | 0.25722 ^{\{ 10 \}} | 0.24399 ^{\{ 6 \}} | 0.11137 ^{\{ 2 \}} | 0.26111 ^{\{ 12 \}} | 0.26136 ^{\{ 13 \}} | 0.01746 ^{\{ 1 \}} | 0.18426 ^{\{ 3 \}} | 0.2533 ^{\{ 8 \}} | 0.196 ^{\{ 4 \}} | |
D_{abs} | 0.02231 ^{\{ 1 \}} | 0.02454 ^{\{ 6 \}} | 0.02511 ^{\{ 9 \}} | 0.02269 ^{\{ 3 \}} | 0.02562 ^{\{ 10 \}} | 0.02595 ^{\{ 12 \}} | 0.02449 ^{\{ 5 \}} | 0.02497 ^{\{ 7 \}} | 0.02396 ^{\{ 4 \}} | 0.02574 ^{\{ 11 \}} | 0.02509 ^{\{ 8 \}} | 0.02261 ^{\{ 2 \}} | 0.03104 ^{\{ 14 \}} | 0.02611 ^{\{ 13 \}} | 0.03167 ^{\{ 15 \}} | |
D_{max} | 0.03664 ^{\{ 2 \}} | 0.04054 ^{\{ 5.5 \}} | 0.0426 ^{\{ 11 \}} | 0.03686 ^{\{ 3 \}} | 0.04306 ^{\{ 12 \}} | 0.04533 ^{\{ 13 \}} | 0.04054 ^{\{ 5.5 \}} | 0.04099 ^{\{ 7 \}} | 0.03813 ^{\{ 4 \}} | 0.04213 ^{\{ 10 \}} | 0.04108 ^{\{ 8 \}} | 0.03463 ^{\{ 1 \}} | 0.0495 ^{\{ 14 \}} | 0.04206 ^{\{ 9 \}} | 0.05085 ^{\{ 15 \}} | |
\sum Ranks | 27 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 95 ^{\{ 13 \}} | 42 ^{\{ 4 \}} | 103 ^{\{ 14 \}} | 115 ^{\{ 15 \}} | 74.5 ^{\{ 10 \}} | 57 ^{\{ 5 \}} | 20 ^{\{ 2 \}} | 80 ^{\{ 11 \}} | 83 ^{\{ 12 \}} | 9 ^{\{ 1 \}} | 62 ^{\{ 7.5 \}} | 62 ^{\{ 7.5 \}} | 60 ^{\{ 6 \}} | |
200 | BIAS( \hat{\delta} ) | 0.06116 ^{\{ 3 \}} | 0.07841 ^{\{ 11 \}} | 0.08834 ^{\{ 13 \}} | 0.06582 ^{\{ 5 \}} | 0.08967 ^{\{ 14 \}} | 0.10318 ^{\{ 15 \}} | 0.07657 ^{\{ 10 \}} | 0.0701 ^{\{ 7 \}} | 0.05022 ^{\{ 2 \}} | 0.07169 ^{\{ 8 \}} | 0.0803 ^{\{ 12 \}} | 0.02285 ^{\{ 1 \}} | 0.06434 ^{\{ 4 \}} | 0.07255 ^{\{ 9 \}} | 0.06584 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.32425 ^{\{ 5 \}} | 0.40338 ^{\{ 10 \}} | 0.42002 ^{\{ 12 \}} | 0.37519 ^{\{ 7 \}} | 0.44639 ^{\{ 13 \}} | 0.47646 ^{\{ 15 \}} | 0.38899 ^{\{ 8 \}} | 0.36465 ^{\{ 6 \}} | 0.20622 ^{\{ 2 \}} | 0.39983 ^{\{ 9 \}} | 0.45097 ^{\{ 14 \}} | 0.02978 ^{\{ 1 \}} | 0.309 ^{\{ 3 \}} | 0.40885 ^{\{ 11 \}} | 0.32061 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.00603 ^{\{ 3 \}} | 0.0094 ^{\{ 11 \}} | 0.01227 ^{\{ 14 \}} | 0.00684 ^{\{ 4 \}} | 0.0121 ^{\{ 13 \}} | 0.01636 ^{\{ 15 \}} | 0.00916 ^{\{ 10 \}} | 0.00762 ^{\{ 6 \}} | 0.00531 ^{\{ 2 \}} | 0.00804 ^{\{ 8 \}} | 0.0099 ^{\{ 12 \}} | 0.00084 ^{\{ 1 \}} | 0.00725 ^{\{ 5 \}} | 0.00822 ^{\{ 9 \}} | 0.00789 ^{\{ 7 \}} | |
MSE( \hat{\beta} ) | 0.18213 ^{\{ 4 \}} | 0.27797 ^{\{ 9 \}} | 0.29886 ^{\{ 12 \}} | 0.25897 ^{\{ 8 \}} | 0.34454 ^{\{ 14 \}} | 0.37648 ^{\{ 15 \}} | 0.2585 ^{\{ 7 \}} | 0.23044 ^{\{ 6 \}} | 0.13188 ^{\{ 2 \}} | 0.2854 ^{\{ 10 \}} | 0.34418 ^{\{ 13 \}} | 0.00461 ^{\{ 1 \}} | 0.1536 ^{\{ 3 \}} | 0.29853 ^{\{ 11 \}} | 0.18333 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.12232 ^{\{ 3 \}} | 0.15682 ^{\{ 11 \}} | 0.17668 ^{\{ 13 \}} | 0.13164 ^{\{ 5 \}} | 0.17934 ^{\{ 14 \}} | 0.20636 ^{\{ 15 \}} | 0.15314 ^{\{ 10 \}} | 0.1402 ^{\{ 7 \}} | 0.10044 ^{\{ 2 \}} | 0.14338 ^{\{ 8 \}} | 0.16059 ^{\{ 12 \}} | 0.0457 ^{\{ 1 \}} | 0.12868 ^{\{ 4 \}} | 0.14509 ^{\{ 9 \}} | 0.13169 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.16212 ^{\{ 5 \}} | 0.20169 ^{\{ 10 \}} | 0.21001 ^{\{ 12 \}} | 0.18759 ^{\{ 7 \}} | 0.2232 ^{\{ 13 \}} | 0.23823 ^{\{ 15 \}} | 0.1945 ^{\{ 8 \}} | 0.18232 ^{\{ 6 \}} | 0.10311 ^{\{ 2 \}} | 0.19991 ^{\{ 9 \}} | 0.22548 ^{\{ 14 \}} | 0.01489 ^{\{ 1 \}} | 0.1545 ^{\{ 3 \}} | 0.20442 ^{\{ 11 \}} | 0.16031 ^{\{ 4 \}} | |
D_{abs} | 0.01724 ^{\{ 2 \}} | 0.01806 ^{\{ 7 \}} | 0.01844 ^{\{ 10 \}} | 0.01741 ^{\{ 3 \}} | 0.01838 ^{\{ 8 \}} | 0.01843 ^{\{ 9 \}} | 0.01779 ^{\{ 4 \}} | 0.01797 ^{\{ 6 \}} | 0.01789 ^{\{ 5 \}} | 0.02027 ^{\{ 13 \}} | 0.01959 ^{\{ 12 \}} | 0.01648 ^{\{ 1 \}} | 0.02114 ^{\{ 14 \}} | 0.0194 ^{\{ 11 \}} | 0.02249 ^{\{ 15 \}} | |
D_{max} | 0.02791 ^{\{ 2 \}} | 0.0299 ^{\{ 7 \}} | 0.03118 ^{\{ 9 \}} | 0.02827 ^{\{ 3 \}} | 0.03116 ^{\{ 8 \}} | 0.03221 ^{\{ 12 \}} | 0.02954 ^{\{ 6 \}} | 0.0293 ^{\{ 5 \}} | 0.02849 ^{\{ 4 \}} | 0.03291 ^{\{ 13 \}} | 0.03189 ^{\{ 11 \}} | 0.02515 ^{\{ 1 \}} | 0.03433 ^{\{ 14 \}} | 0.03138 ^{\{ 10 \}} | 0.03655 ^{\{ 15 \}} | |
\sum Ranks | 27 ^{\{ 3 \}} | 76 ^{\{ 9 \}} | 95 ^{\{ 12 \}} | 42 ^{\{ 4 \}} | 97 ^{\{ 13 \}} | 111 ^{\{ 15 \}} | 63 ^{\{ 8 \}} | 49 ^{\{ 5 \}} | 21 ^{\{ 2 \}} | 78 ^{\{ 10 \}} | 100 ^{\{ 14 \}} | 8 ^{\{ 1 \}} | 50 ^{\{ 6 \}} | 81 ^{\{ 11 \}} | 62 ^{\{ 7 \}} | |
300 | BIAS( \hat{\delta} ) | 0.04976 ^{\{ 3 \}} | 0.063 ^{\{ 10 \}} | 0.07722 ^{\{ 14 \}} | 0.05537 ^{\{ 5 \}} | 0.07222 ^{\{ 13 \}} | 0.08992 ^{\{ 15 \}} | 0.06346 ^{\{ 11 \}} | 0.05881 ^{\{ 7 \}} | 0.04405 ^{\{ 2 \}} | 0.06157 ^{\{ 9 \}} | 0.06661 ^{\{ 12 \}} | 0.01923 ^{\{ 1 \}} | 0.05374 ^{\{ 4 \}} | 0.05954 ^{\{ 8 \}} | 0.05577 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.26113 ^{\{ 3 \}} | 0.32348 ^{\{ 9 \}} | 0.37643 ^{\{ 14 \}} | 0.30537 ^{\{ 6 \}} | 0.36663 ^{\{ 12 \}} | 0.42261 ^{\{ 15 \}} | 0.32101 ^{\{ 8 \}} | 0.30553 ^{\{ 7 \}} | 0.18919 ^{\{ 2 \}} | 0.33927 ^{\{ 11 \}} | 0.37228 ^{\{ 13 \}} | 0.02946 ^{\{ 1 \}} | 0.27025 ^{\{ 5 \}} | 0.32572 ^{\{ 10 \}} | 0.2684 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.00405 ^{\{ 3 \}} | 0.00614 ^{\{ 10 \}} | 0.00941 ^{\{ 14 \}} | 0.00478 ^{\{ 4 \}} | 0.00824 ^{\{ 13 \}} | 0.01248 ^{\{ 15 \}} | 0.0063 ^{\{ 11 \}} | 0.00566 ^{\{ 7 \}} | 0.00375 ^{\{ 2 \}} | 0.00597 ^{\{ 9 \}} | 0.00674 ^{\{ 12 \}} | 0.00063 ^{\{ 1 \}} | 0.00503 ^{\{ 5 \}} | 0.00548 ^{\{ 6 \}} | 0.00588 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.12154 ^{\{ 3 \}} | 0.17435 ^{\{ 8 \}} | 0.24526 ^{\{ 14 \}} | 0.16177 ^{\{ 7 \}} | 0.2386 ^{\{ 13 \}} | 0.30176 ^{\{ 15 \}} | 0.17584 ^{\{ 9 \}} | 0.15982 ^{\{ 6 \}} | 0.10377 ^{\{ 2 \}} | 0.20948 ^{\{ 11 \}} | 0.23664 ^{\{ 12 \}} | 0.00414 ^{\{ 1 \}} | 0.13235 ^{\{ 4 \}} | 0.18403 ^{\{ 10 \}} | 0.14778 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.09951 ^{\{ 3 \}} | 0.12601 ^{\{ 10 \}} | 0.15444 ^{\{ 14 \}} | 0.11075 ^{\{ 5 \}} | 0.14445 ^{\{ 13 \}} | 0.17985 ^{\{ 15 \}} | 0.12692 ^{\{ 11 \}} | 0.11761 ^{\{ 7 \}} | 0.0881 ^{\{ 2 \}} | 0.12314 ^{\{ 9 \}} | 0.13323 ^{\{ 12 \}} | 0.03846 ^{\{ 1 \}} | 0.10748 ^{\{ 4 \}} | 0.11909 ^{\{ 8 \}} | 0.11153 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.13057 ^{\{ 3 \}} | 0.16174 ^{\{ 9 \}} | 0.18822 ^{\{ 14 \}} | 0.15269 ^{\{ 6 \}} | 0.18332 ^{\{ 12 \}} | 0.2113 ^{\{ 15 \}} | 0.1605 ^{\{ 8 \}} | 0.15276 ^{\{ 7 \}} | 0.09459 ^{\{ 2 \}} | 0.16963 ^{\{ 11 \}} | 0.18614 ^{\{ 13 \}} | 0.01473 ^{\{ 1 \}} | 0.13513 ^{\{ 5 \}} | 0.16286 ^{\{ 10 \}} | 0.1342 ^{\{ 4 \}} | |
D_{abs} | 0.01433 ^{\{ 2 \}} | 0.01475 ^{\{ 5.5 \}} | 0.0156 ^{\{ 10 \}} | 0.01439 ^{\{ 3 \}} | 0.01527 ^{\{ 7 \}} | 0.01648 ^{\{ 12 \}} | 0.01475 ^{\{ 5.5 \}} | 0.01471 ^{\{ 4 \}} | 0.01548 ^{\{ 8 \}} | 0.01669 ^{\{ 13 \}} | 0.01557 ^{\{ 9 \}} | 0.01377 ^{\{ 1 \}} | 0.01746 ^{\{ 14 \}} | 0.01569 ^{\{ 11 \}} | 0.01762 ^{\{ 15 \}} | |
D_{max} | 0.02313 ^{\{ 2 \}} | 0.02441 ^{\{ 5 \}} | 0.02642 ^{\{ 11 \}} | 0.0234 ^{\{ 3 \}} | 0.02566 ^{\{ 10 \}} | 0.02851 ^{\{ 14 \}} | 0.02444 ^{\{ 6 \}} | 0.02405 ^{\{ 4 \}} | 0.02467 ^{\{ 7 \}} | 0.02709 ^{\{ 12 \}} | 0.02545 ^{\{ 8 \}} | 0.02099 ^{\{ 1 \}} | 0.02842 ^{\{ 13 \}} | 0.02547 ^{\{ 9 \}} | 0.02869 ^{\{ 15 \}} | |
\sum Ranks | 22 ^{\{ 2 \}} | 66.5 ^{\{ 8 \}} | 105 ^{\{ 14 \}} | 39 ^{\{ 4 \}} | 93 ^{\{ 13 \}} | 116 ^{\{ 15 \}} | 69.5 ^{\{ 9 \}} | 49 ^{\{ 5 \}} | 27 ^{\{ 3 \}} | 85 ^{\{ 11 \}} | 91 ^{\{ 12 \}} | 8 ^{\{ 1 \}} | 54 ^{\{ 6 \}} | 72 ^{\{ 10 \}} | 63 ^{\{ 7 \}} | |
400 | BIAS( \hat{\delta} ) | 0.04586 ^{\{ 3 \}} | 0.05457 ^{\{ 10 \}} | 0.06718 ^{\{ 14 \}} | 0.04629 ^{\{ 4 \}} | 0.06491 ^{\{ 13 \}} | 0.07564 ^{\{ 15 \}} | 0.05516 ^{\{ 11 \}} | 0.05045 ^{\{ 8 \}} | 0.04056 ^{\{ 2 \}} | 0.05384 ^{\{ 9 \}} | 0.05857 ^{\{ 12 \}} | 0.01681 ^{\{ 1 \}} | 0.04911 ^{\{ 5 \}} | 0.0504 ^{\{ 7 \}} | 0.04983 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.24121 ^{\{ 4 \}} | 0.2717 ^{\{ 9 \}} | 0.32822 ^{\{ 14 \}} | 0.25116 ^{\{ 6 \}} | 0.31726 ^{\{ 12 \}} | 0.35759 ^{\{ 15 \}} | 0.2792 ^{\{ 10 \}} | 0.25723 ^{\{ 7 \}} | 0.17791 ^{\{ 2 \}} | 0.29726 ^{\{ 11 \}} | 0.3189 ^{\{ 13 \}} | 0.02741 ^{\{ 1 \}} | 0.2501 ^{\{ 5 \}} | 0.27113 ^{\{ 8 \}} | 0.23703 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.00335 ^{\{ 4 \}} | 0.00479 ^{\{ 10 \}} | 0.00713 ^{\{ 14 \}} | 0.00334 ^{\{ 3 \}} | 0.00654 ^{\{ 13 \}} | 0.00909 ^{\{ 15 \}} | 0.0047 ^{\{ 9 \}} | 0.00393 ^{\{ 5 \}} | 0.00319 ^{\{ 2 \}} | 0.00461 ^{\{ 8 \}} | 0.00534 ^{\{ 12 \}} | 0.00051 ^{\{ 1 \}} | 0.00433 ^{\{ 7 \}} | 0.00397 ^{\{ 6 \}} | 0.00484 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.09651 ^{\{ 3 \}} | 0.12678 ^{\{ 8 \}} | 0.18883 ^{\{ 14 \}} | 0.10551 ^{\{ 4 \}} | 0.17475 ^{\{ 13 \}} | 0.22525 ^{\{ 15 \}} | 0.12978 ^{\{ 10 \}} | 0.11072 ^{\{ 5 \}} | 0.08312 ^{\{ 2 \}} | 0.15568 ^{\{ 11 \}} | 0.17447 ^{\{ 12 \}} | 0.00395 ^{\{ 1 \}} | 0.12704 ^{\{ 9 \}} | 0.12604 ^{\{ 7 \}} | 0.11073 ^{\{ 6 \}} | |
MRE( \hat{\delta} ) | 0.09172 ^{\{ 3 \}} | 0.10915 ^{\{ 10 \}} | 0.13437 ^{\{ 14 \}} | 0.09257 ^{\{ 4 \}} | 0.12982 ^{\{ 13 \}} | 0.15127 ^{\{ 15 \}} | 0.11032 ^{\{ 11 \}} | 0.10089 ^{\{ 8 \}} | 0.08112 ^{\{ 2 \}} | 0.10768 ^{\{ 9 \}} | 0.11714 ^{\{ 12 \}} | 0.03363 ^{\{ 1 \}} | 0.09822 ^{\{ 5 \}} | 0.10079 ^{\{ 7 \}} | 0.09966 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.12061 ^{\{ 4 \}} | 0.13585 ^{\{ 9 \}} | 0.16411 ^{\{ 14 \}} | 0.12558 ^{\{ 6 \}} | 0.15863 ^{\{ 12 \}} | 0.1788 ^{\{ 15 \}} | 0.1396 ^{\{ 10 \}} | 0.12862 ^{\{ 7 \}} | 0.08896 ^{\{ 2 \}} | 0.14863 ^{\{ 11 \}} | 0.15945 ^{\{ 13 \}} | 0.01371 ^{\{ 1 \}} | 0.12505 ^{\{ 5 \}} | 0.13556 ^{\{ 8 \}} | 0.11852 ^{\{ 3 \}} | |
D_{abs} | 0.01235 ^{\{ 3 \}} | 0.01251 ^{\{ 4 \}} | 0.01316 ^{\{ 7 \}} | 0.01191 ^{\{ 2 \}} | 0.0133 ^{\{ 10 \}} | 0.01357 ^{\{ 11 \}} | 0.01283 ^{\{ 6 \}} | 0.01262 ^{\{ 5 \}} | 0.01327 ^{\{ 9 \}} | 0.01469 ^{\{ 13 \}} | 0.01384 ^{\{ 12 \}} | 0.0118 ^{\{ 1 \}} | 0.01612 ^{\{ 15 \}} | 0.01319 ^{\{ 8 \}} | 0.01595 ^{\{ 14 \}} | |
D_{max} | 0.02003 ^{\{ 3 \}} | 0.02068 ^{\{ 5 \}} | 0.02249 ^{\{ 10 \}} | 0.0194 ^{\{ 2 \}} | 0.02245 ^{\{ 9 \}} | 0.02345 ^{\{ 12 \}} | 0.0212 ^{\{ 6 \}} | 0.02064 ^{\{ 4 \}} | 0.02127 ^{\{ 7 \}} | 0.02382 ^{\{ 13 \}} | 0.02263 ^{\{ 11 \}} | 0.01806 ^{\{ 1 \}} | 0.02613 ^{\{ 15 \}} | 0.0215 ^{\{ 8 \}} | 0.02592 ^{\{ 14 \}} | |
\sum Ranks | 27 ^{\{ 2 \}} | 65 ^{\{ 8 \}} | 101 ^{\{ 14 \}} | 31 ^{\{ 4 \}} | 95 ^{\{ 12 \}} | 113 ^{\{ 15 \}} | 73 ^{\{ 10 \}} | 49 ^{\{ 5 \}} | 28 ^{\{ 3 \}} | 85 ^{\{ 11 \}} | 97 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 66 ^{\{ 9 \}} | 59 ^{\{ 6 \}} | 63 ^{\{ 7 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.24947 ^{\{ 2 \}} | 0.39247 ^{\{ 8 \}} | 0.45109 ^{\{ 13 \}} | 0.34898 ^{\{ 4 \}} | 0.4092 ^{\{ 10 \}} | 0.51093 ^{\{ 15 \}} | 0.3875 ^{\{ 7 \}} | 0.40032 ^{\{ 9 \}} | 0.2568 ^{\{ 3 \}} | 0.36632 ^{\{ 6 \}} | 0.47352 ^{\{ 14 \}} | 0.0773 ^{\{ 1 \}} | 0.42866 ^{\{ 12 \}} | 0.3597 ^{\{ 5 \}} | 0.40978 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.10849 ^{\{ 3 \}} | 0.12312 ^{\{ 5 \}} | 0.12419 ^{\{ 6 \}} | 0.12427 ^{\{ 7 \}} | 0.12917 ^{\{ 11 \}} | 0.13824 ^{\{ 12 \}} | 0.12135 ^{\{ 4 \}} | 0.12589 ^{\{ 8 \}} | 0.10086 ^{\{ 2 \}} | 0.12906 ^{\{ 10 \}} | 0.14169 ^{\{ 15 \}} | 0.05665 ^{\{ 1 \}} | 0.14068 ^{\{ 13 \}} | 0.12644 ^{\{ 9 \}} | 0.14073 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.1048 ^{\{ 2 \}} | 0.25062 ^{\{ 8 \}} | 0.35457 ^{\{ 13 \}} | 0.19139 ^{\{ 4 \}} | 0.26462 ^{\{ 10 \}} | 0.44465 ^{\{ 15 \}} | 0.24711 ^{\{ 7 \}} | 0.26728 ^{\{ 11 \}} | 0.13483 ^{\{ 3 \}} | 0.20392 ^{\{ 6 \}} | 0.39226 ^{\{ 14 \}} | 0.02074 ^{\{ 1 \}} | 0.27898 ^{\{ 12 \}} | 0.19815 ^{\{ 5 \}} | 0.25293 ^{\{ 9 \}} | |
MSE( \hat{\beta} ) | 0.02079 ^{\{ 3 \}} | 0.02301 ^{\{ 6 \}} | 0.02292 ^{\{ 5 \}} | 0.02435 ^{\{ 8 \}} | 0.0252 ^{\{ 10 \}} | 0.02791 ^{\{ 12 \}} | 0.02257 ^{\{ 4 \}} | 0.02416 ^{\{ 7 \}} | 0.01873 ^{\{ 2 \}} | 0.02604 ^{\{ 11 \}} | 0.02917 ^{\{ 13 \}} | 0.00597 ^{\{ 1 \}} | 0.03018 ^{\{ 15 \}} | 0.02461 ^{\{ 9 \}} | 0.02959 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.09979 ^{\{ 2 \}} | 0.15699 ^{\{ 8 \}} | 0.18043 ^{\{ 13 \}} | 0.13959 ^{\{ 4 \}} | 0.16368 ^{\{ 10 \}} | 0.20437 ^{\{ 15 \}} | 0.155 ^{\{ 7 \}} | 0.16013 ^{\{ 9 \}} | 0.10272 ^{\{ 3 \}} | 0.14653 ^{\{ 6 \}} | 0.18941 ^{\{ 14 \}} | 0.03092 ^{\{ 1 \}} | 0.17146 ^{\{ 12 \}} | 0.14388 ^{\{ 5 \}} | 0.16391 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.27123 ^{\{ 3 \}} | 0.3078 ^{\{ 5 \}} | 0.31047 ^{\{ 6 \}} | 0.31068 ^{\{ 7 \}} | 0.32292 ^{\{ 11 \}} | 0.34559 ^{\{ 12 \}} | 0.30337 ^{\{ 4 \}} | 0.31471 ^{\{ 8 \}} | 0.25215 ^{\{ 2 \}} | 0.32265 ^{\{ 10 \}} | 0.35423 ^{\{ 15 \}} | 0.14162 ^{\{ 1 \}} | 0.3517 ^{\{ 13 \}} | 0.31611 ^{\{ 9 \}} | 0.35182 ^{\{ 14 \}} | |
D_{abs} | 0.04159 ^{\{ 1 \}} | 0.0451 ^{\{ 3 \}} | 0.0482 ^{\{ 9 \}} | 0.0454 ^{\{ 5 \}} | 0.04619 ^{\{ 6 \}} | 0.0507 ^{\{ 13 \}} | 0.04517 ^{\{ 4 \}} | 0.04721 ^{\{ 8 \}} | 0.04986 ^{\{ 11.5 \}} | 0.04986 ^{\{ 11.5 \}} | 0.04865 ^{\{ 10 \}} | 0.04264 ^{\{ 2 \}} | 0.05413 ^{\{ 15 \}} | 0.0466 ^{\{ 7 \}} | 0.05336 ^{\{ 14 \}} | |
D_{max} | 0.066 ^{\{ 2 \}} | 0.07344 ^{\{ 4 \}} | 0.08102 ^{\{ 12 \}} | 0.07193 ^{\{ 3 \}} | 0.076 ^{\{ 7 \}} | 0.08624 ^{\{ 14 \}} | 0.07378 ^{\{ 5 \}} | 0.07752 ^{\{ 9 \}} | 0.07682 ^{\{ 8 \}} | 0.07921 ^{\{ 10 \}} | 0.08099 ^{\{ 11 \}} | 0.06259 ^{\{ 1 \}} | 0.08762 ^{\{ 15 \}} | 0.07444 ^{\{ 6 \}} | 0.0855 ^{\{ 13 \}} | |
\sum Ranks | 18 ^{\{ 2 \}} | 47 ^{\{ 6 \}} | 77 ^{\{ 11 \}} | 42 ^{\{ 4.5 \}} | 75 ^{\{ 10 \}} | 108 ^{\{ 15 \}} | 42 ^{\{ 4.5 \}} | 69 ^{\{ 8 \}} | 34.5 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 106 ^{\{ 13 \}} | 9 ^{\{ 1 \}} | 107 ^{\{ 14 \}} | 55 ^{\{ 7 \}} | 100 ^{\{ 12 \}} | |
60 | BIAS( \hat{\delta} ) | 0.20453 ^{\{ 2 \}} | 0.2779 ^{\{ 6 \}} | 0.29563 ^{\{ 10 \}} | 0.27622 ^{\{ 5 \}} | 0.31346 ^{\{ 11 \}} | 0.37466 ^{\{ 15 \}} | 0.28173 ^{\{ 7 \}} | 0.28733 ^{\{ 9 \}} | 0.24988 ^{\{ 3 \}} | 0.28363 ^{\{ 8 \}} | 0.35262 ^{\{ 14 \}} | 0.07199 ^{\{ 1 \}} | 0.33443 ^{\{ 13 \}} | 0.27509 ^{\{ 4 \}} | 0.32222 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.08487 ^{\{ 2 \}} | 0.09474 ^{\{ 5 \}} | 0.09425 ^{\{ 3 \}} | 0.10053 ^{\{ 9 \}} | 0.09961 ^{\{ 8 \}} | 0.10752 ^{\{ 12 \}} | 0.09656 ^{\{ 7 \}} | 0.09498 ^{\{ 6 \}} | 0.09445 ^{\{ 4 \}} | 0.10474 ^{\{ 11 \}} | 0.11801 ^{\{ 13 \}} | 0.04214 ^{\{ 1 \}} | 0.11894 ^{\{ 14 \}} | 0.10253 ^{\{ 10 \}} | 0.1193 ^{\{ 15 \}} | |
MSE( \hat{\delta} ) | 0.07403 ^{\{ 2 \}} | 0.12349 ^{\{ 6 \}} | 0.14155 ^{\{ 10 \}} | 0.11748 ^{\{ 4 \}} | 0.15224 ^{\{ 11 \}} | 0.22331 ^{\{ 15 \}} | 0.12956 ^{\{ 8 \}} | 0.13518 ^{\{ 9 \}} | 0.11795 ^{\{ 5 \}} | 0.1257 ^{\{ 7 \}} | 0.20342 ^{\{ 14 \}} | 0.01871 ^{\{ 1 \}} | 0.17116 ^{\{ 13 \}} | 0.1157 ^{\{ 3 \}} | 0.15442 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.01313 ^{\{ 2 \}} | 0.01465 ^{\{ 5 \}} | 0.0141 ^{\{ 3 \}} | 0.01638 ^{\{ 9 \}} | 0.01589 ^{\{ 8 \}} | 0.01792 ^{\{ 11 \}} | 0.01554 ^{\{ 7 \}} | 0.01446 ^{\{ 4 \}} | 0.01553 ^{\{ 6 \}} | 0.01793 ^{\{ 12 \}} | 0.0212 ^{\{ 13 \}} | 0.00382 ^{\{ 1 \}} | 0.02299 ^{\{ 15 \}} | 0.01724 ^{\{ 10 \}} | 0.02298 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.08181 ^{\{ 2 \}} | 0.11116 ^{\{ 6 \}} | 0.11825 ^{\{ 10 \}} | 0.11049 ^{\{ 5 \}} | 0.12538 ^{\{ 11 \}} | 0.14986 ^{\{ 15 \}} | 0.11269 ^{\{ 7 \}} | 0.11493 ^{\{ 9 \}} | 0.09995 ^{\{ 3 \}} | 0.11345 ^{\{ 8 \}} | 0.14105 ^{\{ 14 \}} | 0.0288 ^{\{ 1 \}} | 0.13377 ^{\{ 13 \}} | 0.11003 ^{\{ 4 \}} | 0.12889 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.21217 ^{\{ 2 \}} | 0.23684 ^{\{ 5 \}} | 0.23563 ^{\{ 3 \}} | 0.25133 ^{\{ 9 \}} | 0.24903 ^{\{ 8 \}} | 0.26879 ^{\{ 12 \}} | 0.24139 ^{\{ 7 \}} | 0.23744 ^{\{ 6 \}} | 0.23612 ^{\{ 4 \}} | 0.26185 ^{\{ 11 \}} | 0.29503 ^{\{ 13 \}} | 0.10535 ^{\{ 1 \}} | 0.29735 ^{\{ 14 \}} | 0.25632 ^{\{ 10 \}} | 0.29825 ^{\{ 15 \}} | |
D_{abs} | 0.03306 ^{\{ 3 \}} | 0.0336 ^{\{ 6 \}} | 0.03402 ^{\{ 7 \}} | 0.03258 ^{\{ 2 \}} | 0.03344 ^{\{ 5 \}} | 0.03621 ^{\{ 9 \}} | 0.03435 ^{\{ 8 \}} | 0.03312 ^{\{ 4 \}} | 0.03754 ^{\{ 12 \}} | 0.03649 ^{\{ 10 \}} | 0.03851 ^{\{ 13 \}} | 0.02949 ^{\{ 1 \}} | 0.03966 ^{\{ 14 \}} | 0.0365 ^{\{ 11 \}} | 0.04037 ^{\{ 15 \}} | |
D_{max} | 0.05248 ^{\{ 2 \}} | 0.05501 ^{\{ 5 \}} | 0.05639 ^{\{ 8 \}} | 0.05259 ^{\{ 3 \}} | 0.05553 ^{\{ 6 \}} | 0.06191 ^{\{ 12 \}} | 0.05603 ^{\{ 7 \}} | 0.05434 ^{\{ 4 \}} | 0.05941 ^{\{ 11 \}} | 0.05857 ^{\{ 10 \}} | 0.06353 ^{\{ 13 \}} | 0.04401 ^{\{ 1 \}} | 0.06477 ^{\{ 14 \}} | 0.05849 ^{\{ 9 \}} | 0.06561 ^{\{ 15 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 44 ^{\{ 3 \}} | 54 ^{\{ 7 \}} | 46 ^{\{ 4 \}} | 68 ^{\{ 10 \}} | 101 ^{\{ 12 \}} | 58 ^{\{ 8 \}} | 51 ^{\{ 6 \}} | 48 ^{\{ 5 \}} | 77 ^{\{ 11 \}} | 107 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 110 ^{\{ 14.5 \}} | 61 ^{\{ 9 \}} | 110 ^{\{ 14.5 \}} | |
100 | BIAS( \hat{\delta} ) | 0.16869 ^{\{ 2 \}} | 0.21955 ^{\{ 6 \}} | 0.25099 ^{\{ 11 \}} | 0.20658 ^{\{ 4 \}} | 0.23978 ^{\{ 10 \}} | 0.26936 ^{\{ 13 \}} | 0.22407 ^{\{ 7 \}} | 0.21601 ^{\{ 5 \}} | 0.20488 ^{\{ 3 \}} | 0.23686 ^{\{ 9 \}} | 0.2921 ^{\{ 15 \}} | 0.06504 ^{\{ 1 \}} | 0.27867 ^{\{ 14 \}} | 0.23071 ^{\{ 8 \}} | 0.26579 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.06441 ^{\{ 2 \}} | 0.0751 ^{\{ 5 \}} | 0.08048 ^{\{ 8 \}} | 0.07758 ^{\{ 6 \}} | 0.08158 ^{\{ 9 \}} | 0.08422 ^{\{ 10 \}} | 0.07438 ^{\{ 4 \}} | 0.07323 ^{\{ 3 \}} | 0.07816 ^{\{ 7 \}} | 0.0888 ^{\{ 12 \}} | 0.10363 ^{\{ 15 \}} | 0.03587 ^{\{ 1 \}} | 0.10299 ^{\{ 14 \}} | 0.08464 ^{\{ 11 \}} | 0.09863 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.04708 ^{\{ 2 \}} | 0.07537 ^{\{ 5 \}} | 0.09757 ^{\{ 11 \}} | 0.0662 ^{\{ 3 \}} | 0.08954 ^{\{ 10 \}} | 0.11225 ^{\{ 13 \}} | 0.07976 ^{\{ 8 \}} | 0.07321 ^{\{ 4 \}} | 0.07788 ^{\{ 6 \}} | 0.08763 ^{\{ 9 \}} | 0.13241 ^{\{ 15 \}} | 0.01303 ^{\{ 1 \}} | 0.11883 ^{\{ 14 \}} | 0.07847 ^{\{ 7 \}} | 0.10997 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.00749 ^{\{ 2 \}} | 0.00915 ^{\{ 4 \}} | 0.0102 ^{\{ 6 \}} | 0.01024 ^{\{ 7 \}} | 0.0109 ^{\{ 8 \}} | 0.01143 ^{\{ 9 \}} | 0.00918 ^{\{ 5 \}} | 0.00855 ^{\{ 3 \}} | 0.01159 ^{\{ 11 \}} | 0.0135 ^{\{ 12 \}} | 0.01684 ^{\{ 13 \}} | 0.00259 ^{\{ 1 \}} | 0.01761 ^{\{ 15 \}} | 0.01144 ^{\{ 10 \}} | 0.01689 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.06747 ^{\{ 2 \}} | 0.08782 ^{\{ 6 \}} | 0.10039 ^{\{ 11 \}} | 0.08263 ^{\{ 4 \}} | 0.09591 ^{\{ 10 \}} | 0.10774 ^{\{ 13 \}} | 0.08963 ^{\{ 7 \}} | 0.0864 ^{\{ 5 \}} | 0.08195 ^{\{ 3 \}} | 0.09474 ^{\{ 9 \}} | 0.11684 ^{\{ 15 \}} | 0.02602 ^{\{ 1 \}} | 0.11147 ^{\{ 14 \}} | 0.09228 ^{\{ 8 \}} | 0.10632 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.16103 ^{\{ 2 \}} | 0.18776 ^{\{ 5 \}} | 0.20121 ^{\{ 8 \}} | 0.19396 ^{\{ 6 \}} | 0.20394 ^{\{ 9 \}} | 0.21056 ^{\{ 10 \}} | 0.18594 ^{\{ 4 \}} | 0.18308 ^{\{ 3 \}} | 0.1954 ^{\{ 7 \}} | 0.22201 ^{\{ 12 \}} | 0.25908 ^{\{ 15 \}} | 0.08969 ^{\{ 1 \}} | 0.25746 ^{\{ 14 \}} | 0.2116 ^{\{ 11 \}} | 0.24657 ^{\{ 13 \}} | |
D_{abs} | 0.02344 ^{\{ 1 \}} | 0.0258 ^{\{ 4 \}} | 0.02729 ^{\{ 8 \}} | 0.02632 ^{\{ 5 \}} | 0.02689 ^{\{ 7 \}} | 0.02795 ^{\{ 9 \}} | 0.02543 ^{\{ 3 \}} | 0.02678 ^{\{ 6 \}} | 0.03037 ^{\{ 13 \}} | 0.02968 ^{\{ 12 \}} | 0.02967 ^{\{ 11 \}} | 0.02442 ^{\{ 2 \}} | 0.03344 ^{\{ 15 \}} | 0.02859 ^{\{ 10 \}} | 0.03216 ^{\{ 14 \}} | |
D_{max} | 0.03772 ^{\{ 2 \}} | 0.04225 ^{\{ 4 \}} | 0.04553 ^{\{ 8 \}} | 0.04245 ^{\{ 5 \}} | 0.0444 ^{\{ 7 \}} | 0.04705 ^{\{ 10 \}} | 0.04196 ^{\{ 3 \}} | 0.04374 ^{\{ 6 \}} | 0.0487 ^{\{ 12 \}} | 0.04793 ^{\{ 11 \}} | 0.04912 ^{\{ 13 \}} | 0.03646 ^{\{ 1 \}} | 0.05452 ^{\{ 15 \}} | 0.04636 ^{\{ 9 \}} | 0.05237 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 39 ^{\{ 4 \}} | 71 ^{\{ 9 \}} | 40 ^{\{ 5 \}} | 70 ^{\{ 8 \}} | 87 ^{\{ 12 \}} | 41 ^{\{ 6 \}} | 35 ^{\{ 3 \}} | 62 ^{\{ 7 \}} | 86 ^{\{ 11 \}} | 112 ^{\{ 14 \}} | 9 ^{\{ 1 \}} | 115 ^{\{ 15 \}} | 74 ^{\{ 10 \}} | 104 ^{\{ 13 \}} | |
200 | BIAS( \hat{\delta} ) | 0.12568 ^{\{ 2 \}} | 0.15802 ^{\{ 6 \}} | 0.16754 ^{\{ 10 \}} | 0.14131 ^{\{ 3 \}} | 0.17205 ^{\{ 11 \}} | 0.19869 ^{\{ 12 \}} | 0.1585 ^{\{ 8 \}} | 0.15009 ^{\{ 5 \}} | 0.14533 ^{\{ 4 \}} | 0.16686 ^{\{ 9 \}} | 0.215 ^{\{ 15 \}} | 0.05087 ^{\{ 1 \}} | 0.20091 ^{\{ 13 \}} | 0.15808 ^{\{ 7 \}} | 0.20236 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.04788 ^{\{ 2 \}} | 0.05346 ^{\{ 5 \}} | 0.05363 ^{\{ 6 \}} | 0.0502 ^{\{ 3 \}} | 0.05533 ^{\{ 9 \}} | 0.06203 ^{\{ 12 \}} | 0.05404 ^{\{ 7 \}} | 0.05068 ^{\{ 4 \}} | 0.05485 ^{\{ 8 \}} | 0.06093 ^{\{ 11 \}} | 0.07871 ^{\{ 15 \}} | 0.02619 ^{\{ 1 \}} | 0.07197 ^{\{ 13 \}} | 0.05806 ^{\{ 10 \}} | 0.07216 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.02442 ^{\{ 2 \}} | 0.0386 ^{\{ 5 \}} | 0.04604 ^{\{ 11 \}} | 0.03106 ^{\{ 3 \}} | 0.04597 ^{\{ 10 \}} | 0.06064 ^{\{ 13 \}} | 0.03893 ^{\{ 6 \}} | 0.03567 ^{\{ 4 \}} | 0.03895 ^{\{ 7 \}} | 0.04485 ^{\{ 9 \}} | 0.06901 ^{\{ 15 \}} | 0.00783 ^{\{ 1 \}} | 0.0603 ^{\{ 12 \}} | 0.03976 ^{\{ 8 \}} | 0.06373 ^{\{ 14 \}} | |
MSE( \hat{\beta} ) | 0.00378 ^{\{ 2 \}} | 0.00458 ^{\{ 5 \}} | 0.00482 ^{\{ 7 \}} | 0.00408 ^{\{ 3 \}} | 0.00482 ^{\{ 7 \}} | 0.00624 ^{\{ 12 \}} | 0.00482 ^{\{ 7 \}} | 0.00409 ^{\{ 4 \}} | 0.00537 ^{\{ 9 \}} | 0.00623 ^{\{ 11 \}} | 0.00978 ^{\{ 15 \}} | 0.00135 ^{\{ 1 \}} | 0.00858 ^{\{ 13 \}} | 0.00572 ^{\{ 10 \}} | 0.00898 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.05027 ^{\{ 2 \}} | 0.06321 ^{\{ 6 \}} | 0.06702 ^{\{ 10 \}} | 0.05652 ^{\{ 3 \}} | 0.06882 ^{\{ 11 \}} | 0.07948 ^{\{ 12 \}} | 0.0634 ^{\{ 8 \}} | 0.06003 ^{\{ 5 \}} | 0.05813 ^{\{ 4 \}} | 0.06675 ^{\{ 9 \}} | 0.086 ^{\{ 15 \}} | 0.02035 ^{\{ 1 \}} | 0.08036 ^{\{ 13 \}} | 0.06323 ^{\{ 7 \}} | 0.08094 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.1197 ^{\{ 2 \}} | 0.13366 ^{\{ 5 \}} | 0.13408 ^{\{ 6 \}} | 0.12549 ^{\{ 3 \}} | 0.13833 ^{\{ 9 \}} | 0.15508 ^{\{ 12 \}} | 0.13511 ^{\{ 7 \}} | 0.12671 ^{\{ 4 \}} | 0.13713 ^{\{ 8 \}} | 0.15232 ^{\{ 11 \}} | 0.19678 ^{\{ 15 \}} | 0.06548 ^{\{ 1 \}} | 0.17993 ^{\{ 13 \}} | 0.14516 ^{\{ 10 \}} | 0.1804 ^{\{ 14 \}} | |
D_{abs} | 0.01719 ^{\{ 2 \}} | 0.01841 ^{\{ 5 \}} | 0.01851 ^{\{ 6 \}} | 0.0174 ^{\{ 3 \}} | 0.01866 ^{\{ 7 \}} | 0.02028 ^{\{ 9 \}} | 0.01868 ^{\{ 8 \}} | 0.01825 ^{\{ 4 \}} | 0.02193 ^{\{ 12 \}} | 0.02128 ^{\{ 11 \}} | 0.02281 ^{\{ 13 \}} | 0.0171 ^{\{ 1 \}} | 0.02382 ^{\{ 15 \}} | 0.02076 ^{\{ 10 \}} | 0.02362 ^{\{ 14 \}} | |
D_{max} | 0.02788 ^{\{ 2 \}} | 0.03027 ^{\{ 5 \}} | 0.03093 ^{\{ 7 \}} | 0.02818 ^{\{ 3 \}} | 0.03111 ^{\{ 8 \}} | 0.03422 ^{\{ 10 \}} | 0.03083 ^{\{ 6 \}} | 0.02987 ^{\{ 4 \}} | 0.03496 ^{\{ 12 \}} | 0.03435 ^{\{ 11 \}} | 0.03748 ^{\{ 13 \}} | 0.02581 ^{\{ 1 \}} | 0.03887 ^{\{ 15 \}} | 0.03338 ^{\{ 9 \}} | 0.03869 ^{\{ 14 \}} | |
\sum Ranks | 16 ^{\{ 2 \}} | 42 ^{\{ 5 \}} | 63 ^{\{ 7 \}} | 24 ^{\{ 3 \}} | 72 ^{\{ 10 \}} | 92 ^{\{ 12 \}} | 57 ^{\{ 6 \}} | 34 ^{\{ 4 \}} | 64 ^{\{ 8 \}} | 82 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 8 ^{\{ 1 \}} | 107 ^{\{ 13 \}} | 71 ^{\{ 9 \}} | 112 ^{\{ 14 \}} | |
300 | BIAS( \hat{\delta} ) | 0.10138 ^{\{ 2 \}} | 0.12388 ^{\{ 6 \}} | 0.13609 ^{\{ 10 \}} | 0.11995 ^{\{ 4 \}} | 0.13703 ^{\{ 11 \}} | 0.15219 ^{\{ 12 \}} | 0.11897 ^{\{ 3 \}} | 0.12185 ^{\{ 5 \}} | 0.13147 ^{\{ 8 \}} | 0.13467 ^{\{ 9 \}} | 0.17695 ^{\{ 15 \}} | 0.04844 ^{\{ 1 \}} | 0.1656 ^{\{ 14 \}} | 0.12572 ^{\{ 7 \}} | 0.15695 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.03889 ^{\{ 2 \}} | 0.04166 ^{\{ 4 \}} | 0.04473 ^{\{ 8 \}} | 0.04246 ^{\{ 6 \}} | 0.04436 ^{\{ 7 \}} | 0.04875 ^{\{ 12 \}} | 0.04023 ^{\{ 3 \}} | 0.04196 ^{\{ 5 \}} | 0.04867 ^{\{ 11 \}} | 0.0481 ^{\{ 10 \}} | 0.06487 ^{\{ 15 \}} | 0.02362 ^{\{ 1 \}} | 0.05683 ^{\{ 14 \}} | 0.04507 ^{\{ 9 \}} | 0.05475 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01642 ^{\{ 2 \}} | 0.02414 ^{\{ 6 \}} | 0.02962 ^{\{ 10 \}} | 0.0216 ^{\{ 3 \}} | 0.02872 ^{\{ 9 \}} | 0.03784 ^{\{ 12 \}} | 0.02234 ^{\{ 4 \}} | 0.02409 ^{\{ 5 \}} | 0.03044 ^{\{ 11 \}} | 0.02838 ^{\{ 8 \}} | 0.04928 ^{\{ 15 \}} | 0.00716 ^{\{ 1 \}} | 0.04261 ^{\{ 14 \}} | 0.02478 ^{\{ 7 \}} | 0.03848 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.00251 ^{\{ 2 \}} | 0.00274 ^{\{ 4 \}} | 0.00319 ^{\{ 8 \}} | 0.00286 ^{\{ 6 \}} | 0.00308 ^{\{ 7 \}} | 0.0039 ^{\{ 11 \}} | 0.00252 ^{\{ 3 \}} | 0.00285 ^{\{ 5 \}} | 0.00411 ^{\{ 12 \}} | 0.00379 ^{\{ 10 \}} | 0.00722 ^{\{ 15 \}} | 0.00119 ^{\{ 1 \}} | 0.00549 ^{\{ 14 \}} | 0.00329 ^{\{ 9 \}} | 0.00507 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.04055 ^{\{ 2 \}} | 0.04955 ^{\{ 6 \}} | 0.05444 ^{\{ 10 \}} | 0.04798 ^{\{ 4 \}} | 0.05481 ^{\{ 11 \}} | 0.06088 ^{\{ 12 \}} | 0.04759 ^{\{ 3 \}} | 0.04874 ^{\{ 5 \}} | 0.05259 ^{\{ 8 \}} | 0.05387 ^{\{ 9 \}} | 0.07078 ^{\{ 15 \}} | 0.01938 ^{\{ 1 \}} | 0.06624 ^{\{ 14 \}} | 0.05029 ^{\{ 7 \}} | 0.06278 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.09723 ^{\{ 2 \}} | 0.10415 ^{\{ 4 \}} | 0.11183 ^{\{ 8 \}} | 0.10614 ^{\{ 6 \}} | 0.11091 ^{\{ 7 \}} | 0.12186 ^{\{ 12 \}} | 0.10057 ^{\{ 3 \}} | 0.10489 ^{\{ 5 \}} | 0.12168 ^{\{ 11 \}} | 0.12025 ^{\{ 10 \}} | 0.16218 ^{\{ 15 \}} | 0.05905 ^{\{ 1 \}} | 0.14207 ^{\{ 14 \}} | 0.11268 ^{\{ 9 \}} | 0.13687 ^{\{ 13 \}} | |
D_{abs} | 0.01417 ^{\{ 1 \}} | 0.01453 ^{\{ 2 \}} | 0.01543 ^{\{ 7 \}} | 0.01481 ^{\{ 5 \}} | 0.01544 ^{\{ 8 \}} | 0.01573 ^{\{ 9 \}} | 0.01455 ^{\{ 3 \}} | 0.0152 ^{\{ 6 \}} | 0.01852 ^{\{ 13 \}} | 0.01696 ^{\{ 11 \}} | 0.0176 ^{\{ 12 \}} | 0.0147 ^{\{ 4 \}} | 0.01974 ^{\{ 15 \}} | 0.01602 ^{\{ 10 \}} | 0.01923 ^{\{ 14 \}} | |
D_{max} | 0.02287 ^{\{ 2 \}} | 0.02379 ^{\{ 4 \}} | 0.02546 ^{\{ 7 \}} | 0.02403 ^{\{ 5 \}} | 0.02569 ^{\{ 8 \}} | 0.0263 ^{\{ 10 \}} | 0.02377 ^{\{ 3 \}} | 0.02487 ^{\{ 6 \}} | 0.02963 ^{\{ 13 \}} | 0.02753 ^{\{ 11 \}} | 0.02918 ^{\{ 12 \}} | 0.02235 ^{\{ 1 \}} | 0.03243 ^{\{ 15 \}} | 0.02583 ^{\{ 9 \}} | 0.03132 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 36 ^{\{ 4 \}} | 68 ^{\{ 8.5 \}} | 39 ^{\{ 5 \}} | 68 ^{\{ 8.5 \}} | 90 ^{\{ 12 \}} | 25 ^{\{ 3 \}} | 42 ^{\{ 6 \}} | 87 ^{\{ 11 \}} | 78 ^{\{ 10 \}} | 114 ^{\{ 14.5 \}} | 11 ^{\{ 1 \}} | 114 ^{\{ 14.5 \}} | 67 ^{\{ 7 \}} | 106 ^{\{ 13 \}} | |
400 | BIAS( \hat{\delta} ) | 0.08539 ^{\{ 2 \}} | 0.10758 ^{\{ 5 \}} | 0.12255 ^{\{ 11 \}} | 0.10162 ^{\{ 3 \}} | 0.12057 ^{\{ 10 \}} | 0.13935 ^{\{ 12 \}} | 0.10965 ^{\{ 6 \}} | 0.11168 ^{\{ 7 \}} | 0.10747 ^{\{ 4 \}} | 0.11634 ^{\{ 9 \}} | 0.15238 ^{\{ 15 \}} | 0.03778 ^{\{ 1 \}} | 0.14254 ^{\{ 13 \}} | 0.11229 ^{\{ 8 \}} | 0.14255 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.03288 ^{\{ 2 \}} | 0.03624 ^{\{ 4 \}} | 0.04047 ^{\{ 10 \}} | 0.03533 ^{\{ 3 \}} | 0.03953 ^{\{ 9 \}} | 0.04295 ^{\{ 12 \}} | 0.03632 ^{\{ 5 \}} | 0.03807 ^{\{ 6 \}} | 0.03897 ^{\{ 7 \}} | 0.04102 ^{\{ 11 \}} | 0.05635 ^{\{ 15 \}} | 0.01866 ^{\{ 1 \}} | 0.0513 ^{\{ 14 \}} | 0.03935 ^{\{ 8 \}} | 0.04946 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01164 ^{\{ 2 \}} | 0.01815 ^{\{ 4 \}} | 0.02321 ^{\{ 11 \}} | 0.01641 ^{\{ 3 \}} | 0.02278 ^{\{ 10 \}} | 0.02969 ^{\{ 12 \}} | 0.01878 ^{\{ 5 \}} | 0.01988 ^{\{ 7 \}} | 0.02066 ^{\{ 8 \}} | 0.02178 ^{\{ 9 \}} | 0.03737 ^{\{ 15 \}} | 0.00425 ^{\{ 1 \}} | 0.03212 ^{\{ 14 \}} | 0.01965 ^{\{ 6 \}} | 0.03198 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.00175 ^{\{ 2 \}} | 0.00205 ^{\{ 4 \}} | 0.00259 ^{\{ 9 \}} | 0.00201 ^{\{ 3 \}} | 0.0025 ^{\{ 8 \}} | 0.00289 ^{\{ 12 \}} | 0.00208 ^{\{ 5 \}} | 0.00229 ^{\{ 6 \}} | 0.00265 ^{\{ 10 \}} | 0.00283 ^{\{ 11 \}} | 0.0054 ^{\{ 15 \}} | 0.00073 ^{\{ 1 \}} | 0.00429 ^{\{ 14 \}} | 0.00241 ^{\{ 7 \}} | 0.00393 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.03416 ^{\{ 2 \}} | 0.04303 ^{\{ 5 \}} | 0.04902 ^{\{ 11 \}} | 0.04065 ^{\{ 3 \}} | 0.04823 ^{\{ 10 \}} | 0.05574 ^{\{ 12 \}} | 0.04386 ^{\{ 6 \}} | 0.04467 ^{\{ 7 \}} | 0.04299 ^{\{ 4 \}} | 0.04654 ^{\{ 9 \}} | 0.06095 ^{\{ 15 \}} | 0.01511 ^{\{ 1 \}} | 0.05701 ^{\{ 13 \}} | 0.04492 ^{\{ 8 \}} | 0.05702 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.08219 ^{\{ 2 \}} | 0.09059 ^{\{ 4 \}} | 0.10117 ^{\{ 10 \}} | 0.08833 ^{\{ 3 \}} | 0.09881 ^{\{ 9 \}} | 0.10736 ^{\{ 12 \}} | 0.0908 ^{\{ 5 \}} | 0.09518 ^{\{ 6 \}} | 0.09744 ^{\{ 7 \}} | 0.10255 ^{\{ 11 \}} | 0.14089 ^{\{ 15 \}} | 0.04664 ^{\{ 1 \}} | 0.12824 ^{\{ 14 \}} | 0.09839 ^{\{ 8 \}} | 0.12364 ^{\{ 13 \}} | |
D_{abs} | 0.01182 ^{\{ 1 \}} | 0.01268 ^{\{ 4 \}} | 0.01351 ^{\{ 7.5 \}} | 0.01208 ^{\{ 3 \}} | 0.01351 ^{\{ 7.5 \}} | 0.01356 ^{\{ 9 \}} | 0.01291 ^{\{ 5 \}} | 0.0134 ^{\{ 6 \}} | 0.0148 ^{\{ 11 \}} | 0.01494 ^{\{ 12 \}} | 0.016 ^{\{ 13 \}} | 0.01204 ^{\{ 2 \}} | 0.01733 ^{\{ 15 \}} | 0.01386 ^{\{ 10 \}} | 0.01687 ^{\{ 14 \}} | |
D_{max} | 0.01903 ^{\{ 2 \}} | 0.02082 ^{\{ 4 \}} | 0.02243 ^{\{ 8 \}} | 0.01969 ^{\{ 3 \}} | 0.02228 ^{\{ 7 \}} | 0.02295 ^{\{ 10 \}} | 0.02117 ^{\{ 5 \}} | 0.02188 ^{\{ 6 \}} | 0.02381 ^{\{ 11 \}} | 0.02425 ^{\{ 12 \}} | 0.02642 ^{\{ 13 \}} | 0.01826 ^{\{ 1 \}} | 0.02832 ^{\{ 15 \}} | 0.02247 ^{\{ 9 \}} | 0.02769 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 34 ^{\{ 4 \}} | 77.5 ^{\{ 10 \}} | 24 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 91 ^{\{ 12 \}} | 42 ^{\{ 5 \}} | 51 ^{\{ 6 \}} | 62 ^{\{ 7 \}} | 84 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 9 ^{\{ 1 \}} | 112 ^{\{ 14 \}} | 64 ^{\{ 8 \}} | 108 ^{\{ 13 \}} |
Parameter | n | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
\delta=0.7, \beta=2.5 | 30 | 4.0 | 7.0 | 11.0 | 6.0 | 12.5 | 15.0 | 12.5 | 10.0 | 2.0 | 3.0 | 14.0 | 1.0 | 8.5 | 8.5 | 5.0 |
60 | 3.0 | 11.0 | 14.0 | 4.0 | 10.0 | 15.0 | 12.0 | 9.0 | 2.0 | 5.0 | 8.0 | 1.0 | 6.0 | 13.0 | 7.0 | |
100 | 3.0 | 11.0 | 13.0 | 4.0 | 12.0 | 15.0 | 9.0 | 7.0 | 2.0 | 6.0 | 14.0 | 1.0 | 5.0 | 10.0 | 8.0 | |
200 | 2.0 | 7.0 | 14.0 | 5.0 | 13.0 | 15.0 | 10.0 | 4.0 | 3.0 | 8.0 | 12.0 | 1.0 | 6.0 | 11.0 | 9.0 | |
300 | 2.0 | 6.0 | 13.0 | 4.0 | 14.0 | 15.0 | 11.0 | 5.0 | 3.0 | 7.0 | 12.0 | 1.0 | 8.5 | 10.0 | 8.5 | |
400 | 2.0 | 9.0 | 14.0 | 4.0 | 12.5 | 15.0 | 7.0 | 5.0 | 3.0 | 11.0 | 12.5 | 1.0 | 6.0 | 8.0 | 10.0 | |
\delta=0.25, \beta=0.75 | 30 | 2.0 | 6.0 | 11.5 | 7.0 | 10.0 | 13.0 | 8.0 | 5.0 | 3.5 | 3.5 | 11.5 | 1.0 | 15.0 | 9.0 | 14.0 |
60 | 2.0 | 7.0 | 10.0 | 5.0 | 11.0 | 13.0 | 8.0 | 4.0 | 3.0 | 6.0 | 12.0 | 1.0 | 15.0 | 9.0 | 14.0 | |
100 | 2.0 | 3.5 | 8.0 | 6.0 | 9.0 | 14.0 | 3.5 | 7.0 | 5.0 | 11.0 | 13.0 | 1.0 | 12.0 | 10.0 | 15.0 | |
200 | 2.0 | 3.0 | 10.0 | 4.5 | 11.0 | 13.0 | 7.0 | 4.5 | 6.0 | 8.0 | 14.0 | 1.0 | 15.0 | 9.0 | 12.0 | |
300 | 1.0 | 7.0 | 12.0 | 3.0 | 11.0 | 14.0 | 5.0 | 6.0 | 4.0 | 9.0 | 15.0 | 2.0 | 13.0 | 8.0 | 10.0 | |
400 | 2.0 | 5.0 | 10.0 | 4.0 | 8.0 | 13.0 | 6.0 | 3.0 | 7.0 | 12.0 | 15.0 | 1.0 | 14.0 | 9.0 | 11.0 | |
\delta=1.5, \beta=1.5 | 30 | 2.0 | 8.0 | 12.0 | 5.0 | 11.0 | 13.0 | 10.0 | 4.0 | 3.0 | 7.0 | 14.0 | 1.0 | 15.0 | 6.0 | 9.0 |
60 | 1.0 | 6.0 | 10.0 | 4.0 | 11.5 | 14.0 | 7.0 | 3.0 | 5.0 | 8.5 | 11.5 | 2.0 | 15.0 | 8.5 | 13.0 | |
100 | 1.0 | 7.0 | 11.0 | 4.0 | 12.0 | 13.0 | 8.0 | 3.0 | 6.0 | 9.0 | 10.0 | 2.0 | 15.0 | 5.0 | 14.0 | |
200 | 1.5 | 5.0 | 10.0 | 3.0 | 11.0 | 14.0 | 6.5 | 4.0 | 6.5 | 9.0 | 12.0 | 1.5 | 15.0 | 8.0 | 13.0 | |
300 | 1.0 | 5.0 | 10.5 | 4.0 | 9.0 | 13.0 | 7.0 | 3.0 | 6.0 | 10.5 | 12.0 | 2.0 | 15.0 | 8.0 | 14.0 | |
400 | 1.0 | 7.0 | 10.0 | 3.0 | 11.0 | 14.0 | 6.0 | 4.0 | 9.0 | 8.0 | 12.0 | 2.0 | 15.0 | 5.0 | 13.0 | |
\delta=0.5, \beta=2.0 | 30 | 3.0 | 9.0 | 14.0 | 5.0 | 8.0 | 15.0 | 7.0 | 6.0 | 2.0 | 4.0 | 13.0 | 1.0 | 11.0 | 12.0 | 10.0 |
60 | 3.0 | 5.0 | 14.0 | 4.0 | 11.0 | 15.0 | 12.0 | 9.0 | 2.0 | 6.0 | 13.0 | 1.0 | 10.0 | 8.0 | 7.0 | |
100 | 3.0 | 9.0 | 13.0 | 4.0 | 14.0 | 15.0 | 10.0 | 5.0 | 2.0 | 11.0 | 12.0 | 1.0 | 7.5 | 7.5 | 6.0 | |
200 | 3.0 | 9.0 | 12.0 | 4.0 | 13.0 | 15.0 | 8.0 | 5.0 | 2.0 | 10.0 | 14.0 | 1.0 | 6.0 | 11.0 | 7.0 | |
300 | 2.0 | 8.0 | 14.0 | 4.0 | 13.0 | 15.0 | 9.0 | 5.0 | 3.0 | 11.0 | 12.0 | 1.0 | 6.0 | 10.0 | 7.0 | |
400 | 2.0 | 8.0 | 14.0 | 4.0 | 12.0 | 15.0 | 10.0 | 5.0 | 3.0 | 11.0 | 13.0 | 1.0 | 9.0 | 6.0 | 7.0 | |
\delta=2.5, \beta=0.4 | 30 | 2.0 | 6.0 | 11.0 | 4.5 | 10.0 | 15.0 | 4.5 | 8.0 | 3.0 | 9.0 | 13.0 | 1.0 | 14.0 | 7.0 | 12.0 |
60 | 2.0 | 3.0 | 7.0 | 4.0 | 10.0 | 12.0 | 8.0 | 6.0 | 5.0 | 11.0 | 13.0 | 1.0 | 14.5 | 9.0 | 14.5 | |
100 | 2.0 | 4.0 | 9.0 | 5.0 | 8.0 | 12.0 | 6.0 | 3.0 | 7.0 | 11.0 | 14.0 | 1.0 | 15.0 | 10.0 | 13.0 | |
200 | 2.0 | 5.0 | 7.0 | 3.0 | 10.0 | 12.0 | 6.0 | 4.0 | 8.0 | 11.0 | 15.0 | 1.0 | 13.0 | 9.0 | 14.0 | |
300 | 2.0 | 4.0 | 8.5 | 5.0 | 8.5 | 12.0 | 3.0 | 6.0 | 11.0 | 10.0 | 14.5 | 1.0 | 14.5 | 7.0 | 13.0 | |
400 | 2.0 | 4.0 | 10.0 | 3.0 | 9.0 | 12.0 | 5.0 | 6.0 | 7.0 | 11.0 | 15.0 | 1.0 | 14.0 | 8.0 | 13.0 | |
\sum Ranks | 62.5 | 194.5 | 337.5 | 129.0 | 326.0 | 416.0 | 232.0 | 158.5 | 134.0 | 257.5 | 386.0 | 35.5 | 348.5 | 259.5 | 323.0 | |
Overall Rank | 2 | 6 | 12 | 3 | 11 | 15 | 7 | 5 | 4 | 8 | 14 | 1 | 13 | 9 | 10 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.49641) | (0.60871) | (-0.53065) | (-0.39129) | (-0.43960) | (-0.29722) | (0.38680) | (-0.56988) | (-0.94070) | (-0.73958) | ||
20 | \hat{E} | -0.53095 | 0.59059 | -0.56088 | -0.40941 | -0.46465 | -0.34499 | 0.45684 | -0.60354 | -0.97991 | -0.76199 |
BIAS | 0.07837 | 0.04604 | 0.07246 | 0.04604 | 0.06003 | 0.05431 | 0.07916 | 0.08133 | 0.08408 | 0.08889 | |
MSE | 0.00993 | 0.00330 | 0.00831 | 0.00330 | 0.00570 | 0.00859 | 0.01919 | 0.01077 | 0.01236 | 0.01358 | |
MRE | 0.15788 | 0.07564 | 0.13656 | 0.11767 | 0.13656 | 0.18274 | 0.20465 | 0.14272 | 0.08938 | 0.12019 | |
60 | \hat{E} | -0.51151 | 0.60061 | -0.54401 | -0.39939 | -0.45067 | -0.33185 | 0.43748 | -0.58617 | -0.96029 | -0.75437 |
BIAS | 0.04763 | 0.02850 | 0.04447 | 0.02850 | 0.03684 | 0.04169 | 0.06052 | 0.04871 | 0.04965 | 0.05278 | |
MSE | 0.00366 | 0.00128 | 0.00315 | 0.00128 | 0.00216 | 0.00597 | 0.01312 | 0.00382 | 0.00409 | 0.00461 | |
MRE | 0.09595 | 0.04683 | 0.08379 | 0.07285 | 0.08379 | 0.14026 | 0.15646 | 0.08548 | 0.05278 | 0.07137 | |
100 | \hat{E} | -0.50365 | 0.60492 | -0.53699 | -0.39508 | -0.44486 | -0.32447 | 0.42653 | -0.57848 | -0.95201 | -0.74854 |
BIAS | 0.03577 | 0.02159 | 0.03354 | 0.02159 | 0.02778 | 0.03496 | 0.05049 | 0.03662 | 0.03749 | 0.04155 | |
MSE | 0.00203 | 0.00073 | 0.00178 | 0.00073 | 0.00122 | 0.00432 | 0.00937 | 0.00212 | 0.00225 | 0.00277 | |
MRE | 0.07206 | 0.03546 | 0.06320 | 0.05517 | 0.06320 | 0.11763 | 0.13054 | 0.06426 | 0.03985 | 0.05618 | |
150 | \hat{E} | -0.50062 | 0.60654 | -0.53431 | -0.39346 | -0.44263 | -0.31955 | 0.41918 | -0.57587 | -0.94935 | -0.74777 |
BIAS | 0.02875 | 0.01742 | 0.02701 | 0.01742 | 0.02237 | 0.02987 | 0.04291 | 0.02945 | 0.03022 | 0.03410 | |
MSE | 0.00131 | 0.00048 | 0.00115 | 0.00048 | 0.00079 | 0.00307 | 0.00658 | 0.00136 | 0.00147 | 0.00187 | |
MRE | 0.05791 | 0.02861 | 0.05090 | 0.04451 | 0.05090 | 0.10051 | 0.11094 | 0.05167 | 0.03213 | 0.04610 | |
200 | \hat{E} | -0.49922 | 0.60731 | -0.53305 | -0.39269 | -0.44159 | -0.31498 | 0.41241 | -0.57418 | -0.94720 | -0.74586 |
BIAS | 0.02534 | 0.01537 | 0.02382 | 0.01537 | 0.01973 | 0.02508 | 0.03583 | 0.02571 | 0.02619 | 0.02975 | |
MSE | 0.00100 | 0.00037 | 0.00088 | 0.00037 | 0.00061 | 0.00205 | 0.00433 | 0.00104 | 0.00110 | 0.00142 | |
MRE | 0.05105 | 0.02525 | 0.04489 | 0.03928 | 0.04489 | 0.08439 | 0.09264 | 0.04512 | 0.02784 | 0.04023 | |
250 | \hat{E} | -0.49872 | 0.60755 | -0.53263 | -0.39245 | -0.44125 | -0.31227 | 0.40843 | -0.57363 | -0.94642 | -0.74544 |
BIAS | 0.02237 | 0.01358 | 0.02104 | 0.01358 | 0.01743 | 0.02194 | 0.03125 | 0.02259 | 0.02298 | 0.02633 | |
MSE | 0.00078 | 0.00029 | 0.00069 | 0.00029 | 0.00047 | 0.00152 | 0.00317 | 0.00080 | 0.00085 | 0.00111 | |
MRE | 0.04507 | 0.02231 | 0.03965 | 0.03471 | 0.03965 | 0.07382 | 0.08078 | 0.03964 | 0.02443 | 0.03560 | |
300 | \hat{E} | -0.49867 | 0.60754 | -0.53262 | -0.39246 | -0.44124 | -0.30972 | 0.40472 | -0.57329 | -0.94570 | -0.74456 |
BIAS | 0.02057 | 0.01249 | 0.01934 | 0.01249 | 0.01602 | 0.01937 | 0.02751 | 0.02078 | 0.02112 | 0.02426 | |
MSE | 0.00066 | 0.00024 | 0.00059 | 0.00024 | 0.00040 | 0.00114 | 0.00237 | 0.00068 | 0.00071 | 0.00094 | |
MRE | 0.04143 | 0.02051 | 0.03645 | 0.03191 | 0.03645 | 0.06516 | 0.07111 | 0.03647 | 0.02246 | 0.03281 | |
400 | \hat{E} | -0.49783 | 0.60800 | -0.53187 | -0.39200 | -0.44061 | -0.30769 | 0.40175 | -0.57248 | -0.94480 | -0.74409 |
BIAS | 0.01785 | 0.01084 | 0.01679 | 0.01084 | 0.01391 | 0.01677 | 0.02374 | 0.01791 | 0.01813 | 0.02087 | |
MSE | 0.00050 | 0.00019 | 0.00045 | 0.00019 | 0.00031 | 0.00078 | 0.00159 | 0.00051 | 0.00053 | 0.00070 | |
MRE | 0.03595 | 0.01782 | 0.03165 | 0.02772 | 0.03165 | 0.05643 | 0.06138 | 0.03143 | 0.01927 | 0.02822 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.06783) | (0.93442) | (-0.08051) | (-0.06558) | (-0.06670) | (-0.55698) | (0.77529) | (-0.11870) | (-0.61333) | (-0.19500) | ||
20 | \hat{E} | -0.08600 | 0.91826 | -0.10119 | -0.08174 | -0.08383 | -0.57359 | 0.81240 | -0.14688 | -0.64206 | -0.20407 |
BIAS | 0.03295 | 0.03015 | 0.03804 | 0.03015 | 0.03152 | 0.13286 | 0.21409 | 0.05684 | 0.06063 | 0.02178 | |
MSE | 0.00181 | 0.00148 | 0.00238 | 0.00148 | 0.00163 | 0.02637 | 0.06957 | 0.00541 | 0.00640 | 0.00083 | |
MRE | 0.48578 | 0.03227 | 0.47257 | 0.45978 | 0.47257 | 0.23853 | 0.27615 | 0.47882 | 0.09886 | 0.11170 | |
60 | \hat{E} | -0.07058 | 0.93218 | -0.08350 | -0.06782 | -0.06918 | -0.54615 | 0.76306 | -0.12237 | -0.61731 | -0.19918 |
BIAS | 0.01987 | 0.01844 | 0.02311 | 0.01844 | 0.01914 | 0.08949 | 0.14217 | 0.03542 | 0.03742 | 0.01242 | |
MSE | 0.00071 | 0.00060 | 0.00095 | 0.00060 | 0.00065 | 0.01272 | 0.03216 | 0.00224 | 0.00257 | 0.00026 | |
MRE | 0.29296 | 0.01973 | 0.28701 | 0.28119 | 0.28701 | 0.16067 | 0.18337 | 0.29838 | 0.06101 | 0.06367 | |
100 | \hat{E} | -0.07079 | 0.93192 | -0.08380 | -0.06808 | -0.06942 | -0.55144 | 0.77021 | -0.12307 | -0.61799 | -0.19801 |
BIAS | 0.01830 | 0.01700 | 0.02129 | 0.01700 | 0.01764 | 0.07792 | 0.12400 | 0.03260 | 0.03419 | 0.00970 | |
MSE | 0.00057 | 0.00049 | 0.00077 | 0.00049 | 0.00053 | 0.00945 | 0.02392 | 0.00179 | 0.00197 | 0.00015 | |
MRE | 0.26984 | 0.01819 | 0.26448 | 0.25923 | 0.26448 | 0.13990 | 0.15994 | 0.27467 | 0.05574 | 0.04975 | |
150 | \hat{E} | -0.06964 | 0.93291 | -0.08251 | -0.06709 | -0.06835 | -0.55277 | 0.77125 | -0.12140 | -0.61630 | -0.19696 |
BIAS | 0.01545 | 0.01438 | 0.01799 | 0.01438 | 0.01490 | 0.06590 | 0.10492 | 0.02758 | 0.02884 | 0.00791 | |
MSE | 0.00039 | 0.00034 | 0.00053 | 0.00034 | 0.00036 | 0.00675 | 0.01708 | 0.00124 | 0.00136 | 0.00010 | |
MRE | 0.22776 | 0.01539 | 0.22349 | 0.21929 | 0.22349 | 0.11832 | 0.13533 | 0.23232 | 0.04703 | 0.04055 | |
200 | \hat{E} | -0.06936 | 0.93313 | -0.08220 | -0.06687 | -0.06810 | -0.55416 | 0.77283 | -0.12100 | -0.61582 | -0.19650 |
BIAS | 0.01350 | 0.01258 | 0.01573 | 0.01258 | 0.01303 | 0.05742 | 0.09143 | 0.02413 | 0.02521 | 0.00679 | |
MSE | 0.00029 | 0.00025 | 0.00039 | 0.00025 | 0.00027 | 0.00513 | 0.01298 | 0.00092 | 0.00101 | 0.00007 | |
MRE | 0.19902 | 0.01346 | 0.19539 | 0.19182 | 0.19539 | 0.10308 | 0.11793 | 0.20332 | 0.04110 | 0.03481 | |
250 | \hat{E} | -0.06880 | 0.93362 | -0.08157 | -0.06638 | -0.06758 | -0.55348 | 0.77135 | -0.12007 | -0.61481 | -0.19632 |
BIAS | 0.01209 | 0.01127 | 0.01409 | 0.01127 | 0.01167 | 0.05144 | 0.08187 | 0.02162 | 0.02254 | 0.00606 | |
MSE | 0.00023 | 0.00020 | 0.00031 | 0.00020 | 0.00021 | 0.00412 | 0.01041 | 0.00073 | 0.00079 | 0.00006 | |
MRE | 0.17817 | 0.01206 | 0.17501 | 0.17189 | 0.17501 | 0.09235 | 0.10559 | 0.18216 | 0.03675 | 0.03107 | |
300 | \hat{E} | -0.06849 | 0.93389 | -0.08122 | -0.06611 | -0.06728 | -0.55377 | 0.77148 | -0.11958 | -0.61429 | -0.19608 |
BIAS | 0.01080 | 0.01008 | 0.01259 | 0.01008 | 0.01043 | 0.04597 | 0.07317 | 0.01930 | 0.02008 | 0.00549 | |
MSE | 0.00018 | 0.00016 | 0.00024 | 0.00016 | 0.00017 | 0.00331 | 0.00836 | 0.00057 | 0.00062 | 0.00005 | |
MRE | 0.15919 | 0.01079 | 0.15641 | 0.15367 | 0.15641 | 0.08253 | 0.09437 | 0.16256 | 0.03273 | 0.02814 | |
400 | \hat{E} | -0.06779 | 0.93451 | -0.08043 | -0.06549 | -0.06663 | -0.55280 | 0.76960 | -0.11845 | -0.61313 | -0.19585 |
BIAS | 0.00922 | 0.00861 | 0.01075 | 0.00861 | 0.00891 | 0.03952 | 0.06287 | 0.01649 | 0.01714 | 0.00468 | |
MSE | 0.00013 | 0.00011 | 0.00018 | 0.00011 | 0.00012 | 0.00246 | 0.00620 | 0.00042 | 0.00045 | 0.00003 | |
MRE | 0.13587 | 0.00921 | 0.13356 | 0.13128 | 0.13356 | 0.07096 | 0.08109 | 0.13893 | 0.02795 | 0.02402 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.50778) | (0.60183) | (-0.54131) | (-0.39817) | (-0.44844) | (-0.41510) | (0.55685) | (-0.56943) | (-0.94546) | (-0.64094) | ||
20 | \hat{E} | -0.53963 | 0.58474 | -0.56950 | -0.41526 | -0.47179 | -0.42231 | 0.56992 | -0.60692 | -0.99220 | -0.68195 |
BIAS | 0.06486 | 0.03768 | 0.05965 | 0.03768 | 0.04941 | 0.06592 | 0.09848 | 0.07228 | 0.08395 | 0.06627 | |
MSE | 0.00716 | 0.00232 | 0.00593 | 0.00232 | 0.00407 | 0.00626 | 0.01412 | 0.00905 | 0.01324 | 0.00793 | |
MRE | 0.12773 | 0.06261 | 0.11019 | 0.09464 | 0.11019 | 0.15882 | 0.17685 | 0.12693 | 0.08879 | 0.10339 | |
60 | \hat{E} | -0.51075 | 0.60061 | -0.54366 | -0.39939 | -0.45038 | -0.39987 | 0.53536 | -0.57260 | -0.94830 | -0.65192 |
BIAS | 0.03416 | 0.02045 | 0.03190 | 0.02045 | 0.02642 | 0.04356 | 0.06430 | 0.03772 | 0.04263 | 0.03160 | |
MSE | 0.00191 | 0.00068 | 0.00165 | 0.00068 | 0.00114 | 0.00303 | 0.00661 | 0.00240 | 0.00323 | 0.00169 | |
MRE | 0.06728 | 0.03398 | 0.05893 | 0.05135 | 0.05893 | 0.10495 | 0.11547 | 0.06624 | 0.04509 | 0.04930 | |
100 | \hat{E} | -0.50817 | 0.60195 | -0.54141 | -0.39805 | -0.44852 | -0.40209 | 0.53840 | -0.56973 | -0.94512 | -0.64743 |
BIAS | 0.02656 | 0.01595 | 0.02484 | 0.01595 | 0.02058 | 0.03806 | 0.05627 | 0.02990 | 0.03474 | 0.02354 | |
MSE | 0.00118 | 0.00042 | 0.00102 | 0.00042 | 0.00070 | 0.00235 | 0.00513 | 0.00154 | 0.00218 | 0.00093 | |
MRE | 0.05231 | 0.02650 | 0.04589 | 0.04005 | 0.04589 | 0.09169 | 0.10105 | 0.05251 | 0.03675 | 0.03672 | |
150 | \hat{E} | -0.50866 | 0.60156 | -0.54195 | -0.39844 | -0.44896 | -0.40642 | 0.54463 | -0.57036 | -0.94611 | -0.64604 |
BIAS | 0.02290 | 0.01375 | 0.02142 | 0.01375 | 0.01774 | 0.03357 | 0.04972 | 0.02612 | 0.03078 | 0.01965 | |
MSE | 0.00085 | 0.00031 | 0.00074 | 0.00031 | 0.00051 | 0.00177 | 0.00388 | 0.00113 | 0.00162 | 0.00064 | |
MRE | 0.04510 | 0.02284 | 0.03956 | 0.03452 | 0.03956 | 0.08087 | 0.08930 | 0.04588 | 0.03256 | 0.03065 | |
200 | \hat{E} | -0.50805 | 0.60185 | -0.54143 | -0.39815 | -0.44853 | -0.40717 | 0.54558 | -0.56956 | -0.94505 | -0.64468 |
BIAS | 0.01964 | 0.01180 | 0.01838 | 0.01180 | 0.01523 | 0.02931 | 0.04342 | 0.02235 | 0.02632 | 0.01691 | |
MSE | 0.00061 | 0.00022 | 0.00054 | 0.00022 | 0.00037 | 0.00134 | 0.00294 | 0.00081 | 0.00115 | 0.00046 | |
MRE | 0.03868 | 0.01961 | 0.03395 | 0.02965 | 0.03395 | 0.07061 | 0.07797 | 0.03925 | 0.02783 | 0.02639 | |
250 | \hat{E} | -0.50768 | 0.60204 | -0.54111 | -0.39796 | -0.44827 | -0.40773 | 0.54631 | -0.56909 | -0.94448 | -0.64387 |
BIAS | 0.01792 | 0.01078 | 0.01678 | 0.01078 | 0.01390 | 0.02639 | 0.03910 | 0.02037 | 0.02394 | 0.01534 | |
MSE | 0.00051 | 0.00018 | 0.00044 | 0.00018 | 0.00030 | 0.00108 | 0.00237 | 0.00066 | 0.00093 | 0.00038 | |
MRE | 0.03529 | 0.01791 | 0.03099 | 0.02707 | 0.03099 | 0.06359 | 0.07022 | 0.03578 | 0.02532 | 0.02394 | |
300 | \hat{E} | -0.50732 | 0.60223 | -0.54079 | -0.39777 | -0.44801 | -0.40808 | 0.54678 | -0.56866 | -0.94398 | -0.64321 |
BIAS | 0.01623 | 0.00976 | 0.01520 | 0.00976 | 0.01259 | 0.02435 | 0.03607 | 0.01864 | 0.02221 | 0.01370 | |
MSE | 0.00042 | 0.00015 | 0.00037 | 0.00015 | 0.00025 | 0.00091 | 0.00200 | 0.00056 | 0.00078 | 0.00030 | |
MRE | 0.03196 | 0.01623 | 0.02807 | 0.02452 | 0.02807 | 0.05866 | 0.06477 | 0.03274 | 0.02349 | 0.02138 | |
400 | \hat{E} | -0.50687 | 0.60247 | -0.54040 | -0.39753 | -0.44768 | -0.40877 | 0.54769 | -0.56812 | -0.94337 | -0.64228 |
BIAS | 0.01396 | 0.00840 | 0.01307 | 0.00840 | 0.01083 | 0.02063 | 0.03056 | 0.01594 | 0.01885 | 0.01196 | |
MSE | 0.00031 | 0.00011 | 0.00027 | 0.00011 | 0.00019 | 0.00065 | 0.00142 | 0.00040 | 0.00055 | 0.00023 | |
MRE | 0.02748 | 0.01396 | 0.02415 | 0.02111 | 0.02415 | 0.04969 | 0.05487 | 0.02800 | 0.01994 | 0.01866 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.24470) | (0.78294) | (-0.27803) | (-0.21706) | (-0.23032) | (-0.93457) | (1.43805) | (-0.42447) | (-0.96243) | (-0.20000) | ||
20 | \hat{E} | -0.22848 | 0.79765 | -0.25933 | -0.20235 | -0.21483 | -0.89872 | 1.37489 | -0.39453 | -0.93073 | -0.20048 |
BIAS | 0.06023 | 0.04755 | 0.06458 | 0.04755 | 0.05350 | 0.08863 | 0.16768 | 0.10001 | 0.12505 | 0.01564 | |
MSE | 0.00513 | 0.00318 | 0.00587 | 0.00318 | 0.00403 | 0.01248 | 0.04391 | 0.01414 | 0.02254 | 0.00041 | |
MRE | 0.24614 | 0.06073 | 0.23228 | 0.21907 | 0.23228 | 0.09483 | 0.11660 | 0.23562 | 0.12993 | 0.07820 | |
60 | \hat{E} | -0.22444 | 0.79973 | -0.25575 | -0.20027 | -0.21187 | -0.90532 | 1.38427 | -0.39031 | -0.92289 | -0.19737 |
BIAS | 0.03918 | 0.03122 | 0.04221 | 0.03122 | 0.03497 | 0.05784 | 0.10973 | 0.06527 | 0.08070 | 0.00931 | |
MSE | 0.00233 | 0.00150 | 0.00272 | 0.00150 | 0.00187 | 0.00543 | 0.01921 | 0.00653 | 0.00994 | 0.00013 | |
MRE | 0.16010 | 0.03987 | 0.15182 | 0.14382 | 0.15182 | 0.06189 | 0.07631 | 0.15377 | 0.08385 | 0.04657 | |
100 | \hat{E} | -0.22910 | 0.79586 | -0.26088 | -0.20414 | -0.21612 | -0.91192 | 1.39608 | -0.39810 | -0.93166 | -0.19806 |
BIAS | 0.03308 | 0.02624 | 0.03556 | 0.02624 | 0.02946 | 0.04549 | 0.08654 | 0.05470 | 0.06766 | 0.00776 | |
MSE | 0.00179 | 0.00113 | 0.00207 | 0.00113 | 0.00142 | 0.00351 | 0.01253 | 0.00486 | 0.00735 | 0.00010 | |
MRE | 0.13519 | 0.03351 | 0.12789 | 0.12087 | 0.12789 | 0.04868 | 0.06018 | 0.12887 | 0.07030 | 0.03881 | |
150 | \hat{E} | -0.23449 | 0.79145 | -0.26676 | -0.20855 | -0.22099 | -0.91935 | 1.40990 | -0.40710 | -0.94232 | -0.19880 |
BIAS | 0.02881 | 0.02270 | 0.03087 | 0.02270 | 0.02557 | 0.03868 | 0.07394 | 0.04755 | 0.05926 | 0.00679 | |
MSE | 0.00132 | 0.00082 | 0.00151 | 0.00082 | 0.00104 | 0.00245 | 0.00889 | 0.00358 | 0.00556 | 0.00007 | |
MRE | 0.11772 | 0.02900 | 0.11102 | 0.10459 | 0.11102 | 0.04138 | 0.05141 | 0.11203 | 0.06158 | 0.03394 | |
200 | \hat{E} | -0.23524 | 0.79074 | -0.26765 | -0.20926 | -0.22173 | -0.92085 | 1.41249 | -0.40845 | -0.94356 | -0.19891 |
BIAS | 0.02538 | 0.02001 | 0.02721 | 0.02001 | 0.02254 | 0.03402 | 0.06510 | 0.04188 | 0.05223 | 0.00608 | |
MSE | 0.00098 | 0.00061 | 0.00113 | 0.00061 | 0.00078 | 0.00184 | 0.00669 | 0.00268 | 0.00415 | 0.00006 | |
MRE | 0.10374 | 0.02556 | 0.09785 | 0.09220 | 0.09785 | 0.03640 | 0.04527 | 0.09866 | 0.05426 | 0.03038 | |
250 | \hat{E} | -0.23646 | 0.78972 | -0.26900 | -0.21028 | -0.22285 | -0.92295 | 1.41637 | -0.41058 | -0.94608 | -0.19896 |
BIAS | 0.02336 | 0.01841 | 0.02504 | 0.01841 | 0.02074 | 0.03051 | 0.05844 | 0.03856 | 0.04798 | 0.00550 | |
MSE | 0.00082 | 0.00051 | 0.00095 | 0.00051 | 0.00065 | 0.00149 | 0.00544 | 0.00224 | 0.00346 | 0.00005 | |
MRE | 0.09548 | 0.02352 | 0.09005 | 0.08483 | 0.09005 | 0.03265 | 0.04064 | 0.09083 | 0.04986 | 0.02752 | |
300 | \hat{E} | -0.23566 | 0.79028 | -0.26819 | -0.20972 | -0.22217 | -0.92243 | 1.41526 | -0.40936 | -0.94439 | -0.19879 |
BIAS | 0.02066 | 0.01630 | 0.02215 | 0.01630 | 0.01835 | 0.02713 | 0.05194 | 0.03399 | 0.04211 | 0.00489 | |
MSE | 0.00066 | 0.00041 | 0.00076 | 0.00041 | 0.00052 | 0.00124 | 0.00450 | 0.00181 | 0.00278 | 0.00004 | |
MRE | 0.08442 | 0.02082 | 0.07967 | 0.07511 | 0.07967 | 0.02903 | 0.03612 | 0.08008 | 0.04375 | 0.02447 | |
400 | \hat{E} | -0.23764 | 0.78865 | -0.27036 | -0.21135 | -0.22397 | -0.92502 | 1.42006 | -0.41266 | -0.94823 | -0.19911 |
BIAS | 0.01779 | 0.01402 | 0.01906 | 0.01402 | 0.01579 | 0.02303 | 0.04416 | 0.02926 | 0.03636 | 0.00412 | |
MSE | 0.00048 | 0.00030 | 0.00055 | 0.00030 | 0.00038 | 0.00085 | 0.00310 | 0.00129 | 0.00200 | 0.00003 | |
MRE | 0.07271 | 0.01790 | 0.06856 | 0.06457 | 0.06856 | 0.02464 | 0.03071 | 0.06893 | 0.03777 | 0.02061 |
Model | \hat{\delta} | SE( \hat{\delta} ) | \hat{\beta} | SE( \hat{\beta} ) |
PUIL | 2.4144 | 0.4321 | 0.0068 | 0.0072 |
UIL | 0.2045 | 0.0326 | ||
ETL | 1.7370 | 0.2896 | 9.7115 | 3.8780 |
Km | 1.5878 | 0.2444 | 21.8673 | 10.2082 |
Be | 3.1127 | 0.9368 | 21.8246 | 7.0422 |
TrG | 14.6813 | 2.3213 |
Model | \hat{\delta} | SE( \hat{\delta} ) | \hat{\beta} | SE( \hat{\beta} ) |
PUIL | 2.9709 | 0.5340 | 0.1091 | 0.0645 |
UIL | 0.9867 | 0.1666 | ||
ETL | 4.6858 | 0.9595 | 4.1306 | 1.5083 |
Km | 3.4039 | 0.6073 | 12.0731 | 5.4978 |
Be | 6.9757 | 2.1638 | 9.3522 | 2.9276 |
TrG | 3.4438 | 0.54452 |
Model | Aic | Caic | Bic | Hqic | A | W | KS | KSp | ShE | DEX | WEX |
PUIL | -71.5426 | -70.8367 | -69.5511 | -71.1538 | 0.4162 | 0.0500 | 0.1259 | 0.9092 | -1.9713 | -4.6321 | -0.4527 |
UIL | -57.9514 | -57.7292 | -56.9557 | -57.7570 | 2.6972 | 0.5309 | 0.3036 | 0.05012 | -0.9807 | -1.9026 | -0.1944 |
ETL | 48.2272 | -47.5213 | -46.2358 | -47.8385 | 2.6147 | 0.4524 | 0.2641 | 0.1229 | -1.2521 | -2.0067 | -0.2121 |
Km | -47.2969 | -46.5910 | -45.3054 | -46.9081 | 2.6889 | 0.4681 | 0.2627 | 0.1265 | -1.2290 | -1.9560 | -0.2031 |
Be | -51.7626 | -51.0567 | -49.7711 | -51.3738 | 2.2611 | 0.3727 | 0.2538 | 0.1521 | -1.3941 | -2.3467 | -0.2561 |
TrG | -51.8497 | -51.6275 | -50.8540 | -51.6553 | 2.5040 | 0.4327 | 0.2709 | 0.1062 | -1.2456 | -2.0362 | -0.2010 |
Model | Aic | Caic | Bic | Hqic | A | W | KS | KSp | ShE | DEX | WEX |
PUIL | -30.0341 | -29.3282 | -28.0427 | -29.6454 | 0.2522 | 0.0425 | 0.1226 | 0.9247 | -0.8583 | -1.4576 | -0.5632 |
UIL | -16.4854 | -16.2631 | -15.4896 | -16.2910 | 2.4384 | 0.4656 | 0.3009 | 0.0535 | -0.2021 | -0.6580 | -0.2946 |
ETL | -23.6156 | -22.9097 | -21.6241 | -23.2268 | 0.8845 | 0.1553 | 0.2110 | 0.3353 | -0.6532 | -1.0934 | -0.4659 |
Km | -22.0935 | -21.3876 | -20.1020 | -21.7047 | 1.0040 | 0.17602 | 0.2151 | 0.3132 | -0.6031 | -1.0336 | -0.4439 |
Be | -24.6329 | -23.9270 | -22.6414 | -24.2441 | 0.7991 | 0.1345 | 0.2038 | 0.3771 | -0.7158 | -1.1654 | -0.4923 |
TrG | -15.3907 | -15.1684 | -14.3949 | -15.1963 | 2.1343 | 0.3959 | 0.2905 | 0.0684 | -0.2401 | -0.6892 | -0.2587 |
Name of the entropy | Reference | Expression |
Shannon | [44] | S(\beta, \delta)=-\displaystyle {\int}_{0}^{1}g(z; \beta, \delta) \log \left[ g(z; \beta, \delta)\right] dz |
Rényi | [45] | R_{\kappa}(\beta, \delta)=\frac{1}{1-\kappa} \log\left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right] |
Exponential | [46] | E_{\kappa}(\beta, \delta)=\left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right]^{\frac{1}{1-\kappa}} |
Havrda and Charvat | [47] | HC_{\kappa}(\beta, \delta)=\frac{1}{2^{1-\kappa}-1} \left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz-1\right] |
Arimoto | [48] | A_{\kappa}(\beta, \delta)=\frac{\kappa}{1-\kappa} \left\lbrace \left[\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right]^{\frac{1}{\kappa}}-1\right\rbrace |
Tsallis | [49] | T_{\kappa}(\beta, \delta)=\frac{1}{\kappa-1} \left[1-\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right] |
Awad and Alawneh 1 | [50] | AA1_{\kappa}(\beta, \delta)=\frac{1}{\kappa-1} \log \left\lbrace\left[\sup\limits_{z\in\mathbb{R}}g(z; \beta, \delta)\right]^{1-\kappa}\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right\rbrace |
Awad and Alawneh 2 | [50] | AA2_{\kappa}(\beta, \delta)=\frac{1}{2^{1-\kappa}-1} \left[\left\lbrace\left[\sup\limits_{z\in\mathbb{R}}g(z; \beta, \delta)\right]^{1-\kappa}\displaystyle {\int}_{0}^{1}g(z; \beta, \delta)^{\kappa}dz\right\rbrace -1\right] |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.22373 ^{\{ 7 \}} | 0.22577 ^{\{ 8 \}} | 0.25312 ^{\{ 14 \}} | 0.20754 ^{\{ 5 \}} | 0.25032 ^{\{ 13 \}} | 0.2571 ^{\{ 15 \}} | 0.2331 ^{\{ 11 \}} | 0.23345 ^{\{ 12 \}} | 0.13758 ^{\{ 2 \}} | 0.18677 ^{\{ 3 \}} | 0.22978 ^{\{ 10 \}} | 0.08802 ^{\{ 1 \}} | 0.22914 ^{\{ 9 \}} | 0.2176 ^{\{ 6 \}} | 0.20036 ^{\{ 4 \}} |
BIAS( \hat{\beta} ) | 0.7744 ^{\{ 7 \}} | 0.80526 ^{\{ 9 \}} | 0.75617 ^{\{ 6 \}} | 0.82699 ^{\{ 11 \}} | 0.77464 ^{\{ 8 \}} | 0.80846 ^{\{ 10 \}} | 0.84149 ^{\{ 13 \}} | 0.83473 ^{\{ 12 \}} | 0.29217 ^{\{ 2 \}} | 0.7215 ^{\{ 4 \}} | 0.85207 ^{\{ 15 \}} | 0.04434 ^{\{ 1 \}} | 0.72321 ^{\{ 5 \}} | 0.84278 ^{\{ 14 \}} | 0.64726 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.07932 ^{\{ 8 \}} | 0.07826 ^{\{ 7 \}} | 0.09935 ^{\{ 14 \}} | 0.06438 ^{\{ 4 \}} | 0.09656 ^{\{ 13 \}} | 0.09952 ^{\{ 15 \}} | 0.08196 ^{\{ 9 \}} | 0.08382 ^{\{ 11 \}} | 0.03691 ^{\{ 2 \}} | 0.05693 ^{\{ 3 \}} | 0.08258 ^{\{ 10 \}} | 0.01303 ^{\{ 1 \}} | 0.08619 ^{\{ 12 \}} | 0.07251 ^{\{ 6 \}} | 0.06923 ^{\{ 5 \}} | |
MSE( \hat{\beta} ) | 0.85507 ^{\{ 7 \}} | 0.93048 ^{\{ 10 \}} | 0.79375 ^{\{ 5 \}} | 1.00678 ^{\{ 13 \}} | 0.83857 ^{\{ 6 \}} | 0.87698 ^{\{ 8 \}} | 0.99586 ^{\{ 12 \}} | 0.98746 ^{\{ 11 \}} | 0.29807 ^{\{ 2 \}} | 0.88098 ^{\{ 9 \}} | 1.04622 ^{\{ 15 \}} | 0.00824 ^{\{ 1 \}} | 0.74237 ^{\{ 4 \}} | 1.02031 ^{\{ 14 \}} | 0.62886 ^{\{ 3 \}} | |
MRE( \hat{\delta} ) | 0.31961 ^{\{ 7 \}} | 0.32252 ^{\{ 8 \}} | 0.36159 ^{\{ 14 \}} | 0.29648 ^{\{ 5 \}} | 0.3576 ^{\{ 13 \}} | 0.36728 ^{\{ 15 \}} | 0.333 ^{\{ 11 \}} | 0.3335 ^{\{ 12 \}} | 0.19654 ^{\{ 2 \}} | 0.26682 ^{\{ 3 \}} | 0.32825 ^{\{ 10 \}} | 0.12574 ^{\{ 1 \}} | 0.32734 ^{\{ 9 \}} | 0.31086 ^{\{ 6 \}} | 0.28622 ^{\{ 4 \}} | |
MRE( \hat{\beta} ) | 0.30976 ^{\{ 7 \}} | 0.3221 ^{\{ 9 \}} | 0.30247 ^{\{ 6 \}} | 0.33079 ^{\{ 11 \}} | 0.30985 ^{\{ 8 \}} | 0.32339 ^{\{ 10 \}} | 0.33659 ^{\{ 13 \}} | 0.33389 ^{\{ 12 \}} | 0.11687 ^{\{ 2 \}} | 0.2886 ^{\{ 4 \}} | 0.34083 ^{\{ 15 \}} | 0.01774 ^{\{ 1 \}} | 0.28928 ^{\{ 5 \}} | 0.33711 ^{\{ 14 \}} | 0.2589 ^{\{ 3 \}} | |
D_{abs} | 0.04005 ^{\{ 1 \}} | 0.04126 ^{\{ 4 \}} | 0.04356 ^{\{ 10 \}} | 0.04111 ^{\{ 2 \}} | 0.04425 ^{\{ 11 \}} | 0.04479 ^{\{ 12 \}} | 0.04172 ^{\{ 6 \}} | 0.0414 ^{\{ 5 \}} | 0.04519 ^{\{ 13 \}} | 0.04312 ^{\{ 9 \}} | 0.04124 ^{\{ 3 \}} | 0.04269 ^{\{ 8 \}} | 0.06087 ^{\{ 15 \}} | 0.04232 ^{\{ 7 \}} | 0.05767 ^{\{ 14 \}} | |
D_{max} | 0.06512 ^{\{ 3 \}} | 0.06645 ^{\{ 4 \}} | 0.07113 ^{\{ 12 \}} | 0.06499 ^{\{ 2 \}} | 0.0708 ^{\{ 11 \}} | 0.07279 ^{\{ 13 \}} | 0.06712 ^{\{ 8 \}} | 0.06667 ^{\{ 5 \}} | 0.06964 ^{\{ 10 \}} | 0.06785 ^{\{ 9 \}} | 0.06693 ^{\{ 6 \}} | 0.06424 ^{\{ 1 \}} | 0.09209 ^{\{ 15 \}} | 0.06696 ^{\{ 7 \}} | 0.08699 ^{\{ 14 \}} | |
\sum Ranks | 47 ^{\{ 4 \}} | 59 ^{\{ 7 \}} | 81 ^{\{ 11 \}} | 53 ^{\{ 6 \}} | 83 ^{\{ 12.5 \}} | 98 ^{\{ 15 \}} | 83 ^{\{ 12.5 \}} | 80 ^{\{ 10 \}} | 35 ^{\{ 2 \}} | 44 ^{\{ 3 \}} | 84 ^{\{ 14 \}} | 15 ^{\{ 1 \}} | 74 ^{\{ 8.5 \}} | 74 ^{\{ 8.5 \}} | 50 ^{\{ 5 \}} | |
60 | BIAS( \hat{\delta} ) | 0.17259 ^{\{ 5 \}} | 0.19678 ^{\{ 11 \}} | 0.22989 ^{\{ 15 \}} | 0.17065 ^{\{ 4 \}} | 0.20041 ^{\{ 13 \}} | 0.22807 ^{\{ 14 \}} | 0.19883 ^{\{ 12 \}} | 0.19121 ^{\{ 9 \}} | 0.1085 ^{\{ 2 \}} | 0.16308 ^{\{ 3 \}} | 0.19184 ^{\{ 10 \}} | 0.05999 ^{\{ 1 \}} | 0.18004 ^{\{ 7 \}} | 0.18312 ^{\{ 8 \}} | 0.17361 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.68127 ^{\{ 5 \}} | 0.78854 ^{\{ 14 \}} | 0.7418 ^{\{ 8 \}} | 0.74548 ^{\{ 9 \}} | 0.71544 ^{\{ 7 \}} | 0.77753 ^{\{ 13 \}} | 0.7739 ^{\{ 12 \}} | 0.74844 ^{\{ 10 \}} | 0.27546 ^{\{ 2 \}} | 0.69382 ^{\{ 6 \}} | 0.75274 ^{\{ 11 \}} | 0.03594 ^{\{ 1 \}} | 0.58942 ^{\{ 3 \}} | 0.80753 ^{\{ 15 \}} | 0.59255 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.04735 ^{\{ 5 \}} | 0.05737 ^{\{ 9 \}} | 0.0815 ^{\{ 15 \}} | 0.04443 ^{\{ 4 \}} | 0.06477 ^{\{ 13 \}} | 0.08028 ^{\{ 14 \}} | 0.06309 ^{\{ 12 \}} | 0.05805 ^{\{ 10 \}} | 0.02402 ^{\{ 2 \}} | 0.04275 ^{\{ 3 \}} | 0.05944 ^{\{ 11 \}} | 0.00602 ^{\{ 1 \}} | 0.05663 ^{\{ 8 \}} | 0.05151 ^{\{ 6 \}} | 0.05354 ^{\{ 7 \}} | |
MSE( \hat{\beta} ) | 0.71243 ^{\{ 5 \}} | 0.92878 ^{\{ 14 \}} | 0.76945 ^{\{ 7 \}} | 0.87146 ^{\{ 12 \}} | 0.74299 ^{\{ 6 \}} | 0.85999 ^{\{ 10 \}} | 0.91218 ^{\{ 13 \}} | 0.84322 ^{\{ 9 \}} | 0.26265 ^{\{ 2 \}} | 0.81005 ^{\{ 8 \}} | 0.86264 ^{\{ 11 \}} | 0.00554 ^{\{ 1 \}} | 0.50974 ^{\{ 3 \}} | 0.99557 ^{\{ 15 \}} | 0.5388 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.24656 ^{\{ 5 \}} | 0.28111 ^{\{ 11 \}} | 0.32842 ^{\{ 15 \}} | 0.24378 ^{\{ 4 \}} | 0.2863 ^{\{ 13 \}} | 0.32581 ^{\{ 14 \}} | 0.28404 ^{\{ 12 \}} | 0.27316 ^{\{ 9 \}} | 0.155 ^{\{ 2 \}} | 0.23297 ^{\{ 3 \}} | 0.27405 ^{\{ 10 \}} | 0.08571 ^{\{ 1 \}} | 0.25719 ^{\{ 7 \}} | 0.26159 ^{\{ 8 \}} | 0.24802 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.27251 ^{\{ 5 \}} | 0.31542 ^{\{ 14 \}} | 0.29672 ^{\{ 8 \}} | 0.29819 ^{\{ 9 \}} | 0.28618 ^{\{ 7 \}} | 0.31101 ^{\{ 13 \}} | 0.30956 ^{\{ 12 \}} | 0.29938 ^{\{ 10 \}} | 0.11018 ^{\{ 2 \}} | 0.27753 ^{\{ 6 \}} | 0.3011 ^{\{ 11 \}} | 0.01437 ^{\{ 1 \}} | 0.23577 ^{\{ 3 \}} | 0.32301 ^{\{ 15 \}} | 0.23702 ^{\{ 4 \}} | |
D_{abs} | 0.0295 ^{\{ 2 \}} | 0.03057 ^{\{ 5 \}} | 0.03189 ^{\{ 13 \}} | 0.02811 ^{\{ 1 \}} | 0.03141 ^{\{ 11 \}} | 0.03108 ^{\{ 9 \}} | 0.03081 ^{\{ 6 \}} | 0.03091 ^{\{ 8 \}} | 0.03083 ^{\{ 7 \}} | 0.0314 ^{\{ 10 \}} | 0.02956 ^{\{ 3 \}} | 0.02965 ^{\{ 4 \}} | 0.04047 ^{\{ 14 \}} | 0.03166 ^{\{ 12 \}} | 0.04048 ^{\{ 15 \}} | |
D_{max} | 0.04787 ^{\{ 3 \}} | 0.04999 ^{\{ 6 \}} | 0.05351 ^{\{ 13 \}} | 0.04551 ^{\{ 2 \}} | 0.05135 ^{\{ 11 \}} | 0.05238 ^{\{ 12 \}} | 0.05037 ^{\{ 7 \}} | 0.05048 ^{\{ 9 \}} | 0.04829 ^{\{ 4 \}} | 0.05038 ^{\{ 8 \}} | 0.04885 ^{\{ 5 \}} | 0.04457 ^{\{ 1 \}} | 0.06312 ^{\{ 14 \}} | 0.05093 ^{\{ 10 \}} | 0.06349 ^{\{ 15 \}} | |
\sum Ranks | 35 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 94 ^{\{ 14 \}} | 45 ^{\{ 4 \}} | 81 ^{\{ 10 \}} | 99 ^{\{ 15 \}} | 86 ^{\{ 12 \}} | 74 ^{\{ 9 \}} | 23 ^{\{ 2 \}} | 47 ^{\{ 5 \}} | 72 ^{\{ 8 \}} | 11 ^{\{ 1 \}} | 59 ^{\{ 6 \}} | 89 ^{\{ 13 \}} | 61 ^{\{ 7 \}} | |
100 | BIAS( \hat{\delta} ) | 0.14625 ^{\{ 7 \}} | 0.16744 ^{\{ 12 \}} | 0.19457 ^{\{ 14 \}} | 0.14162 ^{\{ 5 \}} | 0.185 ^{\{ 13 \}} | 0.21524 ^{\{ 15 \}} | 0.16618 ^{\{ 11 \}} | 0.15543 ^{\{ 9 \}} | 0.0977 ^{\{ 2 \}} | 0.13859 ^{\{ 4 \}} | 0.15888 ^{\{ 10 \}} | 0.0488 ^{\{ 1 \}} | 0.13599 ^{\{ 3 \}} | 0.15167 ^{\{ 8 \}} | 0.14603 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.59238 ^{\{ 5 \}} | 0.68527 ^{\{ 13 \}} | 0.68361 ^{\{ 12 \}} | 0.67297 ^{\{ 9 \}} | 0.67575 ^{\{ 10 \}} | 0.75015 ^{\{ 15 \}} | 0.66326 ^{\{ 8 \}} | 0.62369 ^{\{ 6 \}} | 0.27316 ^{\{ 2 \}} | 0.64513 ^{\{ 7 \}} | 0.70079 ^{\{ 14 \}} | 0.03289 ^{\{ 1 \}} | 0.48933 ^{\{ 3 \}} | 0.67978 ^{\{ 11 \}} | 0.52629 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.03434 ^{\{ 6 \}} | 0.04318 ^{\{ 12 \}} | 0.0602 ^{\{ 14 \}} | 0.02976 ^{\{ 3 \}} | 0.05233 ^{\{ 13 \}} | 0.07238 ^{\{ 15 \}} | 0.04301 ^{\{ 11 \}} | 0.0378 ^{\{ 8 \}} | 0.02044 ^{\{ 2 \}} | 0.03082 ^{\{ 4 \}} | 0.03962 ^{\{ 10 \}} | 0.0038 ^{\{ 1 \}} | 0.03378 ^{\{ 5 \}} | 0.03602 ^{\{ 7 \}} | 0.03945 ^{\{ 9 \}} | |
MSE( \hat{\beta} ) | 0.56392 ^{\{ 5 \}} | 0.7543 ^{\{ 13 \}} | 0.69846 ^{\{ 9 \}} | 0.74207 ^{\{ 12 \}} | 0.66928 ^{\{ 7 \}} | 0.84187 ^{\{ 15 \}} | 0.68905 ^{\{ 8 \}} | 0.62999 ^{\{ 6 \}} | 0.25309 ^{\{ 2 \}} | 0.73522 ^{\{ 10 \}} | 0.78489 ^{\{ 14 \}} | 0.00524 ^{\{ 1 \}} | 0.37739 ^{\{ 3 \}} | 0.74173 ^{\{ 11 \}} | 0.48398 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.20892 ^{\{ 7 \}} | 0.2392 ^{\{ 12 \}} | 0.27796 ^{\{ 14 \}} | 0.20231 ^{\{ 5 \}} | 0.26429 ^{\{ 13 \}} | 0.30748 ^{\{ 15 \}} | 0.2374 ^{\{ 11 \}} | 0.22204 ^{\{ 9 \}} | 0.13957 ^{\{ 2 \}} | 0.19798 ^{\{ 4 \}} | 0.22697 ^{\{ 10 \}} | 0.06972 ^{\{ 1 \}} | 0.19427 ^{\{ 3 \}} | 0.21668 ^{\{ 8 \}} | 0.20861 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.23695 ^{\{ 5 \}} | 0.27411 ^{\{ 13 \}} | 0.27344 ^{\{ 12 \}} | 0.26919 ^{\{ 9 \}} | 0.2703 ^{\{ 10 \}} | 0.30006 ^{\{ 15 \}} | 0.26531 ^{\{ 8 \}} | 0.24948 ^{\{ 6 \}} | 0.10926 ^{\{ 2 \}} | 0.25805 ^{\{ 7 \}} | 0.28032 ^{\{ 14 \}} | 0.01315 ^{\{ 1 \}} | 0.19573 ^{\{ 3 \}} | 0.27191 ^{\{ 11 \}} | 0.21052 ^{\{ 4 \}} | |
D_{abs} | 0.02341 ^{\{ 2 \}} | 0.02377 ^{\{ 4 \}} | 0.02389 ^{\{ 5 \}} | 0.02292 ^{\{ 1 \}} | 0.02502 ^{\{ 8 \}} | 0.02535 ^{\{ 10 \}} | 0.02347 ^{\{ 3 \}} | 0.024 ^{\{ 6 \}} | 0.02561 ^{\{ 12 \}} | 0.02526 ^{\{ 9 \}} | 0.02537 ^{\{ 11 \}} | 0.02447 ^{\{ 7 \}} | 0.03101 ^{\{ 15 \}} | 0.02653 ^{\{ 13 \}} | 0.03041 ^{\{ 14 \}} | |
D_{max} | 0.03833 ^{\{ 3 \}} | 0.03924 ^{\{ 5 \}} | 0.04063 ^{\{ 9 \}} | 0.03709 ^{\{ 2 \}} | 0.0422 ^{\{ 11 \}} | 0.04366 ^{\{ 13 \}} | 0.03885 ^{\{ 4 \}} | 0.03946 ^{\{ 6 \}} | 0.04025 ^{\{ 7 \}} | 0.04062 ^{\{ 8 \}} | 0.04131 ^{\{ 10 \}} | 0.03704 ^{\{ 1 \}} | 0.0492 ^{\{ 15 \}} | 0.04265 ^{\{ 12 \}} | 0.04866 ^{\{ 14 \}} | |
\sum Ranks | 40 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 89 ^{\{ 13 \}} | 46 ^{\{ 4 \}} | 85 ^{\{ 12 \}} | 113 ^{\{ 15 \}} | 64 ^{\{ 9 \}} | 56 ^{\{ 7 \}} | 31 ^{\{ 2 \}} | 53 ^{\{ 6 \}} | 93 ^{\{ 14 \}} | 14 ^{\{ 1 \}} | 50 ^{\{ 5 \}} | 81 ^{\{ 10 \}} | 61 ^{\{ 8 \}} | |
200 | BIAS( \hat{\delta} ) | 0.09648 ^{\{ 3 \}} | 0.12482 ^{\{ 10 \}} | 0.15103 ^{\{ 14 \}} | 0.11384 ^{\{ 6 \}} | 0.1425 ^{\{ 13 \}} | 0.17269 ^{\{ 15 \}} | 0.13156 ^{\{ 12 \}} | 0.11484 ^{\{ 7 \}} | 0.07786 ^{\{ 2 \}} | 0.11322 ^{\{ 5 \}} | 0.12492 ^{\{ 11 \}} | 0.03359 ^{\{ 1 \}} | 0.11157 ^{\{ 4 \}} | 0.11946 ^{\{ 8 \}} | 0.12256 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.42875 ^{\{ 3 \}} | 0.5295 ^{\{ 7 \}} | 0.61287 ^{\{ 14 \}} | 0.54979 ^{\{ 8 \}} | 0.59034 ^{\{ 13 \}} | 0.63661 ^{\{ 15 \}} | 0.55697 ^{\{ 10 \}} | 0.50632 ^{\{ 6 \}} | 0.25968 ^{\{ 2 \}} | 0.55546 ^{\{ 9 \}} | 0.58785 ^{\{ 12 \}} | 0.02817 ^{\{ 1 \}} | 0.437 ^{\{ 4 \}} | 0.5818 ^{\{ 11 \}} | 0.45147 ^{\{ 5 \}} | |
MSE( \hat{\delta} ) | 0.01468 ^{\{ 3 \}} | 0.02419 ^{\{ 10 \}} | 0.03546 ^{\{ 14 \}} | 0.01995 ^{\{ 5 \}} | 0.03181 ^{\{ 13 \}} | 0.04731 ^{\{ 15 \}} | 0.02608 ^{\{ 11 \}} | 0.02091 ^{\{ 6 \}} | 0.01268 ^{\{ 2 \}} | 0.01992 ^{\{ 4 \}} | 0.02357 ^{\{ 9 \}} | 0.00198 ^{\{ 1 \}} | 0.023 ^{\{ 8 \}} | 0.02177 ^{\{ 7 \}} | 0.02768 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.31499 ^{\{ 3 \}} | 0.46709 ^{\{ 7 \}} | 0.61924 ^{\{ 14 \}} | 0.53484 ^{\{ 9 \}} | 0.58357 ^{\{ 12 \}} | 0.65048 ^{\{ 15 \}} | 0.5114 ^{\{ 8 \}} | 0.44203 ^{\{ 6 \}} | 0.2079 ^{\{ 2 \}} | 0.55213 ^{\{ 10 \}} | 0.56088 ^{\{ 11 \}} | 0.00473 ^{\{ 1 \}} | 0.34757 ^{\{ 4 \}} | 0.58871 ^{\{ 13 \}} | 0.37455 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.13782 ^{\{ 3 \}} | 0.17831 ^{\{ 10 \}} | 0.21575 ^{\{ 14 \}} | 0.16263 ^{\{ 6 \}} | 0.20357 ^{\{ 13 \}} | 0.2467 ^{\{ 15 \}} | 0.18794 ^{\{ 12 \}} | 0.16405 ^{\{ 7 \}} | 0.11122 ^{\{ 2 \}} | 0.16174 ^{\{ 5 \}} | 0.17846 ^{\{ 11 \}} | 0.04798 ^{\{ 1 \}} | 0.15939 ^{\{ 4 \}} | 0.17065 ^{\{ 8 \}} | 0.17509 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.1715 ^{\{ 3 \}} | 0.2118 ^{\{ 7 \}} | 0.24515 ^{\{ 14 \}} | 0.21992 ^{\{ 8 \}} | 0.23613 ^{\{ 13 \}} | 0.25465 ^{\{ 15 \}} | 0.22279 ^{\{ 10 \}} | 0.20253 ^{\{ 6 \}} | 0.10387 ^{\{ 2 \}} | 0.22218 ^{\{ 9 \}} | 0.23514 ^{\{ 12 \}} | 0.01127 ^{\{ 1 \}} | 0.1748 ^{\{ 4 \}} | 0.23272 ^{\{ 11 \}} | 0.18059 ^{\{ 5 \}} | |
D_{abs} | 0.01667 ^{\{ 2 \}} | 0.0172 ^{\{ 5 \}} | 0.01824 ^{\{ 9 \}} | 0.01714 ^{\{ 4 \}} | 0.01785 ^{\{ 8 \}} | 0.01886 ^{\{ 11 \}} | 0.01762 ^{\{ 6 \}} | 0.01696 ^{\{ 3 \}} | 0.01783 ^{\{ 7 \}} | 0.02031 ^{\{ 13 \}} | 0.01851 ^{\{ 10 \}} | 0.01629 ^{\{ 1 \}} | 0.02275 ^{\{ 15 \}} | 0.01899 ^{\{ 12 \}} | 0.02256 ^{\{ 14 \}} | |
D_{max} | 0.0269 ^{\{ 2 \}} | 0.02854 ^{\{ 6 \}} | 0.03108 ^{\{ 11 \}} | 0.02791 ^{\{ 4 \}} | 0.03005 ^{\{ 8 \}} | 0.03272 ^{\{ 13 \}} | 0.0292 ^{\{ 7 \}} | 0.02788 ^{\{ 3 \}} | 0.02821 ^{\{ 5 \}} | 0.03259 ^{\{ 12 \}} | 0.03021 ^{\{ 9 \}} | 0.02461 ^{\{ 1 \}} | 0.03658 ^{\{ 15 \}} | 0.03087 ^{\{ 10 \}} | 0.03649 ^{\{ 14 \}} | |
\sum Ranks | 22 ^{\{ 2 \}} | 62 ^{\{ 7 \}} | 104 ^{\{ 14 \}} | 50 ^{\{ 5 \}} | 93 ^{\{ 13 \}} | 114 ^{\{ 15 \}} | 76 ^{\{ 10 \}} | 44 ^{\{ 4 \}} | 24 ^{\{ 3 \}} | 67 ^{\{ 8 \}} | 85 ^{\{ 12 \}} | 8 ^{\{ 1 \}} | 58 ^{\{ 6 \}} | 80 ^{\{ 11 \}} | 73 ^{\{ 9 \}} | |
300 | BIAS( \hat{\delta} ) | 0.08469 ^{\{ 3 \}} | 0.10079 ^{\{ 10 \}} | 0.1252 ^{\{ 14 \}} | 0.09076 ^{\{ 4 \}} | 0.12392 ^{\{ 13 \}} | 0.14567 ^{\{ 15 \}} | 0.10608 ^{\{ 12 \}} | 0.09495 ^{\{ 6 \}} | 0.06793 ^{\{ 2 \}} | 0.09395 ^{\{ 5 \}} | 0.10567 ^{\{ 11 \}} | 0.02746 ^{\{ 1 \}} | 0.09748 ^{\{ 7 \}} | 0.10026 ^{\{ 8 \}} | 0.10057 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.36972 ^{\{ 3 \}} | 0.43786 ^{\{ 8 \}} | 0.50879 ^{\{ 13 \}} | 0.43499 ^{\{ 7 \}} | 0.53725 ^{\{ 14 \}} | 0.57736 ^{\{ 15 \}} | 0.45602 ^{\{ 10 \}} | 0.42746 ^{\{ 6 \}} | 0.24286 ^{\{ 2 \}} | 0.44069 ^{\{ 9 \}} | 0.5055 ^{\{ 12 \}} | 0.0244 ^{\{ 1 \}} | 0.38919 ^{\{ 5 \}} | 0.46693 ^{\{ 11 \}} | 0.3866 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.01148 ^{\{ 3 \}} | 0.01591 ^{\{ 8 \}} | 0.02484 ^{\{ 14 \}} | 0.01253 ^{\{ 4 \}} | 0.02397 ^{\{ 13 \}} | 0.03315 ^{\{ 15 \}} | 0.01778 ^{\{ 10 \}} | 0.01434 ^{\{ 6 \}} | 0.00976 ^{\{ 2 \}} | 0.01432 ^{\{ 5 \}} | 0.01768 ^{\{ 9 \}} | 0.00126 ^{\{ 1 \}} | 0.01805 ^{\{ 11 \}} | 0.01511 ^{\{ 7 \}} | 0.01817 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.24338 ^{\{ 3 \}} | 0.33897 ^{\{ 7 \}} | 0.44168 ^{\{ 12 \}} | 0.32176 ^{\{ 6 \}} | 0.51632 ^{\{ 14 \}} | 0.54731 ^{\{ 15 \}} | 0.35494 ^{\{ 9 \}} | 0.34003 ^{\{ 8 \}} | 0.17342 ^{\{ 2 \}} | 0.36794 ^{\{ 10 \}} | 0.44555 ^{\{ 13 \}} | 0.00288 ^{\{ 1 \}} | 0.29013 ^{\{ 5 \}} | 0.37214 ^{\{ 11 \}} | 0.26368 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.12099 ^{\{ 3 \}} | 0.14399 ^{\{ 10 \}} | 0.17886 ^{\{ 14 \}} | 0.12966 ^{\{ 4 \}} | 0.17703 ^{\{ 13 \}} | 0.2081 ^{\{ 15 \}} | 0.15154 ^{\{ 12 \}} | 0.13565 ^{\{ 6 \}} | 0.09705 ^{\{ 2 \}} | 0.13422 ^{\{ 5 \}} | 0.15096 ^{\{ 11 \}} | 0.03922 ^{\{ 1 \}} | 0.13925 ^{\{ 7 \}} | 0.14323 ^{\{ 8 \}} | 0.14368 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.14789 ^{\{ 3 \}} | 0.17514 ^{\{ 8 \}} | 0.20352 ^{\{ 13 \}} | 0.174 ^{\{ 7 \}} | 0.2149 ^{\{ 14 \}} | 0.23094 ^{\{ 15 \}} | 0.18241 ^{\{ 10 \}} | 0.17099 ^{\{ 6 \}} | 0.09715 ^{\{ 2 \}} | 0.17628 ^{\{ 9 \}} | 0.2022 ^{\{ 12 \}} | 0.00976 ^{\{ 1 \}} | 0.15568 ^{\{ 5 \}} | 0.18677 ^{\{ 11 \}} | 0.15464 ^{\{ 4 \}} | |
D_{abs} | 0.01342 ^{\{ 1 \}} | 0.0141 ^{\{ 5 \}} | 0.01471 ^{\{ 7 \}} | 0.01402 ^{\{ 4 \}} | 0.01492 ^{\{ 8 \}} | 0.01569 ^{\{ 12 \}} | 0.015 ^{\{ 9 \}} | 0.01378 ^{\{ 3 \}} | 0.01434 ^{\{ 6 \}} | 0.01559 ^{\{ 11 \}} | 0.01595 ^{\{ 13 \}} | 0.01376 ^{\{ 2 \}} | 0.01845 ^{\{ 15 \}} | 0.01547 ^{\{ 10 \}} | 0.0179 ^{\{ 14 \}} | |
D_{max} | 0.02182 ^{\{ 2 \}} | 0.02339 ^{\{ 6 \}} | 0.025 ^{\{ 8 \}} | 0.0228 ^{\{ 5 \}} | 0.02526 ^{\{ 10 \}} | 0.02734 ^{\{ 13 \}} | 0.02465 ^{\{ 7 \}} | 0.02256 ^{\{ 3 \}} | 0.02277 ^{\{ 4 \}} | 0.02521 ^{\{ 9 \}} | 0.02581 ^{\{ 12 \}} | 0.02079 ^{\{ 1 \}} | 0.03004 ^{\{ 15 \}} | 0.02528 ^{\{ 11 \}} | 0.029 ^{\{ 14 \}} | |
\sum Ranks | 21 ^{\{ 2 \}} | 62 ^{\{ 6 \}} | 95 ^{\{ 13 \}} | 41 ^{\{ 4 \}} | 99 ^{\{ 14 \}} | 115 ^{\{ 15 \}} | 79 ^{\{ 11 \}} | 44 ^{\{ 5 \}} | 22 ^{\{ 3 \}} | 63 ^{\{ 7 \}} | 93 ^{\{ 12 \}} | 9 ^{\{ 1 \}} | 70 ^{\{ 8.5 \}} | 77 ^{\{ 10 \}} | 70 ^{\{ 8.5 \}} | |
400 | BIAS( \hat{\delta} ) | 0.07174 ^{\{ 3 \}} | 0.09293 ^{\{ 11 \}} | 0.10943 ^{\{ 13 \}} | 0.07801 ^{\{ 4 \}} | 0.11145 ^{\{ 14 \}} | 0.13305 ^{\{ 15 \}} | 0.09175 ^{\{ 10 \}} | 0.08463 ^{\{ 6 \}} | 0.0625 ^{\{ 2 \}} | 0.08834 ^{\{ 8 \}} | 0.09467 ^{\{ 12 \}} | 0.02457 ^{\{ 1 \}} | 0.08264 ^{\{ 5 \}} | 0.08637 ^{\{ 7 \}} | 0.09147 ^{\{ 9 \}} |
BIAS( \hat{\beta} ) | 0.31908 ^{\{ 3 \}} | 0.40392 ^{\{ 9 \}} | 0.47059 ^{\{ 14 \}} | 0.35775 ^{\{ 5 \}} | 0.46071 ^{\{ 13 \}} | 0.53833 ^{\{ 15 \}} | 0.40286 ^{\{ 8 \}} | 0.38381 ^{\{ 7 \}} | 0.2324 ^{\{ 2 \}} | 0.4171 ^{\{ 11 \}} | 0.45466 ^{\{ 12 \}} | 0.02287 ^{\{ 1 \}} | 0.33301 ^{\{ 4 \}} | 0.40844 ^{\{ 10 \}} | 0.38044 ^{\{ 6 \}} | |
MSE( \hat{\delta} ) | 0.00819 ^{\{ 3 \}} | 0.01379 ^{\{ 10 \}} | 0.01888 ^{\{ 13 \}} | 0.00948 ^{\{ 4 \}} | 0.01913 ^{\{ 14 \}} | 0.02688 ^{\{ 15 \}} | 0.01303 ^{\{ 9 \}} | 0.01121 ^{\{ 5 \}} | 0.00806 ^{\{ 2 \}} | 0.01219 ^{\{ 7 \}} | 0.01427 ^{\{ 12 \}} | 0.00096 ^{\{ 1 \}} | 0.01264 ^{\{ 8 \}} | 0.01176 ^{\{ 6 \}} | 0.01408 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.17003 ^{\{ 3 \}} | 0.27561 ^{\{ 8 \}} | 0.39787 ^{\{ 14 \}} | 0.22918 ^{\{ 5 \}} | 0.3543 ^{\{ 12 \}} | 0.49644 ^{\{ 15 \}} | 0.28197 ^{\{ 9 \}} | 0.25617 ^{\{ 7 \}} | 0.15577 ^{\{ 2 \}} | 0.30793 ^{\{ 11 \}} | 0.38113 ^{\{ 13 \}} | 0.00249 ^{\{ 1 \}} | 0.21587 ^{\{ 4 \}} | 0.30226 ^{\{ 10 \}} | 0.24672 ^{\{ 6 \}} | |
MRE( \hat{\delta} ) | 0.10249 ^{\{ 3 \}} | 0.13276 ^{\{ 11 \}} | 0.15633 ^{\{ 13 \}} | 0.11145 ^{\{ 4 \}} | 0.15921 ^{\{ 14 \}} | 0.19008 ^{\{ 15 \}} | 0.13108 ^{\{ 10 \}} | 0.1209 ^{\{ 6 \}} | 0.08929 ^{\{ 2 \}} | 0.1262 ^{\{ 8 \}} | 0.13524 ^{\{ 12 \}} | 0.0351 ^{\{ 1 \}} | 0.11806 ^{\{ 5 \}} | 0.12339 ^{\{ 7 \}} | 0.13067 ^{\{ 9 \}} | |
MRE( \hat{\beta} ) | 0.12763 ^{\{ 3 \}} | 0.16157 ^{\{ 9 \}} | 0.18824 ^{\{ 14 \}} | 0.1431 ^{\{ 5 \}} | 0.18428 ^{\{ 13 \}} | 0.21533 ^{\{ 15 \}} | 0.16114 ^{\{ 8 \}} | 0.15352 ^{\{ 7 \}} | 0.09296 ^{\{ 2 \}} | 0.16684 ^{\{ 11 \}} | 0.18186 ^{\{ 12 \}} | 0.00915 ^{\{ 1 \}} | 0.1332 ^{\{ 4 \}} | 0.16337 ^{\{ 10 \}} | 0.15218 ^{\{ 6 \}} | |
D_{abs} | 0.01202 ^{\{ 1 \}} | 0.01289 ^{\{ 5 \}} | 0.01357 ^{\{ 9 \}} | 0.01216 ^{\{ 3 \}} | 0.01333 ^{\{ 8 \}} | 0.01381 ^{\{ 11 \}} | 0.01248 ^{\{ 4 \}} | 0.01302 ^{\{ 7 \}} | 0.01293 ^{\{ 6 \}} | 0.01404 ^{\{ 12 \}} | 0.01407 ^{\{ 13 \}} | 0.01203 ^{\{ 2 \}} | 0.01585 ^{\{ 15 \}} | 0.01372 ^{\{ 10 \}} | 0.01554 ^{\{ 14 \}} | |
D_{max} | 0.01947 ^{\{ 2 \}} | 0.02129 ^{\{ 7 \}} | 0.02291 ^{\{ 12 \}} | 0.01985 ^{\{ 3 \}} | 0.0227 ^{\{ 9 \}} | 0.02414 ^{\{ 13 \}} | 0.02063 ^{\{ 5 \}} | 0.02113 ^{\{ 6 \}} | 0.02061 ^{\{ 4 \}} | 0.02288 ^{\{ 10 \}} | 0.02289 ^{\{ 11 \}} | 0.01819 ^{\{ 1 \}} | 0.02568 ^{\{ 15 \}} | 0.02227 ^{\{ 8 \}} | 0.02537 ^{\{ 14 \}} | |
\sum Ranks | 21 ^{\{ 2 \}} | 70 ^{\{ 9 \}} | 102 ^{\{ 14 \}} | 33 ^{\{ 4 \}} | 97 ^{\{ 12.5 \}} | 114 ^{\{ 15 \}} | 63 ^{\{ 7 \}} | 51 ^{\{ 5 \}} | 22 ^{\{ 3 \}} | 78 ^{\{ 11 \}} | 97 ^{\{ 12.5 \}} | 9 ^{\{ 1 \}} | 60 ^{\{ 6 \}} | 68 ^{\{ 8 \}} | 75 ^{\{ 10 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.03792 ^{\{ 2 \}} | 0.04631 ^{\{ 7 \}} | 0.05819 ^{\{ 14 \}} | 0.04377 ^{\{ 5 \}} | 0.05299 ^{\{ 10 \}} | 0.06162 ^{\{ 15 \}} | 0.04975 ^{\{ 9 \}} | 0.0476 ^{\{ 8 \}} | 0.03992 ^{\{ 3 \}} | 0.04343 ^{\{ 4 \}} | 0.05571 ^{\{ 11.5 \}} | 0.0272 ^{\{ 1 \}} | 0.05571 ^{\{ 11.5 \}} | 0.04408 ^{\{ 6 \}} | 0.05653 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.20078 ^{\{ 3 \}} | 0.22619 ^{\{ 6 \}} | 0.24363 ^{\{ 12 \}} | 0.23866 ^{\{ 10 \}} | 0.23765 ^{\{ 9 \}} | 0.23982 ^{\{ 11 \}} | 0.23231 ^{\{ 7 \}} | 0.21986 ^{\{ 5 \}} | 0.16668 ^{\{ 2 \}} | 0.20972 ^{\{ 4 \}} | 0.25183 ^{\{ 14 \}} | 0.09301 ^{\{ 1 \}} | 0.26527 ^{\{ 15 \}} | 0.23274 ^{\{ 8 \}} | 0.24674 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00219 ^{\{ 2 \}} | 0.00339 ^{\{ 7 \}} | 0.00555 ^{\{ 14 \}} | 0.00291 ^{\{ 4 \}} | 0.00433 ^{\{ 10 \}} | 0.00612 ^{\{ 15 \}} | 0.00397 ^{\{ 9 \}} | 0.00366 ^{\{ 8 \}} | 0.00287 ^{\{ 3 \}} | 0.00308 ^{\{ 5 \}} | 0.00528 ^{\{ 13 \}} | 0.0014 ^{\{ 1 \}} | 0.00479 ^{\{ 11 \}} | 0.00312 ^{\{ 6 \}} | 0.00498 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.0644 ^{\{ 3 \}} | 0.07884 ^{\{ 6 \}} | 0.08607 ^{\{ 10 \}} | 0.08968 ^{\{ 12 \}} | 0.08623 ^{\{ 11 \}} | 0.08533 ^{\{ 8 \}} | 0.08135 ^{\{ 7 \}} | 0.07377 ^{\{ 4 \}} | 0.05803 ^{\{ 2 \}} | 0.07457 ^{\{ 5 \}} | 0.09151 ^{\{ 14 \}} | 0.02381 ^{\{ 1 \}} | 0.10724 ^{\{ 15 \}} | 0.08546 ^{\{ 9 \}} | 0.08994 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.15167 ^{\{ 2 \}} | 0.18523 ^{\{ 7 \}} | 0.23277 ^{\{ 14 \}} | 0.17509 ^{\{ 5 \}} | 0.21197 ^{\{ 10 \}} | 0.24649 ^{\{ 15 \}} | 0.199 ^{\{ 9 \}} | 0.19039 ^{\{ 8 \}} | 0.15967 ^{\{ 3 \}} | 0.1737 ^{\{ 4 \}} | 0.22284 ^{\{ 11.5 \}} | 0.10879 ^{\{ 1 \}} | 0.22284 ^{\{ 11.5 \}} | 0.17631 ^{\{ 6 \}} | 0.22614 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.26771 ^{\{ 3 \}} | 0.30159 ^{\{ 6 \}} | 0.32484 ^{\{ 12 \}} | 0.31822 ^{\{ 10 \}} | 0.31687 ^{\{ 9 \}} | 0.31975 ^{\{ 11 \}} | 0.30975 ^{\{ 7 \}} | 0.29315 ^{\{ 5 \}} | 0.22223 ^{\{ 2 \}} | 0.27962 ^{\{ 4 \}} | 0.33577 ^{\{ 14 \}} | 0.12402 ^{\{ 1 \}} | 0.35369 ^{\{ 15 \}} | 0.31032 ^{\{ 8 \}} | 0.32898 ^{\{ 13 \}} | |
D_{abs} | 0.03978 ^{\{ 1 \}} | 0.04375 ^{\{ 5 \}} | 0.04853 ^{\{ 12 \}} | 0.04421 ^{\{ 7 \}} | 0.04553 ^{\{ 8 \}} | 0.04857 ^{\{ 13 \}} | 0.04381 ^{\{ 6 \}} | 0.04289 ^{\{ 2 \}} | 0.04619 ^{\{ 9 \}} | 0.04303 ^{\{ 3 \}} | 0.04796 ^{\{ 11 \}} | 0.0432 ^{\{ 4 \}} | 0.05475 ^{\{ 15 \}} | 0.04735 ^{\{ 10 \}} | 0.05436 ^{\{ 14 \}} | |
D_{max} | 0.06553 ^{\{ 1 \}} | 0.07131 ^{\{ 6 \}} | 0.08238 ^{\{ 12 \}} | 0.07085 ^{\{ 4 \}} | 0.07553 ^{\{ 10 \}} | 0.08325 ^{\{ 13 \}} | 0.0722 ^{\{ 7 \}} | 0.07117 ^{\{ 5 \}} | 0.07504 ^{\{ 8 \}} | 0.07035 ^{\{ 3 \}} | 0.07991 ^{\{ 11 \}} | 0.06891 ^{\{ 2 \}} | 0.08996 ^{\{ 15 \}} | 0.07547 ^{\{ 9 \}} | 0.0896 ^{\{ 14 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 50 ^{\{ 6 \}} | 100 ^{\{ 11.5 \}} | 57 ^{\{ 7 \}} | 77 ^{\{ 10 \}} | 101 ^{\{ 13 \}} | 61 ^{\{ 8 \}} | 45 ^{\{ 5 \}} | 32 ^{\{ 3.5 \}} | 32 ^{\{ 3.5 \}} | 100 ^{\{ 11.5 \}} | 12 ^{\{ 1 \}} | 109 ^{\{ 15 \}} | 62 ^{\{ 9 \}} | 105 ^{\{ 14 \}} | |
60 | BIAS( \hat{\delta} ) | 0.02673 ^{\{ 2 \}} | 0.03656 ^{\{ 8 \}} | 0.04072 ^{\{ 12 \}} | 0.03364 ^{\{ 4 \}} | 0.04024 ^{\{ 11 \}} | 0.04471 ^{\{ 14 \}} | 0.03721 ^{\{ 9 \}} | 0.03459 ^{\{ 6 \}} | 0.02974 ^{\{ 3 \}} | 0.03401 ^{\{ 5 \}} | 0.03937 ^{\{ 10 \}} | 0.02163 ^{\{ 1 \}} | 0.04522 ^{\{ 15 \}} | 0.03572 ^{\{ 7 \}} | 0.04219 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.15368 ^{\{ 3 \}} | 0.1865 ^{\{ 8 \}} | 0.19022 ^{\{ 9 \}} | 0.18644 ^{\{ 7 \}} | 0.20018 ^{\{ 11 \}} | 0.20036 ^{\{ 12 \}} | 0.18376 ^{\{ 6 \}} | 0.17252 ^{\{ 4 \}} | 0.13065 ^{\{ 2 \}} | 0.17919 ^{\{ 5 \}} | 0.20222 ^{\{ 14 \}} | 0.08068 ^{\{ 1 \}} | 0.22493 ^{\{ 15 \}} | 0.19677 ^{\{ 10 \}} | 0.20108 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00113 ^{\{ 2 \}} | 0.00214 ^{\{ 8 \}} | 0.00273 ^{\{ 12 \}} | 0.00169 ^{\{ 4 \}} | 0.00252 ^{\{ 10 \}} | 0.00318 ^{\{ 15 \}} | 0.00217 ^{\{ 9 \}} | 0.00193 ^{\{ 7 \}} | 0.0016 ^{\{ 3 \}} | 0.00189 ^{\{ 6 \}} | 0.00253 ^{\{ 11 \}} | 0.00092 ^{\{ 1 \}} | 0.00314 ^{\{ 14 \}} | 0.00187 ^{\{ 5 \}} | 0.00278 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.04175 ^{\{ 3 \}} | 0.05644 ^{\{ 6 \}} | 0.05686 ^{\{ 7 \}} | 0.05842 ^{\{ 9 \}} | 0.06424 ^{\{ 14 \}} | 0.06282 ^{\{ 13 \}} | 0.05305 ^{\{ 5 \}} | 0.04927 ^{\{ 4 \}} | 0.03699 ^{\{ 2 \}} | 0.05786 ^{\{ 8 \}} | 0.06276 ^{\{ 12 \}} | 0.01743 ^{\{ 1 \}} | 0.07899 ^{\{ 15 \}} | 0.06166 ^{\{ 10 \}} | 0.06264 ^{\{ 11 \}} | |
MRE( \hat{\delta} ) | 0.10693 ^{\{ 2 \}} | 0.14623 ^{\{ 8 \}} | 0.16287 ^{\{ 12 \}} | 0.13457 ^{\{ 4 \}} | 0.16097 ^{\{ 11 \}} | 0.17885 ^{\{ 14 \}} | 0.14882 ^{\{ 9 \}} | 0.13837 ^{\{ 6 \}} | 0.11896 ^{\{ 3 \}} | 0.13604 ^{\{ 5 \}} | 0.15748 ^{\{ 10 \}} | 0.08653 ^{\{ 1 \}} | 0.18086 ^{\{ 15 \}} | 0.14289 ^{\{ 7 \}} | 0.16877 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.20491 ^{\{ 3 \}} | 0.24867 ^{\{ 8 \}} | 0.25362 ^{\{ 9 \}} | 0.24859 ^{\{ 7 \}} | 0.2669 ^{\{ 11 \}} | 0.26715 ^{\{ 12 \}} | 0.24501 ^{\{ 6 \}} | 0.23003 ^{\{ 4 \}} | 0.17419 ^{\{ 2 \}} | 0.23892 ^{\{ 5 \}} | 0.26963 ^{\{ 14 \}} | 0.10757 ^{\{ 1 \}} | 0.2999 ^{\{ 15 \}} | 0.26236 ^{\{ 10 \}} | 0.2681 ^{\{ 13 \}} | |
D_{abs} | 0.02865 ^{\{ 1 \}} | 0.03235 ^{\{ 3 \}} | 0.03403 ^{\{ 10 \}} | 0.03266 ^{\{ 6 \}} | 0.0338 ^{\{ 8 \}} | 0.03443 ^{\{ 11 \}} | 0.03265 ^{\{ 5 \}} | 0.03189 ^{\{ 2 \}} | 0.03385 ^{\{ 9 \}} | 0.03286 ^{\{ 7 \}} | 0.03487 ^{\{ 12 \}} | 0.03252 ^{\{ 4 \}} | 0.04035 ^{\{ 14 \}} | 0.03501 ^{\{ 13 \}} | 0.04078 ^{\{ 15 \}} | |
D_{max} | 0.04675 ^{\{ 1 \}} | 0.05334 ^{\{ 5 \}} | 0.0571 ^{\{ 11 \}} | 0.05293 ^{\{ 4 \}} | 0.05628 ^{\{ 9 \}} | 0.05832 ^{\{ 13 \}} | 0.05371 ^{\{ 7 \}} | 0.05256 ^{\{ 3 \}} | 0.05479 ^{\{ 8 \}} | 0.05366 ^{\{ 6 \}} | 0.05734 ^{\{ 12 \}} | 0.05239 ^{\{ 2 \}} | 0.06677 ^{\{ 14 \}} | 0.05649 ^{\{ 10 \}} | 0.06719 ^{\{ 15 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 54 ^{\{ 7 \}} | 82 ^{\{ 10 \}} | 45 ^{\{ 5 \}} | 85 ^{\{ 11 \}} | 104 ^{\{ 13 \}} | 56 ^{\{ 8 \}} | 36 ^{\{ 4 \}} | 32 ^{\{ 3 \}} | 47 ^{\{ 6 \}} | 95 ^{\{ 12 \}} | 12 ^{\{ 1 \}} | 117 ^{\{ 15 \}} | 72 ^{\{ 9 \}} | 106 ^{\{ 14 \}} | |
100 | BIAS( \hat{\delta} ) | 0.0231 ^{\{ 2 \}} | 0.02843 ^{\{ 6 \}} | 0.03137 ^{\{ 10 \}} | 0.02728 ^{\{ 4 \}} | 0.0316 ^{\{ 11 \}} | 0.03625 ^{\{ 15 \}} | 0.02813 ^{\{ 5 \}} | 0.02848 ^{\{ 7 \}} | 0.02475 ^{\{ 3 \}} | 0.02932 ^{\{ 9 \}} | 0.03253 ^{\{ 12 \}} | 0.01602 ^{\{ 1 \}} | 0.03265 ^{\{ 13 \}} | 0.029 ^{\{ 8 \}} | 0.03311 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.1315 ^{\{ 3 \}} | 0.14273 ^{\{ 5 \}} | 0.14892 ^{\{ 7 \}} | 0.15193 ^{\{ 8 \}} | 0.15411 ^{\{ 9 \}} | 0.16605 ^{\{ 14 \}} | 0.14096 ^{\{ 4 \}} | 0.14515 ^{\{ 6 \}} | 0.1185 ^{\{ 2 \}} | 0.15562 ^{\{ 10 \}} | 0.1731 ^{\{ 15 \}} | 0.06434 ^{\{ 1 \}} | 0.16197 ^{\{ 12 \}} | 0.16008 ^{\{ 11 \}} | 0.16502 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.00086 ^{\{ 2 \}} | 0.00126 ^{\{ 5 \}} | 0.00157 ^{\{ 11 \}} | 0.00112 ^{\{ 4 \}} | 0.00156 ^{\{ 10 \}} | 0.00208 ^{\{ 15 \}} | 0.00127 ^{\{ 6 \}} | 0.00128 ^{\{ 7 \}} | 0.00107 ^{\{ 3 \}} | 0.00136 ^{\{ 9 \}} | 0.00166 ^{\{ 12.5 \}} | 0.00051 ^{\{ 1 \}} | 0.00166 ^{\{ 12.5 \}} | 0.00129 ^{\{ 8 \}} | 0.00177 ^{\{ 14 \}} | |
MSE( \hat{\beta} ) | 0.03144 ^{\{ 3 \}} | 0.03254 ^{\{ 4 \}} | 0.03644 ^{\{ 7 \}} | 0.03847 ^{\{ 8 \}} | 0.03959 ^{\{ 9 \}} | 0.04527 ^{\{ 14 \}} | 0.03344 ^{\{ 5 \}} | 0.03458 ^{\{ 6 \}} | 0.0301 ^{\{ 2 \}} | 0.04335 ^{\{ 12 \}} | 0.04773 ^{\{ 15 \}} | 0.01114 ^{\{ 1 \}} | 0.04165 ^{\{ 10 \}} | 0.04281 ^{\{ 11 \}} | 0.04409 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.09238 ^{\{ 2 \}} | 0.11374 ^{\{ 6 \}} | 0.12547 ^{\{ 10 \}} | 0.10911 ^{\{ 4 \}} | 0.12639 ^{\{ 11 \}} | 0.14502 ^{\{ 15 \}} | 0.1125 ^{\{ 5 \}} | 0.11391 ^{\{ 7 \}} | 0.09899 ^{\{ 3 \}} | 0.11729 ^{\{ 9 \}} | 0.13012 ^{\{ 12 \}} | 0.0641 ^{\{ 1 \}} | 0.13062 ^{\{ 13 \}} | 0.11599 ^{\{ 8 \}} | 0.13244 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.17533 ^{\{ 3 \}} | 0.1903 ^{\{ 5 \}} | 0.19856 ^{\{ 7 \}} | 0.20257 ^{\{ 8 \}} | 0.20548 ^{\{ 9 \}} | 0.2214 ^{\{ 14 \}} | 0.18795 ^{\{ 4 \}} | 0.19354 ^{\{ 6 \}} | 0.158 ^{\{ 2 \}} | 0.20749 ^{\{ 10 \}} | 0.2308 ^{\{ 15 \}} | 0.08578 ^{\{ 1 \}} | 0.21596 ^{\{ 12 \}} | 0.21345 ^{\{ 11 \}} | 0.22003 ^{\{ 13 \}} | |
D_{abs} | 0.02498 ^{\{ 2 \}} | 0.02503 ^{\{ 3 \}} | 0.02656 ^{\{ 7 \}} | 0.02588 ^{\{ 5 \}} | 0.02596 ^{\{ 6 \}} | 0.02722 ^{\{ 11 \}} | 0.02554 ^{\{ 4 \}} | 0.02663 ^{\{ 8 \}} | 0.02832 ^{\{ 12 \}} | 0.02699 ^{\{ 9 \}} | 0.02868 ^{\{ 13 \}} | 0.02409 ^{\{ 1 \}} | 0.03054 ^{\{ 14 \}} | 0.02703 ^{\{ 10 \}} | 0.03077 ^{\{ 15 \}} | |
D_{max} | 0.04055 ^{\{ 2 \}} | 0.04137 ^{\{ 3 \}} | 0.04456 ^{\{ 10 \}} | 0.04216 ^{\{ 5 \}} | 0.04346 ^{\{ 6 \}} | 0.04638 ^{\{ 12 \}} | 0.04211 ^{\{ 4 \}} | 0.04374 ^{\{ 7 \}} | 0.04578 ^{\{ 11 \}} | 0.04428 ^{\{ 9 \}} | 0.04702 ^{\{ 13 \}} | 0.03867 ^{\{ 1 \}} | 0.0502 ^{\{ 14 \}} | 0.04408 ^{\{ 8 \}} | 0.05067 ^{\{ 15 \}} | |
\sum Ranks | 19 ^{\{ 2 \}} | 37 ^{\{ 3.5 \}} | 69 ^{\{ 8 \}} | 46 ^{\{ 6 \}} | 71 ^{\{ 9 \}} | 110 ^{\{ 14 \}} | 37 ^{\{ 3.5 \}} | 54 ^{\{ 7 \}} | 38 ^{\{ 5 \}} | 77 ^{\{ 11 \}} | 107.5 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 100.5 ^{\{ 12 \}} | 75 ^{\{ 10 \}} | 111 ^{\{ 15 \}} | |
200 | BIAS( \hat{\delta} ) | 0.0173 ^{\{ 2 \}} | 0.01938 ^{\{ 7 \}} | 0.02284 ^{\{ 11 \}} | 0.019 ^{\{ 4 \}} | 0.02223 ^{\{ 10 \}} | 0.02627 ^{\{ 15 \}} | 0.02055 ^{\{ 9 \}} | 0.01934 ^{\{ 6 \}} | 0.01838 ^{\{ 3 \}} | 0.01911 ^{\{ 5 \}} | 0.02522 ^{\{ 13 \}} | 0.01175 ^{\{ 1 \}} | 0.02527 ^{\{ 14 \}} | 0.01943 ^{\{ 8 \}} | 0.02293 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.09413 ^{\{ 3 \}} | 0.09871 ^{\{ 5 \}} | 0.10837 ^{\{ 11 \}} | 0.1018 ^{\{ 7 \}} | 0.10854 ^{\{ 12 \}} | 0.11908 ^{\{ 13 \}} | 0.10176 ^{\{ 6 \}} | 0.0986 ^{\{ 4 \}} | 0.08773 ^{\{ 2 \}} | 0.1022 ^{\{ 8 \}} | 0.14097 ^{\{ 15 \}} | 0.04869 ^{\{ 1 \}} | 0.13008 ^{\{ 14 \}} | 0.10478 ^{\{ 9 \}} | 0.10707 ^{\{ 10 \}} | |
MSE( \hat{\delta} ) | 0.00047 ^{\{ 2 \}} | 0.00059 ^{\{ 5.5 \}} | 0.00082 ^{\{ 11 \}} | 0.00057 ^{\{ 3 \}} | 0.00078 ^{\{ 10 \}} | 0.00108 ^{\{ 15 \}} | 0.00065 ^{\{ 9 \}} | 0.00058 ^{\{ 4 \}} | 0.00061 ^{\{ 7 \}} | 0.00062 ^{\{ 8 \}} | 0.00098 ^{\{ 13.5 \}} | 0.00029 ^{\{ 1 \}} | 0.00098 ^{\{ 13.5 \}} | 0.00059 ^{\{ 5.5 \}} | 0.00091 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.01471 ^{\{ 2 \}} | 0.01633 ^{\{ 5 \}} | 0.01863 ^{\{ 9 \}} | 0.01774 ^{\{ 7 \}} | 0.01979 ^{\{ 11 \}} | 0.0231 ^{\{ 13 \}} | 0.01716 ^{\{ 6 \}} | 0.01534 ^{\{ 3 \}} | 0.01628 ^{\{ 4 \}} | 0.0195 ^{\{ 10 \}} | 0.03311 ^{\{ 15 \}} | 0.00672 ^{\{ 1 \}} | 0.02761 ^{\{ 14 \}} | 0.01843 ^{\{ 8 \}} | 0.02024 ^{\{ 12 \}} | |
MRE( \hat{\delta} ) | 0.06921 ^{\{ 2 \}} | 0.07753 ^{\{ 7 \}} | 0.09134 ^{\{ 11 \}} | 0.07601 ^{\{ 4 \}} | 0.08891 ^{\{ 10 \}} | 0.10508 ^{\{ 15 \}} | 0.0822 ^{\{ 9 \}} | 0.07736 ^{\{ 6 \}} | 0.07351 ^{\{ 3 \}} | 0.07643 ^{\{ 5 \}} | 0.10088 ^{\{ 13 \}} | 0.04701 ^{\{ 1 \}} | 0.10108 ^{\{ 14 \}} | 0.07774 ^{\{ 8 \}} | 0.09172 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.1255 ^{\{ 3 \}} | 0.13161 ^{\{ 5 \}} | 0.1445 ^{\{ 11 \}} | 0.13574 ^{\{ 7 \}} | 0.14471 ^{\{ 12 \}} | 0.15878 ^{\{ 13 \}} | 0.13568 ^{\{ 6 \}} | 0.13146 ^{\{ 4 \}} | 0.11697 ^{\{ 2 \}} | 0.13626 ^{\{ 8 \}} | 0.18796 ^{\{ 15 \}} | 0.06491 ^{\{ 1 \}} | 0.17344 ^{\{ 14 \}} | 0.1397 ^{\{ 9 \}} | 0.14275 ^{\{ 10 \}} | |
D_{abs} | 0.01803 ^{\{ 3 \}} | 0.01795 ^{\{ 2 \}} | 0.01849 ^{\{ 6 \}} | 0.01827 ^{\{ 4 \}} | 0.01852 ^{\{ 7 \}} | 0.01943 ^{\{ 9 \}} | 0.01842 ^{\{ 5 \}} | 0.01853 ^{\{ 8 \}} | 0.01976 ^{\{ 12 \}} | 0.01957 ^{\{ 10 \}} | 0.02149 ^{\{ 13 \}} | 0.0176 ^{\{ 1 \}} | 0.0237 ^{\{ 15 \}} | 0.01974 ^{\{ 11 \}} | 0.02287 ^{\{ 14 \}} | |
D_{max} | 0.02919 ^{\{ 2 \}} | 0.02952 ^{\{ 3 \}} | 0.0311 ^{\{ 8 \}} | 0.02977 ^{\{ 4 \}} | 0.03109 ^{\{ 7 \}} | 0.03335 ^{\{ 12 \}} | 0.0306 ^{\{ 6 \}} | 0.03031 ^{\{ 5 \}} | 0.03239 ^{\{ 11 \}} | 0.03179 ^{\{ 9 \}} | 0.03524 ^{\{ 13 \}} | 0.02807 ^{\{ 1 \}} | 0.03898 ^{\{ 15 \}} | 0.03199 ^{\{ 10 \}} | 0.03733 ^{\{ 14 \}} | |
\sum Ranks | 19 ^{\{ 2 \}} | 39.5 ^{\{ 3 \}} | 78 ^{\{ 10 \}} | 40 ^{\{ 4.5 \}} | 79 ^{\{ 11 \}} | 105 ^{\{ 13 \}} | 56 ^{\{ 7 \}} | 40 ^{\{ 4.5 \}} | 44 ^{\{ 6 \}} | 63 ^{\{ 8 \}} | 110.5 ^{\{ 14 \}} | 8 ^{\{ 1 \}} | 113.5 ^{\{ 15 \}} | 68.5 ^{\{ 9 \}} | 96 ^{\{ 12 \}} | |
300 | BIAS( \hat{\delta} ) | 0.01419 ^{\{ 2 \}} | 0.01736 ^{\{ 9 \}} | 0.01883 ^{\{ 13 \}} | 0.01549 ^{\{ 4 \}} | 0.01842 ^{\{ 12 \}} | 0.02076 ^{\{ 15 \}} | 0.01633 ^{\{ 7 \}} | 0.01636 ^{\{ 8 \}} | 0.01534 ^{\{ 3 \}} | 0.01629 ^{\{ 6 \}} | 0.01982 ^{\{ 14 \}} | 0.00983 ^{\{ 1 \}} | 0.01821 ^{\{ 11 \}} | 0.01577 ^{\{ 5 \}} | 0.01778 ^{\{ 10 \}} |
BIAS( \hat{\beta} ) | 0.0731 ^{\{ 2 \}} | 0.08699 ^{\{ 9 \}} | 0.08814 ^{\{ 11 \}} | 0.08227 ^{\{ 4 \}} | 0.09008 ^{\{ 12 \}} | 0.09593 ^{\{ 14 \}} | 0.08242 ^{\{ 5 \}} | 0.08312 ^{\{ 6 \}} | 0.07557 ^{\{ 3 \}} | 0.08758 ^{\{ 10 \}} | 0.10944 ^{\{ 15 \}} | 0.03688 ^{\{ 1 \}} | 0.09063 ^{\{ 13 \}} | 0.08574 ^{\{ 8 \}} | 0.08526 ^{\{ 7 \}} | |
MSE( \hat{\delta} ) | 0.00031 ^{\{ 2 \}} | 0.00048 ^{\{ 9 \}} | 0.00056 ^{\{ 12 \}} | 0.00038 ^{\{ 3 \}} | 0.00054 ^{\{ 10.5 \}} | 0.00067 ^{\{ 15 \}} | 0.00044 ^{\{ 8 \}} | 0.00041 ^{\{ 5.5 \}} | 0.00041 ^{\{ 5.5 \}} | 0.00043 ^{\{ 7 \}} | 0.00062 ^{\{ 14 \}} | 2e-04 ^{\{ 1 \}} | 0.00057 ^{\{ 13 \}} | 4e-04 ^{\{ 4 \}} | 0.00054 ^{\{ 10.5 \}} | |
MSE( \hat{\beta} ) | 0.00848 ^{\{ 2 \}} | 0.01219 ^{\{ 8 \}} | 0.01264 ^{\{ 10 \}} | 0.01114 ^{\{ 4 \}} | 0.01321 ^{\{ 11 \}} | 0.01472 ^{\{ 14 \}} | 0.01129 ^{\{ 5 \}} | 0.0111 ^{\{ 3 \}} | 0.01187 ^{\{ 6 \}} | 0.01322 ^{\{ 12 \}} | 0.02008 ^{\{ 15 \}} | 0.00412 ^{\{ 1 \}} | 0.01441 ^{\{ 13 \}} | 0.01218 ^{\{ 7 \}} | 0.01225 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.05677 ^{\{ 2 \}} | 0.06944 ^{\{ 9 \}} | 0.07531 ^{\{ 13 \}} | 0.06196 ^{\{ 4 \}} | 0.07367 ^{\{ 12 \}} | 0.08303 ^{\{ 15 \}} | 0.06531 ^{\{ 7 \}} | 0.06545 ^{\{ 8 \}} | 0.06137 ^{\{ 3 \}} | 0.06515 ^{\{ 6 \}} | 0.07927 ^{\{ 14 \}} | 0.03933 ^{\{ 1 \}} | 0.07285 ^{\{ 11 \}} | 0.06308 ^{\{ 5 \}} | 0.07112 ^{\{ 10 \}} | |
MRE( \hat{\beta} ) | 0.09747 ^{\{ 2 \}} | 0.11599 ^{\{ 9 \}} | 0.11753 ^{\{ 11 \}} | 0.10969 ^{\{ 4 \}} | 0.12011 ^{\{ 12 \}} | 0.12791 ^{\{ 14 \}} | 0.1099 ^{\{ 5 \}} | 0.11082 ^{\{ 6 \}} | 0.10076 ^{\{ 3 \}} | 0.11678 ^{\{ 10 \}} | 0.14592 ^{\{ 15 \}} | 0.04918 ^{\{ 1 \}} | 0.12084 ^{\{ 13 \}} | 0.11432 ^{\{ 8 \}} | 0.11368 ^{\{ 7 \}} | |
D_{abs} | 0.01447 ^{\{ 2 \}} | 0.01529 ^{\{ 7 \}} | 0.01535 ^{\{ 8 \}} | 0.01484 ^{\{ 3 \}} | 0.01519 ^{\{ 6 \}} | 0.01601 ^{\{ 9 \}} | 0.01493 ^{\{ 4 \}} | 0.01504 ^{\{ 5 \}} | 0.01637 ^{\{ 11 \}} | 0.01677 ^{\{ 12 \}} | 0.01742 ^{\{ 13 \}} | 0.01433 ^{\{ 1 \}} | 0.01906 ^{\{ 15 \}} | 0.0163 ^{\{ 10 \}} | 0.0188 ^{\{ 14 \}} | |
D_{max} | 0.02349 ^{\{ 2 \}} | 0.02522 ^{\{ 6 \}} | 0.02584 ^{\{ 8 \}} | 0.02421 ^{\{ 3 \}} | 0.02533 ^{\{ 7 \}} | 0.02724 ^{\{ 12 \}} | 0.02471 ^{\{ 4 \}} | 0.02483 ^{\{ 5 \}} | 0.02669 ^{\{ 10 \}} | 0.02712 ^{\{ 11 \}} | 0.02847 ^{\{ 13 \}} | 0.02302 ^{\{ 1 \}} | 0.03095 ^{\{ 15 \}} | 0.02641 ^{\{ 9 \}} | 0.0306 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 1 \}} | 64 ^{\{ 7 \}} | 84 ^{\{ 12 \}} | 28 ^{\{ 3 \}} | 80.5 ^{\{ 11 \}} | 106 ^{\{ 14 \}} | 43 ^{\{ 5 \}} | 44.5 ^{\{ 6 \}} | 42.5 ^{\{ 4 \}} | 72 ^{\{ 9 \}} | 111 ^{\{ 15 \}} | 21 ^{\{ 2 \}} | 102 ^{\{ 13 \}} | 67 ^{\{ 8 \}} | 79.5 ^{\{ 10 \}} | |
400 | BIAS( \hat{\delta} ) | 0.0123 ^{\{ 2 \}} | 0.01445 ^{\{ 7 \}} | 0.01582 ^{\{ 12 \}} | 0.01337 ^{\{ 4 \}} | 0.01565 ^{\{ 11 \}} | 0.01778 ^{\{ 14 \}} | 0.01429 ^{\{ 6 \}} | 0.01336 ^{\{ 3 \}} | 0.01397 ^{\{ 5 \}} | 0.01482 ^{\{ 9 \}} | 0.01814 ^{\{ 15 \}} | 0.0086 ^{\{ 1 \}} | 0.01606 ^{\{ 13 \}} | 0.01477 ^{\{ 8 \}} | 0.01543 ^{\{ 10 \}} |
BIAS( \hat{\beta} ) | 0.06611 ^{\{ 2 \}} | 0.07122 ^{\{ 6 \}} | 0.07645 ^{\{ 9 \}} | 0.07072 ^{\{ 5 \}} | 0.07554 ^{\{ 8 \}} | 0.08205 ^{\{ 14 \}} | 0.07194 ^{\{ 7 \}} | 0.06821 ^{\{ 3 \}} | 0.07008 ^{\{ 4 \}} | 0.07952 ^{\{ 13 \}} | 0.10105 ^{\{ 15 \}} | 0.03501 ^{\{ 1 \}} | 0.07876 ^{\{ 10 \}} | 0.07925 ^{\{ 12 \}} | 0.07879 ^{\{ 11 \}} | |
MSE( \hat{\delta} ) | 0.00023 ^{\{ 2 \}} | 0.00032 ^{\{ 6 \}} | 4e-04 ^{\{ 12 \}} | 0.00027 ^{\{ 3.5 \}} | 0.00039 ^{\{ 11 \}} | 0.00049 ^{\{ 14 \}} | 0.00032 ^{\{ 6 \}} | 0.00027 ^{\{ 3.5 \}} | 0.00035 ^{\{ 8 \}} | 0.00036 ^{\{ 9 \}} | 5e-04 ^{\{ 15 \}} | 0.00016 ^{\{ 1 \}} | 0.00044 ^{\{ 13 \}} | 0.00032 ^{\{ 6 \}} | 0.00037 ^{\{ 10 \}} | |
MSE( \hat{\beta} ) | 0.00715 ^{\{ 2 \}} | 0.00794 ^{\{ 5 \}} | 0.00928 ^{\{ 8 \}} | 0.00783 ^{\{ 4 \}} | 0.00904 ^{\{ 7 \}} | 0.01058 ^{\{ 12 \}} | 0.00843 ^{\{ 6 \}} | 0.00727 ^{\{ 3 \}} | 0.00986 ^{\{ 11 \}} | 0.011 ^{\{ 14 \}} | 0.0167 ^{\{ 15 \}} | 0.00343 ^{\{ 1 \}} | 0.01096 ^{\{ 13 \}} | 0.00975 ^{\{ 10 \}} | 0.00967 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.0492 ^{\{ 2 \}} | 0.05779 ^{\{ 7 \}} | 0.06329 ^{\{ 12 \}} | 0.05346 ^{\{ 4 \}} | 0.06258 ^{\{ 11 \}} | 0.07111 ^{\{ 14 \}} | 0.05715 ^{\{ 6 \}} | 0.05345 ^{\{ 3 \}} | 0.05587 ^{\{ 5 \}} | 0.05929 ^{\{ 9 \}} | 0.07256 ^{\{ 15 \}} | 0.0344 ^{\{ 1 \}} | 0.06425 ^{\{ 13 \}} | 0.0591 ^{\{ 8 \}} | 0.06173 ^{\{ 10 \}} | |
MRE( \hat{\beta} ) | 0.08815 ^{\{ 2 \}} | 0.09495 ^{\{ 6 \}} | 0.10194 ^{\{ 9 \}} | 0.0943 ^{\{ 5 \}} | 0.10072 ^{\{ 8 \}} | 0.1094 ^{\{ 14 \}} | 0.09591 ^{\{ 7 \}} | 0.09095 ^{\{ 3 \}} | 0.09344 ^{\{ 4 \}} | 0.10603 ^{\{ 13 \}} | 0.13473 ^{\{ 15 \}} | 0.04668 ^{\{ 1 \}} | 0.10501 ^{\{ 10 \}} | 0.10567 ^{\{ 12 \}} | 0.10505 ^{\{ 11 \}} | |
D_{abs} | 0.01235 ^{\{ 2 \}} | 0.01243 ^{\{ 3 \}} | 0.0136 ^{\{ 8 \}} | 0.01295 ^{\{ 6 \}} | 0.01324 ^{\{ 7 \}} | 0.0137 ^{\{ 9 \}} | 0.01286 ^{\{ 5 \}} | 0.01258 ^{\{ 4 \}} | 0.01459 ^{\{ 11 \}} | 0.01484 ^{\{ 12 \}} | 0.01561 ^{\{ 13 \}} | 0.0121 ^{\{ 1 \}} | 0.01697 ^{\{ 15 \}} | 0.01421 ^{\{ 10 \}} | 0.01637 ^{\{ 14 \}} | |
D_{max} | 0.02003 ^{\{ 2 \}} | 0.02073 ^{\{ 4 \}} | 0.02267 ^{\{ 8 \}} | 0.0211 ^{\{ 5 \}} | 0.02216 ^{\{ 7 \}} | 0.02335 ^{\{ 10 \}} | 0.02123 ^{\{ 6 \}} | 0.02064 ^{\{ 3 \}} | 0.02367 ^{\{ 11 \}} | 0.02415 ^{\{ 12 \}} | 0.02563 ^{\{ 13 \}} | 0.01949 ^{\{ 1 \}} | 0.02748 ^{\{ 15 \}} | 0.02321 ^{\{ 9 \}} | 0.02659 ^{\{ 14 \}} | |
\sum Ranks | 16 ^{\{ 2 \}} | 44 ^{\{ 5 \}} | 80 ^{\{ 10 \}} | 36.5 ^{\{ 4 \}} | 70 ^{\{ 8 \}} | 100 ^{\{ 13 \}} | 49 ^{\{ 6 \}} | 25.5 ^{\{ 3 \}} | 59 ^{\{ 7 \}} | 91 ^{\{ 12 \}} | 116 ^{\{ 15 \}} | 8 ^{\{ 1 \}} | 101 ^{\{ 14 \}} | 75 ^{\{ 9 \}} | 89 ^{\{ 11 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.36362 ^{\{ 4 \}} | 0.41591 ^{\{ 10 \}} | 0.46651 ^{\{ 14 \}} | 0.3611 ^{\{ 3 \}} | 0.43973 ^{\{ 13 \}} | 0.47679 ^{\{ 15 \}} | 0.42017 ^{\{ 11 \}} | 0.3771 ^{\{ 7 \}} | 0.31372 ^{\{ 2 \}} | 0.37161 ^{\{ 6 \}} | 0.42351 ^{\{ 12 \}} | 0.18844 ^{\{ 1 \}} | 0.41564 ^{\{ 9 \}} | 0.3691 ^{\{ 5 \}} | 0.40653 ^{\{ 8 \}} |
BIAS( \hat{\beta} ) | 0.44791 ^{\{ 3 \}} | 0.47888 ^{\{ 6 \}} | 0.50494 ^{\{ 11 \}} | 0.48893 ^{\{ 8 \}} | 0.50806 ^{\{ 12 \}} | 0.49815 ^{\{ 10 \}} | 0.50822 ^{\{ 13 \}} | 0.46815 ^{\{ 4 \}} | 0.37554 ^{\{ 2 \}} | 0.47535 ^{\{ 5 \}} | 0.51634 ^{\{ 15 \}} | 0.26964 ^{\{ 1 \}} | 0.51083 ^{\{ 14 \}} | 0.48372 ^{\{ 7 \}} | 0.49761 ^{\{ 9 \}} | |
MSE( \hat{\delta} ) | 0.21037 ^{\{ 4 \}} | 0.26991 ^{\{ 10 \}} | 0.33182 ^{\{ 14 \}} | 0.20202 ^{\{ 3 \}} | 0.29159 ^{\{ 13 \}} | 0.35861 ^{\{ 15 \}} | 0.27624 ^{\{ 11 \}} | 0.22664 ^{\{ 7 \}} | 0.17428 ^{\{ 2 \}} | 0.21641 ^{\{ 5 \}} | 0.2846 ^{\{ 12 \}} | 0.05936 ^{\{ 1 \}} | 0.26835 ^{\{ 9 \}} | 0.21794 ^{\{ 6 \}} | 0.25878 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.29849 ^{\{ 3 \}} | 0.33823 ^{\{ 5 \}} | 0.36564 ^{\{ 11 \}} | 0.36137 ^{\{ 10 \}} | 0.37711 ^{\{ 13 \}} | 0.3567 ^{\{ 8 \}} | 0.3741 ^{\{ 12 \}} | 0.32343 ^{\{ 4 \}} | 0.25591 ^{\{ 2 \}} | 0.35154 ^{\{ 7 \}} | 0.38344 ^{\{ 15 \}} | 0.12377 ^{\{ 1 \}} | 0.38059 ^{\{ 14 \}} | 0.35106 ^{\{ 6 \}} | 0.35683 ^{\{ 9 \}} | |
MRE( \hat{\delta} ) | 0.24241 ^{\{ 4 \}} | 0.27727 ^{\{ 10 \}} | 0.31101 ^{\{ 14 \}} | 0.24073 ^{\{ 3 \}} | 0.29315 ^{\{ 13 \}} | 0.31786 ^{\{ 15 \}} | 0.28011 ^{\{ 11 \}} | 0.2514 ^{\{ 7 \}} | 0.20915 ^{\{ 2 \}} | 0.24774 ^{\{ 6 \}} | 0.28234 ^{\{ 12 \}} | 0.12563 ^{\{ 1 \}} | 0.27709 ^{\{ 9 \}} | 0.24606 ^{\{ 5 \}} | 0.27102 ^{\{ 8 \}} | |
MRE( \hat{\beta} ) | 0.29861 ^{\{ 3 \}} | 0.31926 ^{\{ 6 \}} | 0.33662 ^{\{ 11 \}} | 0.32595 ^{\{ 8 \}} | 0.33871 ^{\{ 12 \}} | 0.3321 ^{\{ 10 \}} | 0.33882 ^{\{ 13 \}} | 0.3121 ^{\{ 4 \}} | 0.25036 ^{\{ 2 \}} | 0.3169 ^{\{ 5 \}} | 0.34423 ^{\{ 15 \}} | 0.17976 ^{\{ 1 \}} | 0.34056 ^{\{ 14 \}} | 0.32248 ^{\{ 7 \}} | 0.33174 ^{\{ 9 \}} | |
D_{abs} | 0.03944 ^{\{ 1 \}} | 0.04453 ^{\{ 8 \}} | 0.04527 ^{\{ 10 \}} | 0.04151 ^{\{ 3 \}} | 0.04361 ^{\{ 5 \}} | 0.04561 ^{\{ 12 \}} | 0.04232 ^{\{ 4 \}} | 0.04094 ^{\{ 2 \}} | 0.04552 ^{\{ 11 \}} | 0.04852 ^{\{ 13 \}} | 0.04426 ^{\{ 7 \}} | 0.04386 ^{\{ 6 \}} | 0.05483 ^{\{ 15 \}} | 0.04475 ^{\{ 9 \}} | 0.05458 ^{\{ 14 \}} | |
D_{max} | 0.06565 ^{\{ 1 \}} | 0.07283 ^{\{ 8 \}} | 0.07644 ^{\{ 11 \}} | 0.06681 ^{\{ 2 \}} | 0.0725 ^{\{ 7 \}} | 0.07674 ^{\{ 12 \}} | 0.06966 ^{\{ 5 \}} | 0.06724 ^{\{ 3 \}} | 0.07288 ^{\{ 9 \}} | 0.07755 ^{\{ 13 \}} | 0.07318 ^{\{ 10 \}} | 0.06774 ^{\{ 4 \}} | 0.0875 ^{\{ 15 \}} | 0.07206 ^{\{ 6 \}} | 0.08689 ^{\{ 14 \}} | |
\sum Ranks | 23 ^{\{ 2 \}} | 63 ^{\{ 8 \}} | 96 ^{\{ 12 \}} | 40 ^{\{ 5 \}} | 88 ^{\{ 11 \}} | 97 ^{\{ 13 \}} | 80 ^{\{ 10 \}} | 38 ^{\{ 4 \}} | 32 ^{\{ 3 \}} | 60 ^{\{ 7 \}} | 98 ^{\{ 14 \}} | 16 ^{\{ 1 \}} | 99 ^{\{ 15 \}} | 51 ^{\{ 6 \}} | 79 ^{\{ 9 \}} | |
60 | BIAS( \hat{\delta} ) | 0.25193 ^{\{ 2 \}} | 0.31471 ^{\{ 8 \}} | 0.35735 ^{\{ 14 \}} | 0.27976 ^{\{ 4 \}} | 0.34812 ^{\{ 12 \}} | 0.40958 ^{\{ 15 \}} | 0.32601 ^{\{ 10 \}} | 0.28841 ^{\{ 5 \}} | 0.26064 ^{\{ 3 \}} | 0.29235 ^{\{ 7 \}} | 0.32535 ^{\{ 9 \}} | 0.17636 ^{\{ 1 \}} | 0.35405 ^{\{ 13 \}} | 0.28887 ^{\{ 6 \}} | 0.34765 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.34938 ^{\{ 2 \}} | 0.40316 ^{\{ 5 \}} | 0.41962 ^{\{ 7 \}} | 0.412 ^{\{ 6 \}} | 0.42457 ^{\{ 9 \}} | 0.45051 ^{\{ 14 \}} | 0.42184 ^{\{ 8 \}} | 0.38627 ^{\{ 4 \}} | 0.34945 ^{\{ 3 \}} | 0.42834 ^{\{ 10 \}} | 0.44339 ^{\{ 12 \}} | 0.24504 ^{\{ 1 \}} | 0.46935 ^{\{ 15 \}} | 0.43185 ^{\{ 11 \}} | 0.44997 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.10131 ^{\{ 2 \}} | 0.14878 ^{\{ 8 \}} | 0.20227 ^{\{ 14 \}} | 0.12128 ^{\{ 4 \}} | 0.19801 ^{\{ 13 \}} | 0.25959 ^{\{ 15 \}} | 0.16855 ^{\{ 10 \}} | 0.12991 ^{\{ 5 \}} | 0.12121 ^{\{ 3 \}} | 0.13012 ^{\{ 6 \}} | 0.1646 ^{\{ 9 \}} | 0.05318 ^{\{ 1 \}} | 0.19114 ^{\{ 11 \}} | 0.13018 ^{\{ 7 \}} | 0.1939 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.19887 ^{\{ 2 \}} | 0.24543 ^{\{ 5 \}} | 0.2617 ^{\{ 6 \}} | 0.28333 ^{\{ 9 \}} | 0.28214 ^{\{ 8 \}} | 0.30154 ^{\{ 13 \}} | 0.27339 ^{\{ 7 \}} | 0.23646 ^{\{ 4 \}} | 0.23071 ^{\{ 3 \}} | 0.29922 ^{\{ 11 \}} | 0.30099 ^{\{ 12 \}} | 0.1013 ^{\{ 1 \}} | 0.3262 ^{\{ 15 \}} | 0.29724 ^{\{ 10 \}} | 0.31345 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.16795 ^{\{ 2 \}} | 0.20981 ^{\{ 8 \}} | 0.23823 ^{\{ 14 \}} | 0.18651 ^{\{ 4 \}} | 0.23208 ^{\{ 12 \}} | 0.27306 ^{\{ 15 \}} | 0.21734 ^{\{ 10 \}} | 0.19227 ^{\{ 5 \}} | 0.17376 ^{\{ 3 \}} | 0.1949 ^{\{ 7 \}} | 0.2169 ^{\{ 9 \}} | 0.11757 ^{\{ 1 \}} | 0.23603 ^{\{ 13 \}} | 0.19258 ^{\{ 6 \}} | 0.23177 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.23292 ^{\{ 2 \}} | 0.26877 ^{\{ 5 \}} | 0.27975 ^{\{ 7 \}} | 0.27467 ^{\{ 6 \}} | 0.28305 ^{\{ 9 \}} | 0.30034 ^{\{ 14 \}} | 0.28122 ^{\{ 8 \}} | 0.25751 ^{\{ 4 \}} | 0.23297 ^{\{ 3 \}} | 0.28556 ^{\{ 10 \}} | 0.2956 ^{\{ 12 \}} | 0.16336 ^{\{ 1 \}} | 0.3129 ^{\{ 15 \}} | 0.2879 ^{\{ 11 \}} | 0.29998 ^{\{ 13 \}} | |
D_{abs} | 0.02961 ^{\{ 1 \}} | 0.03051 ^{\{ 2 \}} | 0.03132 ^{\{ 6 \}} | 0.03103 ^{\{ 5 \}} | 0.03319 ^{\{ 9 \}} | 0.03268 ^{\{ 7 \}} | 0.03101 ^{\{ 4 \}} | 0.03063 ^{\{ 3 \}} | 0.03576 ^{\{ 13 \}} | 0.03372 ^{\{ 11 \}} | 0.03366 ^{\{ 10 \}} | 0.03275 ^{\{ 8 \}} | 0.03914 ^{\{ 14 \}} | 0.03378 ^{\{ 12 \}} | 0.03955 ^{\{ 15 \}} | |
D_{max} | 0.04842 ^{\{ 1 \}} | 0.05047 ^{\{ 4 \}} | 0.05302 ^{\{ 7 \}} | 0.04997 ^{\{ 2 \}} | 0.05541 ^{\{ 11 \}} | 0.05606 ^{\{ 12 \}} | 0.05152 ^{\{ 6 \}} | 0.05036 ^{\{ 3 \}} | 0.0573 ^{\{ 13 \}} | 0.05469 ^{\{ 9 \}} | 0.05517 ^{\{ 10 \}} | 0.0514 ^{\{ 5 \}} | 0.06451 ^{\{ 14 \}} | 0.05445 ^{\{ 8 \}} | 0.06483 ^{\{ 15 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 45 ^{\{ 6 \}} | 75 ^{\{ 10 \}} | 40 ^{\{ 4 \}} | 83 ^{\{ 11.5 \}} | 105 ^{\{ 14 \}} | 63 ^{\{ 7 \}} | 33 ^{\{ 3 \}} | 44 ^{\{ 5 \}} | 71 ^{\{ 8.5 \}} | 83 ^{\{ 11.5 \}} | 19 ^{\{ 2 \}} | 110 ^{\{ 15 \}} | 71 ^{\{ 8.5 \}} | 104 ^{\{ 13 \}} | |
100 | BIAS( \hat{\delta} ) | 0.20656 ^{\{ 2 \}} | 0.25391 ^{\{ 8 \}} | 0.30172 ^{\{ 13 \}} | 0.23433 ^{\{ 5 \}} | 0.29934 ^{\{ 12 \}} | 0.327 ^{\{ 15 \}} | 0.25672 ^{\{ 9 \}} | 0.23765 ^{\{ 6 \}} | 0.2224 ^{\{ 3 \}} | 0.24672 ^{\{ 7 \}} | 0.26161 ^{\{ 10 \}} | 0.16346 ^{\{ 1 \}} | 0.30254 ^{\{ 14 \}} | 0.23314 ^{\{ 4 \}} | 0.29396 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.28605 ^{\{ 2 \}} | 0.34848 ^{\{ 6 \}} | 0.37222 ^{\{ 10 \}} | 0.35822 ^{\{ 8 \}} | 0.38959 ^{\{ 13 \}} | 0.38082 ^{\{ 12 \}} | 0.34956 ^{\{ 7 \}} | 0.32677 ^{\{ 4 \}} | 0.30141 ^{\{ 3 \}} | 0.36475 ^{\{ 9 \}} | 0.37444 ^{\{ 11 \}} | 0.22382 ^{\{ 1 \}} | 0.40691 ^{\{ 15 \}} | 0.34202 ^{\{ 5 \}} | 0.40473 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.06736 ^{\{ 2 \}} | 0.10079 ^{\{ 8 \}} | 0.14079 ^{\{ 14 \}} | 0.08235 ^{\{ 3 \}} | 0.13993 ^{\{ 13 \}} | 0.17078 ^{\{ 15 \}} | 0.10453 ^{\{ 9 \}} | 0.09235 ^{\{ 6 \}} | 0.08912 ^{\{ 5 \}} | 0.09267 ^{\{ 7 \}} | 0.10813 ^{\{ 10 \}} | 0.0469 ^{\{ 1 \}} | 0.13908 ^{\{ 12 \}} | 0.08462 ^{\{ 4 \}} | 0.13209 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.14544 ^{\{ 2 \}} | 0.19976 ^{\{ 6 \}} | 0.22305 ^{\{ 9 \}} | 0.21696 ^{\{ 8 \}} | 0.23768 ^{\{ 13 \}} | 0.23128 ^{\{ 12 \}} | 0.20607 ^{\{ 7 \}} | 0.18067 ^{\{ 4 \}} | 0.1778 ^{\{ 3 \}} | 0.22367 ^{\{ 10 \}} | 0.22523 ^{\{ 11 \}} | 0.08729 ^{\{ 1 \}} | 0.26214 ^{\{ 15 \}} | 0.1935 ^{\{ 5 \}} | 0.25864 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.13771 ^{\{ 2 \}} | 0.16927 ^{\{ 8 \}} | 0.20115 ^{\{ 13 \}} | 0.15622 ^{\{ 5 \}} | 0.19956 ^{\{ 12 \}} | 0.218 ^{\{ 15 \}} | 0.17115 ^{\{ 9 \}} | 0.15844 ^{\{ 6 \}} | 0.14827 ^{\{ 3 \}} | 0.16448 ^{\{ 7 \}} | 0.17441 ^{\{ 10 \}} | 0.10897 ^{\{ 1 \}} | 0.2017 ^{\{ 14 \}} | 0.15543 ^{\{ 4 \}} | 0.19598 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.1907 ^{\{ 2 \}} | 0.23232 ^{\{ 6 \}} | 0.24815 ^{\{ 10 \}} | 0.23882 ^{\{ 8 \}} | 0.25972 ^{\{ 13 \}} | 0.25388 ^{\{ 12 \}} | 0.23304 ^{\{ 7 \}} | 0.21785 ^{\{ 4 \}} | 0.20094 ^{\{ 3 \}} | 0.24317 ^{\{ 9 \}} | 0.24963 ^{\{ 11 \}} | 0.14921 ^{\{ 1 \}} | 0.27127 ^{\{ 15 \}} | 0.22801 ^{\{ 5 \}} | 0.26982 ^{\{ 14 \}} | |
D_{abs} | 0.02235 ^{\{ 1 \}} | 0.02482 ^{\{ 3 \}} | 0.02569 ^{\{ 6 \}} | 0.02462 ^{\{ 2 \}} | 0.02622 ^{\{ 10 \}} | 0.02597 ^{\{ 8 \}} | 0.0254 ^{\{ 5 \}} | 0.02484 ^{\{ 4 \}} | 0.02732 ^{\{ 13 \}} | 0.02672 ^{\{ 12 \}} | 0.02663 ^{\{ 11 \}} | 0.02587 ^{\{ 7 \}} | 0.03105 ^{\{ 14 \}} | 0.02607 ^{\{ 9 \}} | 0.03134 ^{\{ 15 \}} | |
D_{max} | 0.03675 ^{\{ 1 \}} | 0.04106 ^{\{ 4 \}} | 0.04372 ^{\{ 10 \}} | 0.04 ^{\{ 2 \}} | 0.04399 ^{\{ 12 \}} | 0.04501 ^{\{ 13 \}} | 0.04203 ^{\{ 6 \}} | 0.04065 ^{\{ 3 \}} | 0.04393 ^{\{ 11 \}} | 0.04357 ^{\{ 8 \}} | 0.04363 ^{\{ 9 \}} | 0.0414 ^{\{ 5 \}} | 0.05119 ^{\{ 14 \}} | 0.04208 ^{\{ 7 \}} | 0.05143 ^{\{ 15 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 49 ^{\{ 7 \}} | 85 ^{\{ 11 \}} | 41 ^{\{ 4 \}} | 98 ^{\{ 12 \}} | 102 ^{\{ 13 \}} | 59 ^{\{ 8 \}} | 37 ^{\{ 3 \}} | 44 ^{\{ 6 \}} | 69 ^{\{ 9 \}} | 83 ^{\{ 10 \}} | 18 ^{\{ 2 \}} | 113 ^{\{ 15 \}} | 43 ^{\{ 5 \}} | 105 ^{\{ 14 \}} | |
200 | BIAS( \hat{\delta} ) | 0.15288 ^{\{ 2 \}} | 0.18956 ^{\{ 8 \}} | 0.21534 ^{\{ 11 \}} | 0.16331 ^{\{ 3 \}} | 0.21778 ^{\{ 12 \}} | 0.26177 ^{\{ 15 \}} | 0.1897 ^{\{ 9 \}} | 0.1732 ^{\{ 4 \}} | 0.17595 ^{\{ 5 \}} | 0.18315 ^{\{ 7 \}} | 0.20027 ^{\{ 10 \}} | 0.13041 ^{\{ 1 \}} | 0.2255 ^{\{ 14 \}} | 0.18271 ^{\{ 6 \}} | 0.22008 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.2161 ^{\{ 2 \}} | 0.25397 ^{\{ 6 \}} | 0.27965 ^{\{ 10 \}} | 0.24415 ^{\{ 4 \}} | 0.28824 ^{\{ 11 \}} | 0.32319 ^{\{ 15 \}} | 0.25894 ^{\{ 7 \}} | 0.23866 ^{\{ 3 \}} | 0.24499 ^{\{ 5 \}} | 0.27244 ^{\{ 9 \}} | 0.29863 ^{\{ 12 \}} | 0.1761 ^{\{ 1 \}} | 0.32229 ^{\{ 14 \}} | 0.26751 ^{\{ 8 \}} | 0.31447 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.03871 ^{\{ 2 \}} | 0.05626 ^{\{ 8 \}} | 0.07407 ^{\{ 12 \}} | 0.0409 ^{\{ 3 \}} | 0.07383 ^{\{ 11 \}} | 0.10743 ^{\{ 15 \}} | 0.05719 ^{\{ 9 \}} | 0.04689 ^{\{ 4 \}} | 0.0535 ^{\{ 7 \}} | 0.05235 ^{\{ 6 \}} | 0.06256 ^{\{ 10 \}} | 0.02939 ^{\{ 1 \}} | 0.07872 ^{\{ 14 \}} | 0.05169 ^{\{ 5 \}} | 0.07812 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.08188 ^{\{ 2 \}} | 0.11144 ^{\{ 5 \}} | 0.13504 ^{\{ 10 \}} | 0.10336 ^{\{ 4 \}} | 0.14447 ^{\{ 11 \}} | 0.17092 ^{\{ 14 \}} | 0.11355 ^{\{ 6 \}} | 0.09586 ^{\{ 3 \}} | 0.11854 ^{\{ 7 \}} | 0.13004 ^{\{ 9 \}} | 0.15599 ^{\{ 12 \}} | 0.0546 ^{\{ 1 \}} | 0.17622 ^{\{ 15 \}} | 0.12213 ^{\{ 8 \}} | 0.16875 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.10192 ^{\{ 2 \}} | 0.12638 ^{\{ 8 \}} | 0.14356 ^{\{ 11 \}} | 0.10888 ^{\{ 3 \}} | 0.14519 ^{\{ 12 \}} | 0.17452 ^{\{ 15 \}} | 0.12646 ^{\{ 9 \}} | 0.11547 ^{\{ 4 \}} | 0.1173 ^{\{ 5 \}} | 0.1221 ^{\{ 7 \}} | 0.13351 ^{\{ 10 \}} | 0.08694 ^{\{ 1 \}} | 0.15034 ^{\{ 14 \}} | 0.1218 ^{\{ 6 \}} | 0.14672 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.14407 ^{\{ 2 \}} | 0.16931 ^{\{ 6 \}} | 0.18643 ^{\{ 10 \}} | 0.16277 ^{\{ 4 \}} | 0.19216 ^{\{ 11 \}} | 0.21546 ^{\{ 15 \}} | 0.17263 ^{\{ 7 \}} | 0.1591 ^{\{ 3 \}} | 0.16333 ^{\{ 5 \}} | 0.18163 ^{\{ 9 \}} | 0.19909 ^{\{ 12 \}} | 0.1174 ^{\{ 1 \}} | 0.21486 ^{\{ 14 \}} | 0.17834 ^{\{ 8 \}} | 0.20965 ^{\{ 13 \}} | |
D_{abs} | 0.01708 ^{\{ 1 \}} | 0.01778 ^{\{ 4 \}} | 0.01841 ^{\{ 7 \}} | 0.01749 ^{\{ 2.5 \}} | 0.01868 ^{\{ 8 \}} | 0.01953 ^{\{ 9 \}} | 0.01788 ^{\{ 6 \}} | 0.01749 ^{\{ 2.5 \}} | 0.02059 ^{\{ 13 \}} | 0.02042 ^{\{ 12 \}} | 0.02035 ^{\{ 11 \}} | 0.01781 ^{\{ 5 \}} | 0.02273 ^{\{ 14 \}} | 0.01969 ^{\{ 10 \}} | 0.02289 ^{\{ 15 \}} | |
D_{max} | 0.02783 ^{\{ 1 \}} | 0.02966 ^{\{ 5 \}} | 0.03114 ^{\{ 7 \}} | 0.02835 ^{\{ 2 \}} | 0.03144 ^{\{ 8 \}} | 0.03378 ^{\{ 13 \}} | 0.02968 ^{\{ 6 \}} | 0.02876 ^{\{ 4 \}} | 0.03344 ^{\{ 12 \}} | 0.03311 ^{\{ 10 \}} | 0.03321 ^{\{ 11 \}} | 0.02872 ^{\{ 3 \}} | 0.03777 ^{\{ 15 \}} | 0.03208 ^{\{ 9 \}} | 0.03745 ^{\{ 14 \}} | |
\sum Ranks | 14 ^{\{ 1.5 \}} | 50 ^{\{ 5 \}} | 78 ^{\{ 10 \}} | 25.5 ^{\{ 3 \}} | 84 ^{\{ 11 \}} | 111 ^{\{ 14 \}} | 59 ^{\{ 6.5 \}} | 27.5 ^{\{ 4 \}} | 59 ^{\{ 6.5 \}} | 69 ^{\{ 9 \}} | 88 ^{\{ 12 \}} | 14 ^{\{ 1.5 \}} | 114 ^{\{ 15 \}} | 60 ^{\{ 8 \}} | 107 ^{\{ 13 \}} | |
300 | BIAS( \hat{\delta} ) | 0.12756 ^{\{ 2 \}} | 0.14717 ^{\{ 6 \}} | 0.17874 ^{\{ 12 \}} | 0.13713 ^{\{ 3 \}} | 0.17341 ^{\{ 11 \}} | 0.21053 ^{\{ 15 \}} | 0.16111 ^{\{ 9 \}} | 0.13772 ^{\{ 4 \}} | 0.1435 ^{\{ 5 \}} | 0.15587 ^{\{ 8 \}} | 0.16547 ^{\{ 10 \}} | 0.11423 ^{\{ 1 \}} | 0.19156 ^{\{ 14 \}} | 0.14743 ^{\{ 7 \}} | 0.18039 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.17764 ^{\{ 2 \}} | 0.19923 ^{\{ 4 \}} | 0.22686 ^{\{ 9 \}} | 0.20548 ^{\{ 6 \}} | 0.22944 ^{\{ 10 \}} | 0.26697 ^{\{ 13 \}} | 0.21258 ^{\{ 7 \}} | 0.18859 ^{\{ 3 \}} | 0.20381 ^{\{ 5 \}} | 0.23225 ^{\{ 11 \}} | 0.25014 ^{\{ 12 \}} | 0.14872 ^{\{ 1 \}} | 0.27192 ^{\{ 15 \}} | 0.21783 ^{\{ 8 \}} | 0.2676 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.02609 ^{\{ 2 \}} | 0.03343 ^{\{ 5 \}} | 0.04979 ^{\{ 12 \}} | 0.0295 ^{\{ 3 \}} | 0.0464 ^{\{ 11 \}} | 0.06938 ^{\{ 15 \}} | 0.04051 ^{\{ 9 \}} | 0.03057 ^{\{ 4 \}} | 0.03482 ^{\{ 7 \}} | 0.03819 ^{\{ 8 \}} | 0.0425 ^{\{ 10 \}} | 0.02264 ^{\{ 1 \}} | 0.05705 ^{\{ 14 \}} | 0.03431 ^{\{ 6 \}} | 0.05061 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.05098 ^{\{ 2 \}} | 0.06578 ^{\{ 4 \}} | 0.09046 ^{\{ 10 \}} | 0.0717 ^{\{ 5 \}} | 0.08987 ^{\{ 9 \}} | 0.12315 ^{\{ 13 \}} | 0.07547 ^{\{ 6 \}} | 0.06013 ^{\{ 3 \}} | 0.07747 ^{\{ 7 \}} | 0.09348 ^{\{ 11 \}} | 0.10805 ^{\{ 12 \}} | 0.04003 ^{\{ 1 \}} | 0.12352 ^{\{ 14 \}} | 0.08226 ^{\{ 8 \}} | 0.1243 ^{\{ 15 \}} | |
MRE( \hat{\delta} ) | 0.08504 ^{\{ 2 \}} | 0.09811 ^{\{ 6 \}} | 0.11916 ^{\{ 12 \}} | 0.09142 ^{\{ 3 \}} | 0.1156 ^{\{ 11 \}} | 0.14035 ^{\{ 15 \}} | 0.10741 ^{\{ 9 \}} | 0.09181 ^{\{ 4 \}} | 0.09567 ^{\{ 5 \}} | 0.10392 ^{\{ 8 \}} | 0.11031 ^{\{ 10 \}} | 0.07615 ^{\{ 1 \}} | 0.1277 ^{\{ 14 \}} | 0.09829 ^{\{ 7 \}} | 0.12026 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.11842 ^{\{ 2 \}} | 0.13282 ^{\{ 4 \}} | 0.15124 ^{\{ 9 \}} | 0.13699 ^{\{ 6 \}} | 0.15296 ^{\{ 10 \}} | 0.17798 ^{\{ 13 \}} | 0.14172 ^{\{ 7 \}} | 0.12573 ^{\{ 3 \}} | 0.13587 ^{\{ 5 \}} | 0.15484 ^{\{ 11 \}} | 0.16676 ^{\{ 12 \}} | 0.09915 ^{\{ 1 \}} | 0.18128 ^{\{ 15 \}} | 0.14522 ^{\{ 8 \}} | 0.1784 ^{\{ 14 \}} | |
D_{abs} | 0.01393 ^{\{ 1 \}} | 0.01439 ^{\{ 3 \}} | 0.01544 ^{\{ 8 \}} | 0.01449 ^{\{ 4 \}} | 0.01487 ^{\{ 6 \}} | 0.01613 ^{\{ 9 \}} | 0.0152 ^{\{ 7 \}} | 0.01395 ^{\{ 2 \}} | 0.0174 ^{\{ 13 \}} | 0.01678 ^{\{ 12 \}} | 0.01636 ^{\{ 11 \}} | 0.01461 ^{\{ 5 \}} | 0.01959 ^{\{ 15 \}} | 0.0163 ^{\{ 10 \}} | 0.01954 ^{\{ 14 \}} | |
D_{max} | 0.02275 ^{\{ 1 \}} | 0.02384 ^{\{ 5 \}} | 0.02607 ^{\{ 8 \}} | 0.02357 ^{\{ 3 \}} | 0.02515 ^{\{ 6 \}} | 0.0278 ^{\{ 12 \}} | 0.02531 ^{\{ 7 \}} | 0.02298 ^{\{ 2 \}} | 0.02812 ^{\{ 13 \}} | 0.02723 ^{\{ 11 \}} | 0.02671 ^{\{ 10 \}} | 0.02372 ^{\{ 4 \}} | 0.03222 ^{\{ 15 \}} | 0.02651 ^{\{ 9 \}} | 0.03184 ^{\{ 14 \}} | |
\sum Ranks | 14 ^{\{ 1 \}} | 37 ^{\{ 5 \}} | 80 ^{\{ 10.5 \}} | 33 ^{\{ 4 \}} | 74 ^{\{ 9 \}} | 105 ^{\{ 13 \}} | 61 ^{\{ 7 \}} | 25 ^{\{ 3 \}} | 60 ^{\{ 6 \}} | 80 ^{\{ 10.5 \}} | 87 ^{\{ 12 \}} | 15 ^{\{ 2 \}} | 116 ^{\{ 15 \}} | 63 ^{\{ 8 \}} | 110 ^{\{ 14 \}} | |
400 | BIAS( \hat{\delta} ) | 0.10866 ^{\{ 2 \}} | 0.1345 ^{\{ 9 \}} | 0.15044 ^{\{ 11 \}} | 0.11647 ^{\{ 3 \}} | 0.15668 ^{\{ 13 \}} | 0.18648 ^{\{ 15 \}} | 0.13412 ^{\{ 8 \}} | 0.11906 ^{\{ 4 \}} | 0.13407 ^{\{ 7 \}} | 0.13152 ^{\{ 6 \}} | 0.1481 ^{\{ 10 \}} | 0.10545 ^{\{ 1 \}} | 0.16093 ^{\{ 14 \}} | 0.12007 ^{\{ 5 \}} | 0.15558 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.15352 ^{\{ 2 \}} | 0.18616 ^{\{ 8 \}} | 0.19179 ^{\{ 9 \}} | 0.16571 ^{\{ 4 \}} | 0.20261 ^{\{ 11 \}} | 0.23735 ^{\{ 15 \}} | 0.18354 ^{\{ 6 \}} | 0.16043 ^{\{ 3 \}} | 0.18538 ^{\{ 7 \}} | 0.19245 ^{\{ 10 \}} | 0.22086 ^{\{ 12 \}} | 0.1359 ^{\{ 1 \}} | 0.2328 ^{\{ 14 \}} | 0.17147 ^{\{ 5 \}} | 0.22366 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01904 ^{\{ 1 \}} | 0.02909 ^{\{ 8 \}} | 0.0361 ^{\{ 11 \}} | 0.02075 ^{\{ 3 \}} | 0.03897 ^{\{ 12 \}} | 0.05496 ^{\{ 15 \}} | 0.02796 ^{\{ 7 \}} | 0.02201 ^{\{ 4 \}} | 0.03078 ^{\{ 9 \}} | 0.02632 ^{\{ 6 \}} | 0.03496 ^{\{ 10 \}} | 0.01912 ^{\{ 2 \}} | 0.0414 ^{\{ 14 \}} | 0.02291 ^{\{ 5 \}} | 0.03957 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.03755 ^{\{ 2 \}} | 0.05973 ^{\{ 7 \}} | 0.06171 ^{\{ 9 \}} | 0.04387 ^{\{ 4 \}} | 0.07077 ^{\{ 11 \}} | 0.09576 ^{\{ 15 \}} | 0.05488 ^{\{ 6 \}} | 0.04111 ^{\{ 3 \}} | 0.06503 ^{\{ 10 \}} | 0.06059 ^{\{ 8 \}} | 0.08705 ^{\{ 13 \}} | 0.03291 ^{\{ 1 \}} | 0.09212 ^{\{ 14 \}} | 0.04832 ^{\{ 5 \}} | 0.08653 ^{\{ 12 \}} | |
MRE( \hat{\delta} ) | 0.07244 ^{\{ 2 \}} | 0.08967 ^{\{ 9 \}} | 0.10029 ^{\{ 11 \}} | 0.07765 ^{\{ 3 \}} | 0.10445 ^{\{ 13 \}} | 0.12432 ^{\{ 15 \}} | 0.08942 ^{\{ 8 \}} | 0.07938 ^{\{ 4 \}} | 0.08938 ^{\{ 7 \}} | 0.08768 ^{\{ 6 \}} | 0.09874 ^{\{ 10 \}} | 0.0703 ^{\{ 1 \}} | 0.10729 ^{\{ 14 \}} | 0.08004 ^{\{ 5 \}} | 0.10372 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.10235 ^{\{ 2 \}} | 0.1241 ^{\{ 8 \}} | 0.12786 ^{\{ 9 \}} | 0.11048 ^{\{ 4 \}} | 0.13507 ^{\{ 11 \}} | 0.15823 ^{\{ 15 \}} | 0.12236 ^{\{ 6 \}} | 0.10695 ^{\{ 3 \}} | 0.12359 ^{\{ 7 \}} | 0.1283 ^{\{ 10 \}} | 0.14724 ^{\{ 12 \}} | 0.0906 ^{\{ 1 \}} | 0.1552 ^{\{ 14 \}} | 0.11431 ^{\{ 5 \}} | 0.1491 ^{\{ 13 \}} | |
D_{abs} | 0.01246 ^{\{ 3 \}} | 0.01304 ^{\{ 6 \}} | 0.0132 ^{\{ 7 \}} | 0.01242 ^{\{ 1.5 \}} | 0.0134 ^{\{ 8 \}} | 0.0142 ^{\{ 10 \}} | 0.01274 ^{\{ 4 \}} | 0.01242 ^{\{ 1.5 \}} | 0.01486 ^{\{ 13 \}} | 0.01465 ^{\{ 11 \}} | 0.01477 ^{\{ 12 \}} | 0.01284 ^{\{ 5 \}} | 0.01692 ^{\{ 15 \}} | 0.01356 ^{\{ 9 \}} | 0.01666 ^{\{ 14 \}} | |
D_{max} | 0.0202 ^{\{ 1 \}} | 0.02154 ^{\{ 6 \}} | 0.02231 ^{\{ 8 \}} | 0.02028 ^{\{ 2 \}} | 0.02261 ^{\{ 9 \}} | 0.02465 ^{\{ 13 \}} | 0.02112 ^{\{ 5 \}} | 0.02043 ^{\{ 3 \}} | 0.02406 ^{\{ 11 \}} | 0.02377 ^{\{ 10 \}} | 0.02409 ^{\{ 12 \}} | 0.02086 ^{\{ 4 \}} | 0.02768 ^{\{ 15 \}} | 0.02207 ^{\{ 7 \}} | 0.02724 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 1 \}} | 61 ^{\{ 7 \}} | 75 ^{\{ 10 \}} | 24.5 ^{\{ 3 \}} | 88 ^{\{ 11 \}} | 113 ^{\{ 14 \}} | 50 ^{\{ 6 \}} | 25.5 ^{\{ 4 \}} | 71 ^{\{ 9 \}} | 67 ^{\{ 8 \}} | 91 ^{\{ 12 \}} | 16 ^{\{ 2 \}} | 114 ^{\{ 15 \}} | 46 ^{\{ 5 \}} | 103 ^{\{ 13 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.13624 ^{\{ 4 \}} | 0.15298 ^{\{ 10 \}} | 0.16819 ^{\{ 14 \}} | 0.13657 ^{\{ 5 \}} | 0.15175 ^{\{ 8 \}} | 0.18055 ^{\{ 15 \}} | 0.15251 ^{\{ 9 \}} | 0.14595 ^{\{ 6 \}} | 0.09288 ^{\{ 2 \}} | 0.12666 ^{\{ 3 \}} | 0.15422 ^{\{ 11 \}} | 0.05648 ^{\{ 1 \}} | 0.1601 ^{\{ 12 \}} | 0.14753 ^{\{ 7 \}} | 0.16209 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.57765 ^{\{ 4 \}} | 0.65842 ^{\{ 12 \}} | 0.63719 ^{\{ 9 \}} | 0.64979 ^{\{ 10 \}} | 0.60867 ^{\{ 7 \}} | 0.68047 ^{\{ 14 \}} | 0.65627 ^{\{ 11 \}} | 0.62937 ^{\{ 8 \}} | 0.25245 ^{\{ 2 \}} | 0.57701 ^{\{ 3 \}} | 0.66039 ^{\{ 13 \}} | 0.04775 ^{\{ 1 \}} | 0.5962 ^{\{ 6 \}} | 0.69113 ^{\{ 15 \}} | 0.58347 ^{\{ 5 \}} | |
MSE( \hat{\delta} ) | 0.02921 ^{\{ 5 \}} | 0.03635 ^{\{ 10 \}} | 0.04402 ^{\{ 14 \}} | 0.02809 ^{\{ 4 \}} | 0.03633 ^{\{ 9 \}} | 0.04928 ^{\{ 15 \}} | 0.03627 ^{\{ 8 \}} | 0.0338 ^{\{ 6 \}} | 0.01634 ^{\{ 2 \}} | 0.02598 ^{\{ 3 \}} | 0.03818 ^{\{ 11 \}} | 0.00526 ^{\{ 1 \}} | 0.04053 ^{\{ 12 \}} | 0.03407 ^{\{ 7 \}} | 0.0423 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.49086 ^{\{ 3 \}} | 0.62358 ^{\{ 11 \}} | 0.56339 ^{\{ 7 \}} | 0.62516 ^{\{ 12 \}} | 0.53029 ^{\{ 6 \}} | 0.6451 ^{\{ 14 \}} | 0.61704 ^{\{ 10 \}} | 0.58229 ^{\{ 9 \}} | 0.2075 ^{\{ 2 \}} | 0.56556 ^{\{ 8 \}} | 0.6369 ^{\{ 13 \}} | 0.01233 ^{\{ 1 \}} | 0.51078 ^{\{ 5 \}} | 0.68808 ^{\{ 15 \}} | 0.49454 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.27249 ^{\{ 4 \}} | 0.30596 ^{\{ 10 \}} | 0.33638 ^{\{ 14 \}} | 0.27314 ^{\{ 5 \}} | 0.3035 ^{\{ 8 \}} | 0.36111 ^{\{ 15 \}} | 0.30502 ^{\{ 9 \}} | 0.29189 ^{\{ 6 \}} | 0.18576 ^{\{ 2 \}} | 0.25331 ^{\{ 3 \}} | 0.30843 ^{\{ 11 \}} | 0.11296 ^{\{ 1 \}} | 0.32019 ^{\{ 12 \}} | 0.29507 ^{\{ 7 \}} | 0.32419 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.28882 ^{\{ 4 \}} | 0.32921 ^{\{ 12 \}} | 0.31859 ^{\{ 9 \}} | 0.3249 ^{\{ 10 \}} | 0.30434 ^{\{ 7 \}} | 0.34024 ^{\{ 14 \}} | 0.32814 ^{\{ 11 \}} | 0.31469 ^{\{ 8 \}} | 0.12623 ^{\{ 2 \}} | 0.28851 ^{\{ 3 \}} | 0.33019 ^{\{ 13 \}} | 0.02388 ^{\{ 1 \}} | 0.2981 ^{\{ 6 \}} | 0.34556 ^{\{ 15 \}} | 0.29173 ^{\{ 5 \}} | |
D_{abs} | 0.03768 ^{\{ 1 \}} | 0.04098 ^{\{ 5 \}} | 0.04468 ^{\{ 12 \}} | 0.04059 ^{\{ 2 \}} | 0.04585 ^{\{ 13 \}} | 0.04352 ^{\{ 10 \}} | 0.04069 ^{\{ 3 \}} | 0.04313 ^{\{ 9 \}} | 0.0425 ^{\{ 7 \}} | 0.04216 ^{\{ 6 \}} | 0.04287 ^{\{ 8 \}} | 0.04096 ^{\{ 4 \}} | 0.06514 ^{\{ 15 \}} | 0.04419 ^{\{ 11 \}} | 0.06029 ^{\{ 14 \}} | |
D_{max} | 0.06199 ^{\{ 1 \}} | 0.06671 ^{\{ 5 \}} | 0.07388 ^{\{ 13 \}} | 0.06517 ^{\{ 3 \}} | 0.07369 ^{\{ 12 \}} | 0.07286 ^{\{ 11 \}} | 0.06621 ^{\{ 4 \}} | 0.07003 ^{\{ 8 \}} | 0.06696 ^{\{ 6 \}} | 0.06706 ^{\{ 7 \}} | 0.07019 ^{\{ 9 \}} | 0.0623 ^{\{ 2 \}} | 0.10106 ^{\{ 15 \}} | 0.07066 ^{\{ 10 \}} | 0.09345 ^{\{ 14 \}} | |
\sum Ranks | 26 ^{\{ 3 \}} | 75 ^{\{ 9 \}} | 92 ^{\{ 14 \}} | 51 ^{\{ 5 \}} | 70 ^{\{ 8 \}} | 108 ^{\{ 15 \}} | 65 ^{\{ 7 \}} | 60 ^{\{ 6 \}} | 25 ^{\{ 2 \}} | 36 ^{\{ 4 \}} | 89 ^{\{ 13 \}} | 12 ^{\{ 1 \}} | 83 ^{\{ 11 \}} | 87 ^{\{ 12 \}} | 81 ^{\{ 10 \}} | |
60 | BIAS( \hat{\delta} ) | 0.10659 ^{\{ 3 \}} | 0.12172 ^{\{ 9 \}} | 0.14371 ^{\{ 14 \}} | 0.10716 ^{\{ 4 \}} | 0.13619 ^{\{ 13 \}} | 0.16304 ^{\{ 15 \}} | 0.13084 ^{\{ 11 \}} | 0.11699 ^{\{ 8 \}} | 0.07293 ^{\{ 2 \}} | 0.10752 ^{\{ 5 \}} | 0.12531 ^{\{ 10 \}} | 0.04248 ^{\{ 1 \}} | 0.13535 ^{\{ 12 \}} | 0.11105 ^{\{ 6 \}} | 0.11259 ^{\{ 7 \}} |
BIAS( \hat{\beta} ) | 0.50725 ^{\{ 5 \}} | 0.55301 ^{\{ 7 \}} | 0.6035 ^{\{ 12 \}} | 0.58515 ^{\{ 9 \}} | 0.5899 ^{\{ 11 \}} | 0.64435 ^{\{ 15 \}} | 0.60784 ^{\{ 14 \}} | 0.58714 ^{\{ 10 \}} | 0.23162 ^{\{ 2 \}} | 0.5467 ^{\{ 6 \}} | 0.60699 ^{\{ 13 \}} | 0.03925 ^{\{ 1 \}} | 0.49202 ^{\{ 4 \}} | 0.56928 ^{\{ 8 \}} | 0.4335 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.01774 ^{\{ 4 \}} | 0.02332 ^{\{ 9 \}} | 0.03276 ^{\{ 14 \}} | 0.01729 ^{\{ 3 \}} | 0.0283 ^{\{ 12 \}} | 0.04047 ^{\{ 15 \}} | 0.02684 ^{\{ 11 \}} | 0.0214 ^{\{ 7 \}} | 0.01089 ^{\{ 2 \}} | 0.01838 ^{\{ 5 \}} | 0.02516 ^{\{ 10 \}} | 0.00304 ^{\{ 1 \}} | 0.03231 ^{\{ 13 \}} | 0.02003 ^{\{ 6 \}} | 0.02258 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.41751 ^{\{ 5 \}} | 0.47226 ^{\{ 6 \}} | 0.53147 ^{\{ 10 \}} | 0.56899 ^{\{ 14 \}} | 0.50961 ^{\{ 8 \}} | 0.58852 ^{\{ 15 \}} | 0.55385 ^{\{ 12 \}} | 0.54235 ^{\{ 11 \}} | 0.18176 ^{\{ 2 \}} | 0.50391 ^{\{ 7 \}} | 0.56276 ^{\{ 13 \}} | 0.00761 ^{\{ 1 \}} | 0.36579 ^{\{ 4 \}} | 0.51548 ^{\{ 9 \}} | 0.28438 ^{\{ 3 \}} | |
MRE( \hat{\delta} ) | 0.21318 ^{\{ 3 \}} | 0.24344 ^{\{ 9 \}} | 0.28743 ^{\{ 14 \}} | 0.21432 ^{\{ 4 \}} | 0.27239 ^{\{ 13 \}} | 0.32609 ^{\{ 15 \}} | 0.26168 ^{\{ 11 \}} | 0.23398 ^{\{ 8 \}} | 0.14585 ^{\{ 2 \}} | 0.21505 ^{\{ 5 \}} | 0.25063 ^{\{ 10 \}} | 0.08496 ^{\{ 1 \}} | 0.27071 ^{\{ 12 \}} | 0.22211 ^{\{ 6 \}} | 0.22518 ^{\{ 7 \}} | |
MRE( \hat{\beta} ) | 0.25363 ^{\{ 5 \}} | 0.27651 ^{\{ 7 \}} | 0.30175 ^{\{ 12 \}} | 0.29257 ^{\{ 9 \}} | 0.29495 ^{\{ 11 \}} | 0.32217 ^{\{ 15 \}} | 0.30392 ^{\{ 14 \}} | 0.29357 ^{\{ 10 \}} | 0.11581 ^{\{ 2 \}} | 0.27335 ^{\{ 6 \}} | 0.30349 ^{\{ 13 \}} | 0.01962 ^{\{ 1 \}} | 0.24601 ^{\{ 4 \}} | 0.28464 ^{\{ 8 \}} | 0.21675 ^{\{ 3 \}} | |
D_{abs} | 0.02879 ^{\{ 1 \}} | 0.02974 ^{\{ 2 \}} | 0.03169 ^{\{ 9 \}} | 0.03041 ^{\{ 4 \}} | 0.03128 ^{\{ 7 \}} | 0.03293 ^{\{ 13 \}} | 0.03119 ^{\{ 6 \}} | 0.03093 ^{\{ 5 \}} | 0.03153 ^{\{ 8 \}} | 0.03238 ^{\{ 12 \}} | 0.03207 ^{\{ 10 \}} | 0.03029 ^{\{ 3 \}} | 0.04427 ^{\{ 15 \}} | 0.03231 ^{\{ 11 \}} | 0.04117 ^{\{ 14 \}} | |
D_{max} | 0.04743 ^{\{ 2 \}} | 0.04934 ^{\{ 4 \}} | 0.05375 ^{\{ 12 \}} | 0.0493 ^{\{ 3 \}} | 0.05256 ^{\{ 10 \}} | 0.0566 ^{\{ 13 \}} | 0.05167 ^{\{ 7 \}} | 0.05037 ^{\{ 6 \}} | 0.04949 ^{\{ 5 \}} | 0.05255 ^{\{ 9 \}} | 0.05272 ^{\{ 11 \}} | 0.04604 ^{\{ 1 \}} | 0.06992 ^{\{ 15 \}} | 0.05183 ^{\{ 8 \}} | 0.06489 ^{\{ 14 \}} | |
\sum Ranks | 28 ^{\{ 3 \}} | 53 ^{\{ 5 \}} | 97 ^{\{ 14 \}} | 50 ^{\{ 4 \}} | 85 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 86 ^{\{ 12 \}} | 65 ^{\{ 9 \}} | 25 ^{\{ 2 \}} | 55 ^{\{ 6 \}} | 90 ^{\{ 13 \}} | 10 ^{\{ 1 \}} | 79 ^{\{ 10 \}} | 62 ^{\{ 8 \}} | 59 ^{\{ 7 \}} | |
100 | BIAS( \hat{\delta} ) | 0.08286 ^{\{ 3 \}} | 0.10295 ^{\{ 11 \}} | 0.11725 ^{\{ 14 \}} | 0.08874 ^{\{ 4 \}} | 0.11484 ^{\{ 13 \}} | 0.14099 ^{\{ 15 \}} | 0.10506 ^{\{ 12 \}} | 0.09668 ^{\{ 9 \}} | 0.06209 ^{\{ 2 \}} | 0.09521 ^{\{ 8 \}} | 0.10061 ^{\{ 10 \}} | 0.0325 ^{\{ 1 \}} | 0.09485 ^{\{ 7 \}} | 0.09062 ^{\{ 6 \}} | 0.09056 ^{\{ 5 \}} |
BIAS( \hat{\beta} ) | 0.41544 ^{\{ 5 \}} | 0.51195 ^{\{ 9 \}} | 0.51848 ^{\{ 11 \}} | 0.50035 ^{\{ 7 \}} | 0.55605 ^{\{ 14 \}} | 0.57985 ^{\{ 15 \}} | 0.51445 ^{\{ 10 \}} | 0.48798 ^{\{ 6 \}} | 0.22274 ^{\{ 2 \}} | 0.52223 ^{\{ 12 \}} | 0.52272 ^{\{ 13 \}} | 0.03493 ^{\{ 1 \}} | 0.36852 ^{\{ 3 \}} | 0.5066 ^{\{ 8 \}} | 0.392 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.01073 ^{\{ 3 \}} | 0.01683 ^{\{ 10 \}} | 0.02226 ^{\{ 14 \}} | 0.01217 ^{\{ 4 \}} | 0.02079 ^{\{ 13 \}} | 0.03092 ^{\{ 15 \}} | 0.01737 ^{\{ 12 \}} | 0.01473 ^{\{ 7 \}} | 0.00763 ^{\{ 2 \}} | 0.01427 ^{\{ 6 \}} | 0.01587 ^{\{ 9 \}} | 0.00182 ^{\{ 1 \}} | 0.0169 ^{\{ 11 \}} | 0.0124 ^{\{ 5 \}} | 0.01525 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.29348 ^{\{ 5 \}} | 0.42649 ^{\{ 9 \}} | 0.43143 ^{\{ 11 \}} | 0.42826 ^{\{ 10 \}} | 0.49967 ^{\{ 14 \}} | 0.51432 ^{\{ 15 \}} | 0.42161 ^{\{ 8 \}} | 0.38818 ^{\{ 6 \}} | 0.15605 ^{\{ 2 \}} | 0.47725 ^{\{ 13 \}} | 0.43798 ^{\{ 12 \}} | 0.0066 ^{\{ 1 \}} | 0.21342 ^{\{ 3 \}} | 0.41474 ^{\{ 7 \}} | 0.23478 ^{\{ 4 \}} | |
MRE( \hat{\delta} ) | 0.16573 ^{\{ 3 \}} | 0.2059 ^{\{ 11 \}} | 0.23451 ^{\{ 14 \}} | 0.17748 ^{\{ 4 \}} | 0.22968 ^{\{ 13 \}} | 0.28198 ^{\{ 15 \}} | 0.21011 ^{\{ 12 \}} | 0.19336 ^{\{ 9 \}} | 0.12419 ^{\{ 2 \}} | 0.19042 ^{\{ 8 \}} | 0.20121 ^{\{ 10 \}} | 0.065 ^{\{ 1 \}} | 0.18971 ^{\{ 7 \}} | 0.18125 ^{\{ 6 \}} | 0.18112 ^{\{ 5 \}} | |
MRE( \hat{\beta} ) | 0.20772 ^{\{ 5 \}} | 0.25598 ^{\{ 9 \}} | 0.25924 ^{\{ 11 \}} | 0.25018 ^{\{ 7 \}} | 0.27802 ^{\{ 14 \}} | 0.28993 ^{\{ 15 \}} | 0.25722 ^{\{ 10 \}} | 0.24399 ^{\{ 6 \}} | 0.11137 ^{\{ 2 \}} | 0.26111 ^{\{ 12 \}} | 0.26136 ^{\{ 13 \}} | 0.01746 ^{\{ 1 \}} | 0.18426 ^{\{ 3 \}} | 0.2533 ^{\{ 8 \}} | 0.196 ^{\{ 4 \}} | |
D_{abs} | 0.02231 ^{\{ 1 \}} | 0.02454 ^{\{ 6 \}} | 0.02511 ^{\{ 9 \}} | 0.02269 ^{\{ 3 \}} | 0.02562 ^{\{ 10 \}} | 0.02595 ^{\{ 12 \}} | 0.02449 ^{\{ 5 \}} | 0.02497 ^{\{ 7 \}} | 0.02396 ^{\{ 4 \}} | 0.02574 ^{\{ 11 \}} | 0.02509 ^{\{ 8 \}} | 0.02261 ^{\{ 2 \}} | 0.03104 ^{\{ 14 \}} | 0.02611 ^{\{ 13 \}} | 0.03167 ^{\{ 15 \}} | |
D_{max} | 0.03664 ^{\{ 2 \}} | 0.04054 ^{\{ 5.5 \}} | 0.0426 ^{\{ 11 \}} | 0.03686 ^{\{ 3 \}} | 0.04306 ^{\{ 12 \}} | 0.04533 ^{\{ 13 \}} | 0.04054 ^{\{ 5.5 \}} | 0.04099 ^{\{ 7 \}} | 0.03813 ^{\{ 4 \}} | 0.04213 ^{\{ 10 \}} | 0.04108 ^{\{ 8 \}} | 0.03463 ^{\{ 1 \}} | 0.0495 ^{\{ 14 \}} | 0.04206 ^{\{ 9 \}} | 0.05085 ^{\{ 15 \}} | |
\sum Ranks | 27 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 95 ^{\{ 13 \}} | 42 ^{\{ 4 \}} | 103 ^{\{ 14 \}} | 115 ^{\{ 15 \}} | 74.5 ^{\{ 10 \}} | 57 ^{\{ 5 \}} | 20 ^{\{ 2 \}} | 80 ^{\{ 11 \}} | 83 ^{\{ 12 \}} | 9 ^{\{ 1 \}} | 62 ^{\{ 7.5 \}} | 62 ^{\{ 7.5 \}} | 60 ^{\{ 6 \}} | |
200 | BIAS( \hat{\delta} ) | 0.06116 ^{\{ 3 \}} | 0.07841 ^{\{ 11 \}} | 0.08834 ^{\{ 13 \}} | 0.06582 ^{\{ 5 \}} | 0.08967 ^{\{ 14 \}} | 0.10318 ^{\{ 15 \}} | 0.07657 ^{\{ 10 \}} | 0.0701 ^{\{ 7 \}} | 0.05022 ^{\{ 2 \}} | 0.07169 ^{\{ 8 \}} | 0.0803 ^{\{ 12 \}} | 0.02285 ^{\{ 1 \}} | 0.06434 ^{\{ 4 \}} | 0.07255 ^{\{ 9 \}} | 0.06584 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.32425 ^{\{ 5 \}} | 0.40338 ^{\{ 10 \}} | 0.42002 ^{\{ 12 \}} | 0.37519 ^{\{ 7 \}} | 0.44639 ^{\{ 13 \}} | 0.47646 ^{\{ 15 \}} | 0.38899 ^{\{ 8 \}} | 0.36465 ^{\{ 6 \}} | 0.20622 ^{\{ 2 \}} | 0.39983 ^{\{ 9 \}} | 0.45097 ^{\{ 14 \}} | 0.02978 ^{\{ 1 \}} | 0.309 ^{\{ 3 \}} | 0.40885 ^{\{ 11 \}} | 0.32061 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.00603 ^{\{ 3 \}} | 0.0094 ^{\{ 11 \}} | 0.01227 ^{\{ 14 \}} | 0.00684 ^{\{ 4 \}} | 0.0121 ^{\{ 13 \}} | 0.01636 ^{\{ 15 \}} | 0.00916 ^{\{ 10 \}} | 0.00762 ^{\{ 6 \}} | 0.00531 ^{\{ 2 \}} | 0.00804 ^{\{ 8 \}} | 0.0099 ^{\{ 12 \}} | 0.00084 ^{\{ 1 \}} | 0.00725 ^{\{ 5 \}} | 0.00822 ^{\{ 9 \}} | 0.00789 ^{\{ 7 \}} | |
MSE( \hat{\beta} ) | 0.18213 ^{\{ 4 \}} | 0.27797 ^{\{ 9 \}} | 0.29886 ^{\{ 12 \}} | 0.25897 ^{\{ 8 \}} | 0.34454 ^{\{ 14 \}} | 0.37648 ^{\{ 15 \}} | 0.2585 ^{\{ 7 \}} | 0.23044 ^{\{ 6 \}} | 0.13188 ^{\{ 2 \}} | 0.2854 ^{\{ 10 \}} | 0.34418 ^{\{ 13 \}} | 0.00461 ^{\{ 1 \}} | 0.1536 ^{\{ 3 \}} | 0.29853 ^{\{ 11 \}} | 0.18333 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.12232 ^{\{ 3 \}} | 0.15682 ^{\{ 11 \}} | 0.17668 ^{\{ 13 \}} | 0.13164 ^{\{ 5 \}} | 0.17934 ^{\{ 14 \}} | 0.20636 ^{\{ 15 \}} | 0.15314 ^{\{ 10 \}} | 0.1402 ^{\{ 7 \}} | 0.10044 ^{\{ 2 \}} | 0.14338 ^{\{ 8 \}} | 0.16059 ^{\{ 12 \}} | 0.0457 ^{\{ 1 \}} | 0.12868 ^{\{ 4 \}} | 0.14509 ^{\{ 9 \}} | 0.13169 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.16212 ^{\{ 5 \}} | 0.20169 ^{\{ 10 \}} | 0.21001 ^{\{ 12 \}} | 0.18759 ^{\{ 7 \}} | 0.2232 ^{\{ 13 \}} | 0.23823 ^{\{ 15 \}} | 0.1945 ^{\{ 8 \}} | 0.18232 ^{\{ 6 \}} | 0.10311 ^{\{ 2 \}} | 0.19991 ^{\{ 9 \}} | 0.22548 ^{\{ 14 \}} | 0.01489 ^{\{ 1 \}} | 0.1545 ^{\{ 3 \}} | 0.20442 ^{\{ 11 \}} | 0.16031 ^{\{ 4 \}} | |
D_{abs} | 0.01724 ^{\{ 2 \}} | 0.01806 ^{\{ 7 \}} | 0.01844 ^{\{ 10 \}} | 0.01741 ^{\{ 3 \}} | 0.01838 ^{\{ 8 \}} | 0.01843 ^{\{ 9 \}} | 0.01779 ^{\{ 4 \}} | 0.01797 ^{\{ 6 \}} | 0.01789 ^{\{ 5 \}} | 0.02027 ^{\{ 13 \}} | 0.01959 ^{\{ 12 \}} | 0.01648 ^{\{ 1 \}} | 0.02114 ^{\{ 14 \}} | 0.0194 ^{\{ 11 \}} | 0.02249 ^{\{ 15 \}} | |
D_{max} | 0.02791 ^{\{ 2 \}} | 0.0299 ^{\{ 7 \}} | 0.03118 ^{\{ 9 \}} | 0.02827 ^{\{ 3 \}} | 0.03116 ^{\{ 8 \}} | 0.03221 ^{\{ 12 \}} | 0.02954 ^{\{ 6 \}} | 0.0293 ^{\{ 5 \}} | 0.02849 ^{\{ 4 \}} | 0.03291 ^{\{ 13 \}} | 0.03189 ^{\{ 11 \}} | 0.02515 ^{\{ 1 \}} | 0.03433 ^{\{ 14 \}} | 0.03138 ^{\{ 10 \}} | 0.03655 ^{\{ 15 \}} | |
\sum Ranks | 27 ^{\{ 3 \}} | 76 ^{\{ 9 \}} | 95 ^{\{ 12 \}} | 42 ^{\{ 4 \}} | 97 ^{\{ 13 \}} | 111 ^{\{ 15 \}} | 63 ^{\{ 8 \}} | 49 ^{\{ 5 \}} | 21 ^{\{ 2 \}} | 78 ^{\{ 10 \}} | 100 ^{\{ 14 \}} | 8 ^{\{ 1 \}} | 50 ^{\{ 6 \}} | 81 ^{\{ 11 \}} | 62 ^{\{ 7 \}} | |
300 | BIAS( \hat{\delta} ) | 0.04976 ^{\{ 3 \}} | 0.063 ^{\{ 10 \}} | 0.07722 ^{\{ 14 \}} | 0.05537 ^{\{ 5 \}} | 0.07222 ^{\{ 13 \}} | 0.08992 ^{\{ 15 \}} | 0.06346 ^{\{ 11 \}} | 0.05881 ^{\{ 7 \}} | 0.04405 ^{\{ 2 \}} | 0.06157 ^{\{ 9 \}} | 0.06661 ^{\{ 12 \}} | 0.01923 ^{\{ 1 \}} | 0.05374 ^{\{ 4 \}} | 0.05954 ^{\{ 8 \}} | 0.05577 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.26113 ^{\{ 3 \}} | 0.32348 ^{\{ 9 \}} | 0.37643 ^{\{ 14 \}} | 0.30537 ^{\{ 6 \}} | 0.36663 ^{\{ 12 \}} | 0.42261 ^{\{ 15 \}} | 0.32101 ^{\{ 8 \}} | 0.30553 ^{\{ 7 \}} | 0.18919 ^{\{ 2 \}} | 0.33927 ^{\{ 11 \}} | 0.37228 ^{\{ 13 \}} | 0.02946 ^{\{ 1 \}} | 0.27025 ^{\{ 5 \}} | 0.32572 ^{\{ 10 \}} | 0.2684 ^{\{ 4 \}} | |
MSE( \hat{\delta} ) | 0.00405 ^{\{ 3 \}} | 0.00614 ^{\{ 10 \}} | 0.00941 ^{\{ 14 \}} | 0.00478 ^{\{ 4 \}} | 0.00824 ^{\{ 13 \}} | 0.01248 ^{\{ 15 \}} | 0.0063 ^{\{ 11 \}} | 0.00566 ^{\{ 7 \}} | 0.00375 ^{\{ 2 \}} | 0.00597 ^{\{ 9 \}} | 0.00674 ^{\{ 12 \}} | 0.00063 ^{\{ 1 \}} | 0.00503 ^{\{ 5 \}} | 0.00548 ^{\{ 6 \}} | 0.00588 ^{\{ 8 \}} | |
MSE( \hat{\beta} ) | 0.12154 ^{\{ 3 \}} | 0.17435 ^{\{ 8 \}} | 0.24526 ^{\{ 14 \}} | 0.16177 ^{\{ 7 \}} | 0.2386 ^{\{ 13 \}} | 0.30176 ^{\{ 15 \}} | 0.17584 ^{\{ 9 \}} | 0.15982 ^{\{ 6 \}} | 0.10377 ^{\{ 2 \}} | 0.20948 ^{\{ 11 \}} | 0.23664 ^{\{ 12 \}} | 0.00414 ^{\{ 1 \}} | 0.13235 ^{\{ 4 \}} | 0.18403 ^{\{ 10 \}} | 0.14778 ^{\{ 5 \}} | |
MRE( \hat{\delta} ) | 0.09951 ^{\{ 3 \}} | 0.12601 ^{\{ 10 \}} | 0.15444 ^{\{ 14 \}} | 0.11075 ^{\{ 5 \}} | 0.14445 ^{\{ 13 \}} | 0.17985 ^{\{ 15 \}} | 0.12692 ^{\{ 11 \}} | 0.11761 ^{\{ 7 \}} | 0.0881 ^{\{ 2 \}} | 0.12314 ^{\{ 9 \}} | 0.13323 ^{\{ 12 \}} | 0.03846 ^{\{ 1 \}} | 0.10748 ^{\{ 4 \}} | 0.11909 ^{\{ 8 \}} | 0.11153 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.13057 ^{\{ 3 \}} | 0.16174 ^{\{ 9 \}} | 0.18822 ^{\{ 14 \}} | 0.15269 ^{\{ 6 \}} | 0.18332 ^{\{ 12 \}} | 0.2113 ^{\{ 15 \}} | 0.1605 ^{\{ 8 \}} | 0.15276 ^{\{ 7 \}} | 0.09459 ^{\{ 2 \}} | 0.16963 ^{\{ 11 \}} | 0.18614 ^{\{ 13 \}} | 0.01473 ^{\{ 1 \}} | 0.13513 ^{\{ 5 \}} | 0.16286 ^{\{ 10 \}} | 0.1342 ^{\{ 4 \}} | |
D_{abs} | 0.01433 ^{\{ 2 \}} | 0.01475 ^{\{ 5.5 \}} | 0.0156 ^{\{ 10 \}} | 0.01439 ^{\{ 3 \}} | 0.01527 ^{\{ 7 \}} | 0.01648 ^{\{ 12 \}} | 0.01475 ^{\{ 5.5 \}} | 0.01471 ^{\{ 4 \}} | 0.01548 ^{\{ 8 \}} | 0.01669 ^{\{ 13 \}} | 0.01557 ^{\{ 9 \}} | 0.01377 ^{\{ 1 \}} | 0.01746 ^{\{ 14 \}} | 0.01569 ^{\{ 11 \}} | 0.01762 ^{\{ 15 \}} | |
D_{max} | 0.02313 ^{\{ 2 \}} | 0.02441 ^{\{ 5 \}} | 0.02642 ^{\{ 11 \}} | 0.0234 ^{\{ 3 \}} | 0.02566 ^{\{ 10 \}} | 0.02851 ^{\{ 14 \}} | 0.02444 ^{\{ 6 \}} | 0.02405 ^{\{ 4 \}} | 0.02467 ^{\{ 7 \}} | 0.02709 ^{\{ 12 \}} | 0.02545 ^{\{ 8 \}} | 0.02099 ^{\{ 1 \}} | 0.02842 ^{\{ 13 \}} | 0.02547 ^{\{ 9 \}} | 0.02869 ^{\{ 15 \}} | |
\sum Ranks | 22 ^{\{ 2 \}} | 66.5 ^{\{ 8 \}} | 105 ^{\{ 14 \}} | 39 ^{\{ 4 \}} | 93 ^{\{ 13 \}} | 116 ^{\{ 15 \}} | 69.5 ^{\{ 9 \}} | 49 ^{\{ 5 \}} | 27 ^{\{ 3 \}} | 85 ^{\{ 11 \}} | 91 ^{\{ 12 \}} | 8 ^{\{ 1 \}} | 54 ^{\{ 6 \}} | 72 ^{\{ 10 \}} | 63 ^{\{ 7 \}} | |
400 | BIAS( \hat{\delta} ) | 0.04586 ^{\{ 3 \}} | 0.05457 ^{\{ 10 \}} | 0.06718 ^{\{ 14 \}} | 0.04629 ^{\{ 4 \}} | 0.06491 ^{\{ 13 \}} | 0.07564 ^{\{ 15 \}} | 0.05516 ^{\{ 11 \}} | 0.05045 ^{\{ 8 \}} | 0.04056 ^{\{ 2 \}} | 0.05384 ^{\{ 9 \}} | 0.05857 ^{\{ 12 \}} | 0.01681 ^{\{ 1 \}} | 0.04911 ^{\{ 5 \}} | 0.0504 ^{\{ 7 \}} | 0.04983 ^{\{ 6 \}} |
BIAS( \hat{\beta} ) | 0.24121 ^{\{ 4 \}} | 0.2717 ^{\{ 9 \}} | 0.32822 ^{\{ 14 \}} | 0.25116 ^{\{ 6 \}} | 0.31726 ^{\{ 12 \}} | 0.35759 ^{\{ 15 \}} | 0.2792 ^{\{ 10 \}} | 0.25723 ^{\{ 7 \}} | 0.17791 ^{\{ 2 \}} | 0.29726 ^{\{ 11 \}} | 0.3189 ^{\{ 13 \}} | 0.02741 ^{\{ 1 \}} | 0.2501 ^{\{ 5 \}} | 0.27113 ^{\{ 8 \}} | 0.23703 ^{\{ 3 \}} | |
MSE( \hat{\delta} ) | 0.00335 ^{\{ 4 \}} | 0.00479 ^{\{ 10 \}} | 0.00713 ^{\{ 14 \}} | 0.00334 ^{\{ 3 \}} | 0.00654 ^{\{ 13 \}} | 0.00909 ^{\{ 15 \}} | 0.0047 ^{\{ 9 \}} | 0.00393 ^{\{ 5 \}} | 0.00319 ^{\{ 2 \}} | 0.00461 ^{\{ 8 \}} | 0.00534 ^{\{ 12 \}} | 0.00051 ^{\{ 1 \}} | 0.00433 ^{\{ 7 \}} | 0.00397 ^{\{ 6 \}} | 0.00484 ^{\{ 11 \}} | |
MSE( \hat{\beta} ) | 0.09651 ^{\{ 3 \}} | 0.12678 ^{\{ 8 \}} | 0.18883 ^{\{ 14 \}} | 0.10551 ^{\{ 4 \}} | 0.17475 ^{\{ 13 \}} | 0.22525 ^{\{ 15 \}} | 0.12978 ^{\{ 10 \}} | 0.11072 ^{\{ 5 \}} | 0.08312 ^{\{ 2 \}} | 0.15568 ^{\{ 11 \}} | 0.17447 ^{\{ 12 \}} | 0.00395 ^{\{ 1 \}} | 0.12704 ^{\{ 9 \}} | 0.12604 ^{\{ 7 \}} | 0.11073 ^{\{ 6 \}} | |
MRE( \hat{\delta} ) | 0.09172 ^{\{ 3 \}} | 0.10915 ^{\{ 10 \}} | 0.13437 ^{\{ 14 \}} | 0.09257 ^{\{ 4 \}} | 0.12982 ^{\{ 13 \}} | 0.15127 ^{\{ 15 \}} | 0.11032 ^{\{ 11 \}} | 0.10089 ^{\{ 8 \}} | 0.08112 ^{\{ 2 \}} | 0.10768 ^{\{ 9 \}} | 0.11714 ^{\{ 12 \}} | 0.03363 ^{\{ 1 \}} | 0.09822 ^{\{ 5 \}} | 0.10079 ^{\{ 7 \}} | 0.09966 ^{\{ 6 \}} | |
MRE( \hat{\beta} ) | 0.12061 ^{\{ 4 \}} | 0.13585 ^{\{ 9 \}} | 0.16411 ^{\{ 14 \}} | 0.12558 ^{\{ 6 \}} | 0.15863 ^{\{ 12 \}} | 0.1788 ^{\{ 15 \}} | 0.1396 ^{\{ 10 \}} | 0.12862 ^{\{ 7 \}} | 0.08896 ^{\{ 2 \}} | 0.14863 ^{\{ 11 \}} | 0.15945 ^{\{ 13 \}} | 0.01371 ^{\{ 1 \}} | 0.12505 ^{\{ 5 \}} | 0.13556 ^{\{ 8 \}} | 0.11852 ^{\{ 3 \}} | |
D_{abs} | 0.01235 ^{\{ 3 \}} | 0.01251 ^{\{ 4 \}} | 0.01316 ^{\{ 7 \}} | 0.01191 ^{\{ 2 \}} | 0.0133 ^{\{ 10 \}} | 0.01357 ^{\{ 11 \}} | 0.01283 ^{\{ 6 \}} | 0.01262 ^{\{ 5 \}} | 0.01327 ^{\{ 9 \}} | 0.01469 ^{\{ 13 \}} | 0.01384 ^{\{ 12 \}} | 0.0118 ^{\{ 1 \}} | 0.01612 ^{\{ 15 \}} | 0.01319 ^{\{ 8 \}} | 0.01595 ^{\{ 14 \}} | |
D_{max} | 0.02003 ^{\{ 3 \}} | 0.02068 ^{\{ 5 \}} | 0.02249 ^{\{ 10 \}} | 0.0194 ^{\{ 2 \}} | 0.02245 ^{\{ 9 \}} | 0.02345 ^{\{ 12 \}} | 0.0212 ^{\{ 6 \}} | 0.02064 ^{\{ 4 \}} | 0.02127 ^{\{ 7 \}} | 0.02382 ^{\{ 13 \}} | 0.02263 ^{\{ 11 \}} | 0.01806 ^{\{ 1 \}} | 0.02613 ^{\{ 15 \}} | 0.0215 ^{\{ 8 \}} | 0.02592 ^{\{ 14 \}} | |
\sum Ranks | 27 ^{\{ 2 \}} | 65 ^{\{ 8 \}} | 101 ^{\{ 14 \}} | 31 ^{\{ 4 \}} | 95 ^{\{ 12 \}} | 113 ^{\{ 15 \}} | 73 ^{\{ 10 \}} | 49 ^{\{ 5 \}} | 28 ^{\{ 3 \}} | 85 ^{\{ 11 \}} | 97 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 66 ^{\{ 9 \}} | 59 ^{\{ 6 \}} | 63 ^{\{ 7 \}} |
n | Est. | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
30 | BIAS( \hat{\delta} ) | 0.24947 ^{\{ 2 \}} | 0.39247 ^{\{ 8 \}} | 0.45109 ^{\{ 13 \}} | 0.34898 ^{\{ 4 \}} | 0.4092 ^{\{ 10 \}} | 0.51093 ^{\{ 15 \}} | 0.3875 ^{\{ 7 \}} | 0.40032 ^{\{ 9 \}} | 0.2568 ^{\{ 3 \}} | 0.36632 ^{\{ 6 \}} | 0.47352 ^{\{ 14 \}} | 0.0773 ^{\{ 1 \}} | 0.42866 ^{\{ 12 \}} | 0.3597 ^{\{ 5 \}} | 0.40978 ^{\{ 11 \}} |
BIAS( \hat{\beta} ) | 0.10849 ^{\{ 3 \}} | 0.12312 ^{\{ 5 \}} | 0.12419 ^{\{ 6 \}} | 0.12427 ^{\{ 7 \}} | 0.12917 ^{\{ 11 \}} | 0.13824 ^{\{ 12 \}} | 0.12135 ^{\{ 4 \}} | 0.12589 ^{\{ 8 \}} | 0.10086 ^{\{ 2 \}} | 0.12906 ^{\{ 10 \}} | 0.14169 ^{\{ 15 \}} | 0.05665 ^{\{ 1 \}} | 0.14068 ^{\{ 13 \}} | 0.12644 ^{\{ 9 \}} | 0.14073 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.1048 ^{\{ 2 \}} | 0.25062 ^{\{ 8 \}} | 0.35457 ^{\{ 13 \}} | 0.19139 ^{\{ 4 \}} | 0.26462 ^{\{ 10 \}} | 0.44465 ^{\{ 15 \}} | 0.24711 ^{\{ 7 \}} | 0.26728 ^{\{ 11 \}} | 0.13483 ^{\{ 3 \}} | 0.20392 ^{\{ 6 \}} | 0.39226 ^{\{ 14 \}} | 0.02074 ^{\{ 1 \}} | 0.27898 ^{\{ 12 \}} | 0.19815 ^{\{ 5 \}} | 0.25293 ^{\{ 9 \}} | |
MSE( \hat{\beta} ) | 0.02079 ^{\{ 3 \}} | 0.02301 ^{\{ 6 \}} | 0.02292 ^{\{ 5 \}} | 0.02435 ^{\{ 8 \}} | 0.0252 ^{\{ 10 \}} | 0.02791 ^{\{ 12 \}} | 0.02257 ^{\{ 4 \}} | 0.02416 ^{\{ 7 \}} | 0.01873 ^{\{ 2 \}} | 0.02604 ^{\{ 11 \}} | 0.02917 ^{\{ 13 \}} | 0.00597 ^{\{ 1 \}} | 0.03018 ^{\{ 15 \}} | 0.02461 ^{\{ 9 \}} | 0.02959 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.09979 ^{\{ 2 \}} | 0.15699 ^{\{ 8 \}} | 0.18043 ^{\{ 13 \}} | 0.13959 ^{\{ 4 \}} | 0.16368 ^{\{ 10 \}} | 0.20437 ^{\{ 15 \}} | 0.155 ^{\{ 7 \}} | 0.16013 ^{\{ 9 \}} | 0.10272 ^{\{ 3 \}} | 0.14653 ^{\{ 6 \}} | 0.18941 ^{\{ 14 \}} | 0.03092 ^{\{ 1 \}} | 0.17146 ^{\{ 12 \}} | 0.14388 ^{\{ 5 \}} | 0.16391 ^{\{ 11 \}} | |
MRE( \hat{\beta} ) | 0.27123 ^{\{ 3 \}} | 0.3078 ^{\{ 5 \}} | 0.31047 ^{\{ 6 \}} | 0.31068 ^{\{ 7 \}} | 0.32292 ^{\{ 11 \}} | 0.34559 ^{\{ 12 \}} | 0.30337 ^{\{ 4 \}} | 0.31471 ^{\{ 8 \}} | 0.25215 ^{\{ 2 \}} | 0.32265 ^{\{ 10 \}} | 0.35423 ^{\{ 15 \}} | 0.14162 ^{\{ 1 \}} | 0.3517 ^{\{ 13 \}} | 0.31611 ^{\{ 9 \}} | 0.35182 ^{\{ 14 \}} | |
D_{abs} | 0.04159 ^{\{ 1 \}} | 0.0451 ^{\{ 3 \}} | 0.0482 ^{\{ 9 \}} | 0.0454 ^{\{ 5 \}} | 0.04619 ^{\{ 6 \}} | 0.0507 ^{\{ 13 \}} | 0.04517 ^{\{ 4 \}} | 0.04721 ^{\{ 8 \}} | 0.04986 ^{\{ 11.5 \}} | 0.04986 ^{\{ 11.5 \}} | 0.04865 ^{\{ 10 \}} | 0.04264 ^{\{ 2 \}} | 0.05413 ^{\{ 15 \}} | 0.0466 ^{\{ 7 \}} | 0.05336 ^{\{ 14 \}} | |
D_{max} | 0.066 ^{\{ 2 \}} | 0.07344 ^{\{ 4 \}} | 0.08102 ^{\{ 12 \}} | 0.07193 ^{\{ 3 \}} | 0.076 ^{\{ 7 \}} | 0.08624 ^{\{ 14 \}} | 0.07378 ^{\{ 5 \}} | 0.07752 ^{\{ 9 \}} | 0.07682 ^{\{ 8 \}} | 0.07921 ^{\{ 10 \}} | 0.08099 ^{\{ 11 \}} | 0.06259 ^{\{ 1 \}} | 0.08762 ^{\{ 15 \}} | 0.07444 ^{\{ 6 \}} | 0.0855 ^{\{ 13 \}} | |
\sum Ranks | 18 ^{\{ 2 \}} | 47 ^{\{ 6 \}} | 77 ^{\{ 11 \}} | 42 ^{\{ 4.5 \}} | 75 ^{\{ 10 \}} | 108 ^{\{ 15 \}} | 42 ^{\{ 4.5 \}} | 69 ^{\{ 8 \}} | 34.5 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 106 ^{\{ 13 \}} | 9 ^{\{ 1 \}} | 107 ^{\{ 14 \}} | 55 ^{\{ 7 \}} | 100 ^{\{ 12 \}} | |
60 | BIAS( \hat{\delta} ) | 0.20453 ^{\{ 2 \}} | 0.2779 ^{\{ 6 \}} | 0.29563 ^{\{ 10 \}} | 0.27622 ^{\{ 5 \}} | 0.31346 ^{\{ 11 \}} | 0.37466 ^{\{ 15 \}} | 0.28173 ^{\{ 7 \}} | 0.28733 ^{\{ 9 \}} | 0.24988 ^{\{ 3 \}} | 0.28363 ^{\{ 8 \}} | 0.35262 ^{\{ 14 \}} | 0.07199 ^{\{ 1 \}} | 0.33443 ^{\{ 13 \}} | 0.27509 ^{\{ 4 \}} | 0.32222 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.08487 ^{\{ 2 \}} | 0.09474 ^{\{ 5 \}} | 0.09425 ^{\{ 3 \}} | 0.10053 ^{\{ 9 \}} | 0.09961 ^{\{ 8 \}} | 0.10752 ^{\{ 12 \}} | 0.09656 ^{\{ 7 \}} | 0.09498 ^{\{ 6 \}} | 0.09445 ^{\{ 4 \}} | 0.10474 ^{\{ 11 \}} | 0.11801 ^{\{ 13 \}} | 0.04214 ^{\{ 1 \}} | 0.11894 ^{\{ 14 \}} | 0.10253 ^{\{ 10 \}} | 0.1193 ^{\{ 15 \}} | |
MSE( \hat{\delta} ) | 0.07403 ^{\{ 2 \}} | 0.12349 ^{\{ 6 \}} | 0.14155 ^{\{ 10 \}} | 0.11748 ^{\{ 4 \}} | 0.15224 ^{\{ 11 \}} | 0.22331 ^{\{ 15 \}} | 0.12956 ^{\{ 8 \}} | 0.13518 ^{\{ 9 \}} | 0.11795 ^{\{ 5 \}} | 0.1257 ^{\{ 7 \}} | 0.20342 ^{\{ 14 \}} | 0.01871 ^{\{ 1 \}} | 0.17116 ^{\{ 13 \}} | 0.1157 ^{\{ 3 \}} | 0.15442 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.01313 ^{\{ 2 \}} | 0.01465 ^{\{ 5 \}} | 0.0141 ^{\{ 3 \}} | 0.01638 ^{\{ 9 \}} | 0.01589 ^{\{ 8 \}} | 0.01792 ^{\{ 11 \}} | 0.01554 ^{\{ 7 \}} | 0.01446 ^{\{ 4 \}} | 0.01553 ^{\{ 6 \}} | 0.01793 ^{\{ 12 \}} | 0.0212 ^{\{ 13 \}} | 0.00382 ^{\{ 1 \}} | 0.02299 ^{\{ 15 \}} | 0.01724 ^{\{ 10 \}} | 0.02298 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.08181 ^{\{ 2 \}} | 0.11116 ^{\{ 6 \}} | 0.11825 ^{\{ 10 \}} | 0.11049 ^{\{ 5 \}} | 0.12538 ^{\{ 11 \}} | 0.14986 ^{\{ 15 \}} | 0.11269 ^{\{ 7 \}} | 0.11493 ^{\{ 9 \}} | 0.09995 ^{\{ 3 \}} | 0.11345 ^{\{ 8 \}} | 0.14105 ^{\{ 14 \}} | 0.0288 ^{\{ 1 \}} | 0.13377 ^{\{ 13 \}} | 0.11003 ^{\{ 4 \}} | 0.12889 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.21217 ^{\{ 2 \}} | 0.23684 ^{\{ 5 \}} | 0.23563 ^{\{ 3 \}} | 0.25133 ^{\{ 9 \}} | 0.24903 ^{\{ 8 \}} | 0.26879 ^{\{ 12 \}} | 0.24139 ^{\{ 7 \}} | 0.23744 ^{\{ 6 \}} | 0.23612 ^{\{ 4 \}} | 0.26185 ^{\{ 11 \}} | 0.29503 ^{\{ 13 \}} | 0.10535 ^{\{ 1 \}} | 0.29735 ^{\{ 14 \}} | 0.25632 ^{\{ 10 \}} | 0.29825 ^{\{ 15 \}} | |
D_{abs} | 0.03306 ^{\{ 3 \}} | 0.0336 ^{\{ 6 \}} | 0.03402 ^{\{ 7 \}} | 0.03258 ^{\{ 2 \}} | 0.03344 ^{\{ 5 \}} | 0.03621 ^{\{ 9 \}} | 0.03435 ^{\{ 8 \}} | 0.03312 ^{\{ 4 \}} | 0.03754 ^{\{ 12 \}} | 0.03649 ^{\{ 10 \}} | 0.03851 ^{\{ 13 \}} | 0.02949 ^{\{ 1 \}} | 0.03966 ^{\{ 14 \}} | 0.0365 ^{\{ 11 \}} | 0.04037 ^{\{ 15 \}} | |
D_{max} | 0.05248 ^{\{ 2 \}} | 0.05501 ^{\{ 5 \}} | 0.05639 ^{\{ 8 \}} | 0.05259 ^{\{ 3 \}} | 0.05553 ^{\{ 6 \}} | 0.06191 ^{\{ 12 \}} | 0.05603 ^{\{ 7 \}} | 0.05434 ^{\{ 4 \}} | 0.05941 ^{\{ 11 \}} | 0.05857 ^{\{ 10 \}} | 0.06353 ^{\{ 13 \}} | 0.04401 ^{\{ 1 \}} | 0.06477 ^{\{ 14 \}} | 0.05849 ^{\{ 9 \}} | 0.06561 ^{\{ 15 \}} | |
\sum Ranks | 17 ^{\{ 2 \}} | 44 ^{\{ 3 \}} | 54 ^{\{ 7 \}} | 46 ^{\{ 4 \}} | 68 ^{\{ 10 \}} | 101 ^{\{ 12 \}} | 58 ^{\{ 8 \}} | 51 ^{\{ 6 \}} | 48 ^{\{ 5 \}} | 77 ^{\{ 11 \}} | 107 ^{\{ 13 \}} | 8 ^{\{ 1 \}} | 110 ^{\{ 14.5 \}} | 61 ^{\{ 9 \}} | 110 ^{\{ 14.5 \}} | |
100 | BIAS( \hat{\delta} ) | 0.16869 ^{\{ 2 \}} | 0.21955 ^{\{ 6 \}} | 0.25099 ^{\{ 11 \}} | 0.20658 ^{\{ 4 \}} | 0.23978 ^{\{ 10 \}} | 0.26936 ^{\{ 13 \}} | 0.22407 ^{\{ 7 \}} | 0.21601 ^{\{ 5 \}} | 0.20488 ^{\{ 3 \}} | 0.23686 ^{\{ 9 \}} | 0.2921 ^{\{ 15 \}} | 0.06504 ^{\{ 1 \}} | 0.27867 ^{\{ 14 \}} | 0.23071 ^{\{ 8 \}} | 0.26579 ^{\{ 12 \}} |
BIAS( \hat{\beta} ) | 0.06441 ^{\{ 2 \}} | 0.0751 ^{\{ 5 \}} | 0.08048 ^{\{ 8 \}} | 0.07758 ^{\{ 6 \}} | 0.08158 ^{\{ 9 \}} | 0.08422 ^{\{ 10 \}} | 0.07438 ^{\{ 4 \}} | 0.07323 ^{\{ 3 \}} | 0.07816 ^{\{ 7 \}} | 0.0888 ^{\{ 12 \}} | 0.10363 ^{\{ 15 \}} | 0.03587 ^{\{ 1 \}} | 0.10299 ^{\{ 14 \}} | 0.08464 ^{\{ 11 \}} | 0.09863 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.04708 ^{\{ 2 \}} | 0.07537 ^{\{ 5 \}} | 0.09757 ^{\{ 11 \}} | 0.0662 ^{\{ 3 \}} | 0.08954 ^{\{ 10 \}} | 0.11225 ^{\{ 13 \}} | 0.07976 ^{\{ 8 \}} | 0.07321 ^{\{ 4 \}} | 0.07788 ^{\{ 6 \}} | 0.08763 ^{\{ 9 \}} | 0.13241 ^{\{ 15 \}} | 0.01303 ^{\{ 1 \}} | 0.11883 ^{\{ 14 \}} | 0.07847 ^{\{ 7 \}} | 0.10997 ^{\{ 12 \}} | |
MSE( \hat{\beta} ) | 0.00749 ^{\{ 2 \}} | 0.00915 ^{\{ 4 \}} | 0.0102 ^{\{ 6 \}} | 0.01024 ^{\{ 7 \}} | 0.0109 ^{\{ 8 \}} | 0.01143 ^{\{ 9 \}} | 0.00918 ^{\{ 5 \}} | 0.00855 ^{\{ 3 \}} | 0.01159 ^{\{ 11 \}} | 0.0135 ^{\{ 12 \}} | 0.01684 ^{\{ 13 \}} | 0.00259 ^{\{ 1 \}} | 0.01761 ^{\{ 15 \}} | 0.01144 ^{\{ 10 \}} | 0.01689 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.06747 ^{\{ 2 \}} | 0.08782 ^{\{ 6 \}} | 0.10039 ^{\{ 11 \}} | 0.08263 ^{\{ 4 \}} | 0.09591 ^{\{ 10 \}} | 0.10774 ^{\{ 13 \}} | 0.08963 ^{\{ 7 \}} | 0.0864 ^{\{ 5 \}} | 0.08195 ^{\{ 3 \}} | 0.09474 ^{\{ 9 \}} | 0.11684 ^{\{ 15 \}} | 0.02602 ^{\{ 1 \}} | 0.11147 ^{\{ 14 \}} | 0.09228 ^{\{ 8 \}} | 0.10632 ^{\{ 12 \}} | |
MRE( \hat{\beta} ) | 0.16103 ^{\{ 2 \}} | 0.18776 ^{\{ 5 \}} | 0.20121 ^{\{ 8 \}} | 0.19396 ^{\{ 6 \}} | 0.20394 ^{\{ 9 \}} | 0.21056 ^{\{ 10 \}} | 0.18594 ^{\{ 4 \}} | 0.18308 ^{\{ 3 \}} | 0.1954 ^{\{ 7 \}} | 0.22201 ^{\{ 12 \}} | 0.25908 ^{\{ 15 \}} | 0.08969 ^{\{ 1 \}} | 0.25746 ^{\{ 14 \}} | 0.2116 ^{\{ 11 \}} | 0.24657 ^{\{ 13 \}} | |
D_{abs} | 0.02344 ^{\{ 1 \}} | 0.0258 ^{\{ 4 \}} | 0.02729 ^{\{ 8 \}} | 0.02632 ^{\{ 5 \}} | 0.02689 ^{\{ 7 \}} | 0.02795 ^{\{ 9 \}} | 0.02543 ^{\{ 3 \}} | 0.02678 ^{\{ 6 \}} | 0.03037 ^{\{ 13 \}} | 0.02968 ^{\{ 12 \}} | 0.02967 ^{\{ 11 \}} | 0.02442 ^{\{ 2 \}} | 0.03344 ^{\{ 15 \}} | 0.02859 ^{\{ 10 \}} | 0.03216 ^{\{ 14 \}} | |
D_{max} | 0.03772 ^{\{ 2 \}} | 0.04225 ^{\{ 4 \}} | 0.04553 ^{\{ 8 \}} | 0.04245 ^{\{ 5 \}} | 0.0444 ^{\{ 7 \}} | 0.04705 ^{\{ 10 \}} | 0.04196 ^{\{ 3 \}} | 0.04374 ^{\{ 6 \}} | 0.0487 ^{\{ 12 \}} | 0.04793 ^{\{ 11 \}} | 0.04912 ^{\{ 13 \}} | 0.03646 ^{\{ 1 \}} | 0.05452 ^{\{ 15 \}} | 0.04636 ^{\{ 9 \}} | 0.05237 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 39 ^{\{ 4 \}} | 71 ^{\{ 9 \}} | 40 ^{\{ 5 \}} | 70 ^{\{ 8 \}} | 87 ^{\{ 12 \}} | 41 ^{\{ 6 \}} | 35 ^{\{ 3 \}} | 62 ^{\{ 7 \}} | 86 ^{\{ 11 \}} | 112 ^{\{ 14 \}} | 9 ^{\{ 1 \}} | 115 ^{\{ 15 \}} | 74 ^{\{ 10 \}} | 104 ^{\{ 13 \}} | |
200 | BIAS( \hat{\delta} ) | 0.12568 ^{\{ 2 \}} | 0.15802 ^{\{ 6 \}} | 0.16754 ^{\{ 10 \}} | 0.14131 ^{\{ 3 \}} | 0.17205 ^{\{ 11 \}} | 0.19869 ^{\{ 12 \}} | 0.1585 ^{\{ 8 \}} | 0.15009 ^{\{ 5 \}} | 0.14533 ^{\{ 4 \}} | 0.16686 ^{\{ 9 \}} | 0.215 ^{\{ 15 \}} | 0.05087 ^{\{ 1 \}} | 0.20091 ^{\{ 13 \}} | 0.15808 ^{\{ 7 \}} | 0.20236 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.04788 ^{\{ 2 \}} | 0.05346 ^{\{ 5 \}} | 0.05363 ^{\{ 6 \}} | 0.0502 ^{\{ 3 \}} | 0.05533 ^{\{ 9 \}} | 0.06203 ^{\{ 12 \}} | 0.05404 ^{\{ 7 \}} | 0.05068 ^{\{ 4 \}} | 0.05485 ^{\{ 8 \}} | 0.06093 ^{\{ 11 \}} | 0.07871 ^{\{ 15 \}} | 0.02619 ^{\{ 1 \}} | 0.07197 ^{\{ 13 \}} | 0.05806 ^{\{ 10 \}} | 0.07216 ^{\{ 14 \}} | |
MSE( \hat{\delta} ) | 0.02442 ^{\{ 2 \}} | 0.0386 ^{\{ 5 \}} | 0.04604 ^{\{ 11 \}} | 0.03106 ^{\{ 3 \}} | 0.04597 ^{\{ 10 \}} | 0.06064 ^{\{ 13 \}} | 0.03893 ^{\{ 6 \}} | 0.03567 ^{\{ 4 \}} | 0.03895 ^{\{ 7 \}} | 0.04485 ^{\{ 9 \}} | 0.06901 ^{\{ 15 \}} | 0.00783 ^{\{ 1 \}} | 0.0603 ^{\{ 12 \}} | 0.03976 ^{\{ 8 \}} | 0.06373 ^{\{ 14 \}} | |
MSE( \hat{\beta} ) | 0.00378 ^{\{ 2 \}} | 0.00458 ^{\{ 5 \}} | 0.00482 ^{\{ 7 \}} | 0.00408 ^{\{ 3 \}} | 0.00482 ^{\{ 7 \}} | 0.00624 ^{\{ 12 \}} | 0.00482 ^{\{ 7 \}} | 0.00409 ^{\{ 4 \}} | 0.00537 ^{\{ 9 \}} | 0.00623 ^{\{ 11 \}} | 0.00978 ^{\{ 15 \}} | 0.00135 ^{\{ 1 \}} | 0.00858 ^{\{ 13 \}} | 0.00572 ^{\{ 10 \}} | 0.00898 ^{\{ 14 \}} | |
MRE( \hat{\delta} ) | 0.05027 ^{\{ 2 \}} | 0.06321 ^{\{ 6 \}} | 0.06702 ^{\{ 10 \}} | 0.05652 ^{\{ 3 \}} | 0.06882 ^{\{ 11 \}} | 0.07948 ^{\{ 12 \}} | 0.0634 ^{\{ 8 \}} | 0.06003 ^{\{ 5 \}} | 0.05813 ^{\{ 4 \}} | 0.06675 ^{\{ 9 \}} | 0.086 ^{\{ 15 \}} | 0.02035 ^{\{ 1 \}} | 0.08036 ^{\{ 13 \}} | 0.06323 ^{\{ 7 \}} | 0.08094 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.1197 ^{\{ 2 \}} | 0.13366 ^{\{ 5 \}} | 0.13408 ^{\{ 6 \}} | 0.12549 ^{\{ 3 \}} | 0.13833 ^{\{ 9 \}} | 0.15508 ^{\{ 12 \}} | 0.13511 ^{\{ 7 \}} | 0.12671 ^{\{ 4 \}} | 0.13713 ^{\{ 8 \}} | 0.15232 ^{\{ 11 \}} | 0.19678 ^{\{ 15 \}} | 0.06548 ^{\{ 1 \}} | 0.17993 ^{\{ 13 \}} | 0.14516 ^{\{ 10 \}} | 0.1804 ^{\{ 14 \}} | |
D_{abs} | 0.01719 ^{\{ 2 \}} | 0.01841 ^{\{ 5 \}} | 0.01851 ^{\{ 6 \}} | 0.0174 ^{\{ 3 \}} | 0.01866 ^{\{ 7 \}} | 0.02028 ^{\{ 9 \}} | 0.01868 ^{\{ 8 \}} | 0.01825 ^{\{ 4 \}} | 0.02193 ^{\{ 12 \}} | 0.02128 ^{\{ 11 \}} | 0.02281 ^{\{ 13 \}} | 0.0171 ^{\{ 1 \}} | 0.02382 ^{\{ 15 \}} | 0.02076 ^{\{ 10 \}} | 0.02362 ^{\{ 14 \}} | |
D_{max} | 0.02788 ^{\{ 2 \}} | 0.03027 ^{\{ 5 \}} | 0.03093 ^{\{ 7 \}} | 0.02818 ^{\{ 3 \}} | 0.03111 ^{\{ 8 \}} | 0.03422 ^{\{ 10 \}} | 0.03083 ^{\{ 6 \}} | 0.02987 ^{\{ 4 \}} | 0.03496 ^{\{ 12 \}} | 0.03435 ^{\{ 11 \}} | 0.03748 ^{\{ 13 \}} | 0.02581 ^{\{ 1 \}} | 0.03887 ^{\{ 15 \}} | 0.03338 ^{\{ 9 \}} | 0.03869 ^{\{ 14 \}} | |
\sum Ranks | 16 ^{\{ 2 \}} | 42 ^{\{ 5 \}} | 63 ^{\{ 7 \}} | 24 ^{\{ 3 \}} | 72 ^{\{ 10 \}} | 92 ^{\{ 12 \}} | 57 ^{\{ 6 \}} | 34 ^{\{ 4 \}} | 64 ^{\{ 8 \}} | 82 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 8 ^{\{ 1 \}} | 107 ^{\{ 13 \}} | 71 ^{\{ 9 \}} | 112 ^{\{ 14 \}} | |
300 | BIAS( \hat{\delta} ) | 0.10138 ^{\{ 2 \}} | 0.12388 ^{\{ 6 \}} | 0.13609 ^{\{ 10 \}} | 0.11995 ^{\{ 4 \}} | 0.13703 ^{\{ 11 \}} | 0.15219 ^{\{ 12 \}} | 0.11897 ^{\{ 3 \}} | 0.12185 ^{\{ 5 \}} | 0.13147 ^{\{ 8 \}} | 0.13467 ^{\{ 9 \}} | 0.17695 ^{\{ 15 \}} | 0.04844 ^{\{ 1 \}} | 0.1656 ^{\{ 14 \}} | 0.12572 ^{\{ 7 \}} | 0.15695 ^{\{ 13 \}} |
BIAS( \hat{\beta} ) | 0.03889 ^{\{ 2 \}} | 0.04166 ^{\{ 4 \}} | 0.04473 ^{\{ 8 \}} | 0.04246 ^{\{ 6 \}} | 0.04436 ^{\{ 7 \}} | 0.04875 ^{\{ 12 \}} | 0.04023 ^{\{ 3 \}} | 0.04196 ^{\{ 5 \}} | 0.04867 ^{\{ 11 \}} | 0.0481 ^{\{ 10 \}} | 0.06487 ^{\{ 15 \}} | 0.02362 ^{\{ 1 \}} | 0.05683 ^{\{ 14 \}} | 0.04507 ^{\{ 9 \}} | 0.05475 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01642 ^{\{ 2 \}} | 0.02414 ^{\{ 6 \}} | 0.02962 ^{\{ 10 \}} | 0.0216 ^{\{ 3 \}} | 0.02872 ^{\{ 9 \}} | 0.03784 ^{\{ 12 \}} | 0.02234 ^{\{ 4 \}} | 0.02409 ^{\{ 5 \}} | 0.03044 ^{\{ 11 \}} | 0.02838 ^{\{ 8 \}} | 0.04928 ^{\{ 15 \}} | 0.00716 ^{\{ 1 \}} | 0.04261 ^{\{ 14 \}} | 0.02478 ^{\{ 7 \}} | 0.03848 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.00251 ^{\{ 2 \}} | 0.00274 ^{\{ 4 \}} | 0.00319 ^{\{ 8 \}} | 0.00286 ^{\{ 6 \}} | 0.00308 ^{\{ 7 \}} | 0.0039 ^{\{ 11 \}} | 0.00252 ^{\{ 3 \}} | 0.00285 ^{\{ 5 \}} | 0.00411 ^{\{ 12 \}} | 0.00379 ^{\{ 10 \}} | 0.00722 ^{\{ 15 \}} | 0.00119 ^{\{ 1 \}} | 0.00549 ^{\{ 14 \}} | 0.00329 ^{\{ 9 \}} | 0.00507 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.04055 ^{\{ 2 \}} | 0.04955 ^{\{ 6 \}} | 0.05444 ^{\{ 10 \}} | 0.04798 ^{\{ 4 \}} | 0.05481 ^{\{ 11 \}} | 0.06088 ^{\{ 12 \}} | 0.04759 ^{\{ 3 \}} | 0.04874 ^{\{ 5 \}} | 0.05259 ^{\{ 8 \}} | 0.05387 ^{\{ 9 \}} | 0.07078 ^{\{ 15 \}} | 0.01938 ^{\{ 1 \}} | 0.06624 ^{\{ 14 \}} | 0.05029 ^{\{ 7 \}} | 0.06278 ^{\{ 13 \}} | |
MRE( \hat{\beta} ) | 0.09723 ^{\{ 2 \}} | 0.10415 ^{\{ 4 \}} | 0.11183 ^{\{ 8 \}} | 0.10614 ^{\{ 6 \}} | 0.11091 ^{\{ 7 \}} | 0.12186 ^{\{ 12 \}} | 0.10057 ^{\{ 3 \}} | 0.10489 ^{\{ 5 \}} | 0.12168 ^{\{ 11 \}} | 0.12025 ^{\{ 10 \}} | 0.16218 ^{\{ 15 \}} | 0.05905 ^{\{ 1 \}} | 0.14207 ^{\{ 14 \}} | 0.11268 ^{\{ 9 \}} | 0.13687 ^{\{ 13 \}} | |
D_{abs} | 0.01417 ^{\{ 1 \}} | 0.01453 ^{\{ 2 \}} | 0.01543 ^{\{ 7 \}} | 0.01481 ^{\{ 5 \}} | 0.01544 ^{\{ 8 \}} | 0.01573 ^{\{ 9 \}} | 0.01455 ^{\{ 3 \}} | 0.0152 ^{\{ 6 \}} | 0.01852 ^{\{ 13 \}} | 0.01696 ^{\{ 11 \}} | 0.0176 ^{\{ 12 \}} | 0.0147 ^{\{ 4 \}} | 0.01974 ^{\{ 15 \}} | 0.01602 ^{\{ 10 \}} | 0.01923 ^{\{ 14 \}} | |
D_{max} | 0.02287 ^{\{ 2 \}} | 0.02379 ^{\{ 4 \}} | 0.02546 ^{\{ 7 \}} | 0.02403 ^{\{ 5 \}} | 0.02569 ^{\{ 8 \}} | 0.0263 ^{\{ 10 \}} | 0.02377 ^{\{ 3 \}} | 0.02487 ^{\{ 6 \}} | 0.02963 ^{\{ 13 \}} | 0.02753 ^{\{ 11 \}} | 0.02918 ^{\{ 12 \}} | 0.02235 ^{\{ 1 \}} | 0.03243 ^{\{ 15 \}} | 0.02583 ^{\{ 9 \}} | 0.03132 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 36 ^{\{ 4 \}} | 68 ^{\{ 8.5 \}} | 39 ^{\{ 5 \}} | 68 ^{\{ 8.5 \}} | 90 ^{\{ 12 \}} | 25 ^{\{ 3 \}} | 42 ^{\{ 6 \}} | 87 ^{\{ 11 \}} | 78 ^{\{ 10 \}} | 114 ^{\{ 14.5 \}} | 11 ^{\{ 1 \}} | 114 ^{\{ 14.5 \}} | 67 ^{\{ 7 \}} | 106 ^{\{ 13 \}} | |
400 | BIAS( \hat{\delta} ) | 0.08539 ^{\{ 2 \}} | 0.10758 ^{\{ 5 \}} | 0.12255 ^{\{ 11 \}} | 0.10162 ^{\{ 3 \}} | 0.12057 ^{\{ 10 \}} | 0.13935 ^{\{ 12 \}} | 0.10965 ^{\{ 6 \}} | 0.11168 ^{\{ 7 \}} | 0.10747 ^{\{ 4 \}} | 0.11634 ^{\{ 9 \}} | 0.15238 ^{\{ 15 \}} | 0.03778 ^{\{ 1 \}} | 0.14254 ^{\{ 13 \}} | 0.11229 ^{\{ 8 \}} | 0.14255 ^{\{ 14 \}} |
BIAS( \hat{\beta} ) | 0.03288 ^{\{ 2 \}} | 0.03624 ^{\{ 4 \}} | 0.04047 ^{\{ 10 \}} | 0.03533 ^{\{ 3 \}} | 0.03953 ^{\{ 9 \}} | 0.04295 ^{\{ 12 \}} | 0.03632 ^{\{ 5 \}} | 0.03807 ^{\{ 6 \}} | 0.03897 ^{\{ 7 \}} | 0.04102 ^{\{ 11 \}} | 0.05635 ^{\{ 15 \}} | 0.01866 ^{\{ 1 \}} | 0.0513 ^{\{ 14 \}} | 0.03935 ^{\{ 8 \}} | 0.04946 ^{\{ 13 \}} | |
MSE( \hat{\delta} ) | 0.01164 ^{\{ 2 \}} | 0.01815 ^{\{ 4 \}} | 0.02321 ^{\{ 11 \}} | 0.01641 ^{\{ 3 \}} | 0.02278 ^{\{ 10 \}} | 0.02969 ^{\{ 12 \}} | 0.01878 ^{\{ 5 \}} | 0.01988 ^{\{ 7 \}} | 0.02066 ^{\{ 8 \}} | 0.02178 ^{\{ 9 \}} | 0.03737 ^{\{ 15 \}} | 0.00425 ^{\{ 1 \}} | 0.03212 ^{\{ 14 \}} | 0.01965 ^{\{ 6 \}} | 0.03198 ^{\{ 13 \}} | |
MSE( \hat{\beta} ) | 0.00175 ^{\{ 2 \}} | 0.00205 ^{\{ 4 \}} | 0.00259 ^{\{ 9 \}} | 0.00201 ^{\{ 3 \}} | 0.0025 ^{\{ 8 \}} | 0.00289 ^{\{ 12 \}} | 0.00208 ^{\{ 5 \}} | 0.00229 ^{\{ 6 \}} | 0.00265 ^{\{ 10 \}} | 0.00283 ^{\{ 11 \}} | 0.0054 ^{\{ 15 \}} | 0.00073 ^{\{ 1 \}} | 0.00429 ^{\{ 14 \}} | 0.00241 ^{\{ 7 \}} | 0.00393 ^{\{ 13 \}} | |
MRE( \hat{\delta} ) | 0.03416 ^{\{ 2 \}} | 0.04303 ^{\{ 5 \}} | 0.04902 ^{\{ 11 \}} | 0.04065 ^{\{ 3 \}} | 0.04823 ^{\{ 10 \}} | 0.05574 ^{\{ 12 \}} | 0.04386 ^{\{ 6 \}} | 0.04467 ^{\{ 7 \}} | 0.04299 ^{\{ 4 \}} | 0.04654 ^{\{ 9 \}} | 0.06095 ^{\{ 15 \}} | 0.01511 ^{\{ 1 \}} | 0.05701 ^{\{ 13 \}} | 0.04492 ^{\{ 8 \}} | 0.05702 ^{\{ 14 \}} | |
MRE( \hat{\beta} ) | 0.08219 ^{\{ 2 \}} | 0.09059 ^{\{ 4 \}} | 0.10117 ^{\{ 10 \}} | 0.08833 ^{\{ 3 \}} | 0.09881 ^{\{ 9 \}} | 0.10736 ^{\{ 12 \}} | 0.0908 ^{\{ 5 \}} | 0.09518 ^{\{ 6 \}} | 0.09744 ^{\{ 7 \}} | 0.10255 ^{\{ 11 \}} | 0.14089 ^{\{ 15 \}} | 0.04664 ^{\{ 1 \}} | 0.12824 ^{\{ 14 \}} | 0.09839 ^{\{ 8 \}} | 0.12364 ^{\{ 13 \}} | |
D_{abs} | 0.01182 ^{\{ 1 \}} | 0.01268 ^{\{ 4 \}} | 0.01351 ^{\{ 7.5 \}} | 0.01208 ^{\{ 3 \}} | 0.01351 ^{\{ 7.5 \}} | 0.01356 ^{\{ 9 \}} | 0.01291 ^{\{ 5 \}} | 0.0134 ^{\{ 6 \}} | 0.0148 ^{\{ 11 \}} | 0.01494 ^{\{ 12 \}} | 0.016 ^{\{ 13 \}} | 0.01204 ^{\{ 2 \}} | 0.01733 ^{\{ 15 \}} | 0.01386 ^{\{ 10 \}} | 0.01687 ^{\{ 14 \}} | |
D_{max} | 0.01903 ^{\{ 2 \}} | 0.02082 ^{\{ 4 \}} | 0.02243 ^{\{ 8 \}} | 0.01969 ^{\{ 3 \}} | 0.02228 ^{\{ 7 \}} | 0.02295 ^{\{ 10 \}} | 0.02117 ^{\{ 5 \}} | 0.02188 ^{\{ 6 \}} | 0.02381 ^{\{ 11 \}} | 0.02425 ^{\{ 12 \}} | 0.02642 ^{\{ 13 \}} | 0.01826 ^{\{ 1 \}} | 0.02832 ^{\{ 15 \}} | 0.02247 ^{\{ 9 \}} | 0.02769 ^{\{ 14 \}} | |
\sum Ranks | 15 ^{\{ 2 \}} | 34 ^{\{ 4 \}} | 77.5 ^{\{ 10 \}} | 24 ^{\{ 3 \}} | 70.5 ^{\{ 9 \}} | 91 ^{\{ 12 \}} | 42 ^{\{ 5 \}} | 51 ^{\{ 6 \}} | 62 ^{\{ 7 \}} | 84 ^{\{ 11 \}} | 116 ^{\{ 15 \}} | 9 ^{\{ 1 \}} | 112 ^{\{ 14 \}} | 64 ^{\{ 8 \}} | 108 ^{\{ 13 \}} |
Parameter | n | MLE | ADE | CVME | MPSE | OLSE | RTADE | WLSE | LTADE | MSADE | MSALDE | ADSOE | KE | MSSD | MSSLD | MSLND |
\delta=0.7, \beta=2.5 | 30 | 4.0 | 7.0 | 11.0 | 6.0 | 12.5 | 15.0 | 12.5 | 10.0 | 2.0 | 3.0 | 14.0 | 1.0 | 8.5 | 8.5 | 5.0 |
60 | 3.0 | 11.0 | 14.0 | 4.0 | 10.0 | 15.0 | 12.0 | 9.0 | 2.0 | 5.0 | 8.0 | 1.0 | 6.0 | 13.0 | 7.0 | |
100 | 3.0 | 11.0 | 13.0 | 4.0 | 12.0 | 15.0 | 9.0 | 7.0 | 2.0 | 6.0 | 14.0 | 1.0 | 5.0 | 10.0 | 8.0 | |
200 | 2.0 | 7.0 | 14.0 | 5.0 | 13.0 | 15.0 | 10.0 | 4.0 | 3.0 | 8.0 | 12.0 | 1.0 | 6.0 | 11.0 | 9.0 | |
300 | 2.0 | 6.0 | 13.0 | 4.0 | 14.0 | 15.0 | 11.0 | 5.0 | 3.0 | 7.0 | 12.0 | 1.0 | 8.5 | 10.0 | 8.5 | |
400 | 2.0 | 9.0 | 14.0 | 4.0 | 12.5 | 15.0 | 7.0 | 5.0 | 3.0 | 11.0 | 12.5 | 1.0 | 6.0 | 8.0 | 10.0 | |
\delta=0.25, \beta=0.75 | 30 | 2.0 | 6.0 | 11.5 | 7.0 | 10.0 | 13.0 | 8.0 | 5.0 | 3.5 | 3.5 | 11.5 | 1.0 | 15.0 | 9.0 | 14.0 |
60 | 2.0 | 7.0 | 10.0 | 5.0 | 11.0 | 13.0 | 8.0 | 4.0 | 3.0 | 6.0 | 12.0 | 1.0 | 15.0 | 9.0 | 14.0 | |
100 | 2.0 | 3.5 | 8.0 | 6.0 | 9.0 | 14.0 | 3.5 | 7.0 | 5.0 | 11.0 | 13.0 | 1.0 | 12.0 | 10.0 | 15.0 | |
200 | 2.0 | 3.0 | 10.0 | 4.5 | 11.0 | 13.0 | 7.0 | 4.5 | 6.0 | 8.0 | 14.0 | 1.0 | 15.0 | 9.0 | 12.0 | |
300 | 1.0 | 7.0 | 12.0 | 3.0 | 11.0 | 14.0 | 5.0 | 6.0 | 4.0 | 9.0 | 15.0 | 2.0 | 13.0 | 8.0 | 10.0 | |
400 | 2.0 | 5.0 | 10.0 | 4.0 | 8.0 | 13.0 | 6.0 | 3.0 | 7.0 | 12.0 | 15.0 | 1.0 | 14.0 | 9.0 | 11.0 | |
\delta=1.5, \beta=1.5 | 30 | 2.0 | 8.0 | 12.0 | 5.0 | 11.0 | 13.0 | 10.0 | 4.0 | 3.0 | 7.0 | 14.0 | 1.0 | 15.0 | 6.0 | 9.0 |
60 | 1.0 | 6.0 | 10.0 | 4.0 | 11.5 | 14.0 | 7.0 | 3.0 | 5.0 | 8.5 | 11.5 | 2.0 | 15.0 | 8.5 | 13.0 | |
100 | 1.0 | 7.0 | 11.0 | 4.0 | 12.0 | 13.0 | 8.0 | 3.0 | 6.0 | 9.0 | 10.0 | 2.0 | 15.0 | 5.0 | 14.0 | |
200 | 1.5 | 5.0 | 10.0 | 3.0 | 11.0 | 14.0 | 6.5 | 4.0 | 6.5 | 9.0 | 12.0 | 1.5 | 15.0 | 8.0 | 13.0 | |
300 | 1.0 | 5.0 | 10.5 | 4.0 | 9.0 | 13.0 | 7.0 | 3.0 | 6.0 | 10.5 | 12.0 | 2.0 | 15.0 | 8.0 | 14.0 | |
400 | 1.0 | 7.0 | 10.0 | 3.0 | 11.0 | 14.0 | 6.0 | 4.0 | 9.0 | 8.0 | 12.0 | 2.0 | 15.0 | 5.0 | 13.0 | |
\delta=0.5, \beta=2.0 | 30 | 3.0 | 9.0 | 14.0 | 5.0 | 8.0 | 15.0 | 7.0 | 6.0 | 2.0 | 4.0 | 13.0 | 1.0 | 11.0 | 12.0 | 10.0 |
60 | 3.0 | 5.0 | 14.0 | 4.0 | 11.0 | 15.0 | 12.0 | 9.0 | 2.0 | 6.0 | 13.0 | 1.0 | 10.0 | 8.0 | 7.0 | |
100 | 3.0 | 9.0 | 13.0 | 4.0 | 14.0 | 15.0 | 10.0 | 5.0 | 2.0 | 11.0 | 12.0 | 1.0 | 7.5 | 7.5 | 6.0 | |
200 | 3.0 | 9.0 | 12.0 | 4.0 | 13.0 | 15.0 | 8.0 | 5.0 | 2.0 | 10.0 | 14.0 | 1.0 | 6.0 | 11.0 | 7.0 | |
300 | 2.0 | 8.0 | 14.0 | 4.0 | 13.0 | 15.0 | 9.0 | 5.0 | 3.0 | 11.0 | 12.0 | 1.0 | 6.0 | 10.0 | 7.0 | |
400 | 2.0 | 8.0 | 14.0 | 4.0 | 12.0 | 15.0 | 10.0 | 5.0 | 3.0 | 11.0 | 13.0 | 1.0 | 9.0 | 6.0 | 7.0 | |
\delta=2.5, \beta=0.4 | 30 | 2.0 | 6.0 | 11.0 | 4.5 | 10.0 | 15.0 | 4.5 | 8.0 | 3.0 | 9.0 | 13.0 | 1.0 | 14.0 | 7.0 | 12.0 |
60 | 2.0 | 3.0 | 7.0 | 4.0 | 10.0 | 12.0 | 8.0 | 6.0 | 5.0 | 11.0 | 13.0 | 1.0 | 14.5 | 9.0 | 14.5 | |
100 | 2.0 | 4.0 | 9.0 | 5.0 | 8.0 | 12.0 | 6.0 | 3.0 | 7.0 | 11.0 | 14.0 | 1.0 | 15.0 | 10.0 | 13.0 | |
200 | 2.0 | 5.0 | 7.0 | 3.0 | 10.0 | 12.0 | 6.0 | 4.0 | 8.0 | 11.0 | 15.0 | 1.0 | 13.0 | 9.0 | 14.0 | |
300 | 2.0 | 4.0 | 8.5 | 5.0 | 8.5 | 12.0 | 3.0 | 6.0 | 11.0 | 10.0 | 14.5 | 1.0 | 14.5 | 7.0 | 13.0 | |
400 | 2.0 | 4.0 | 10.0 | 3.0 | 9.0 | 12.0 | 5.0 | 6.0 | 7.0 | 11.0 | 15.0 | 1.0 | 14.0 | 8.0 | 13.0 | |
\sum Ranks | 62.5 | 194.5 | 337.5 | 129.0 | 326.0 | 416.0 | 232.0 | 158.5 | 134.0 | 257.5 | 386.0 | 35.5 | 348.5 | 259.5 | 323.0 | |
Overall Rank | 2 | 6 | 12 | 3 | 11 | 15 | 7 | 5 | 4 | 8 | 14 | 1 | 13 | 9 | 10 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.49641) | (0.60871) | (-0.53065) | (-0.39129) | (-0.43960) | (-0.29722) | (0.38680) | (-0.56988) | (-0.94070) | (-0.73958) | ||
20 | \hat{E} | -0.53095 | 0.59059 | -0.56088 | -0.40941 | -0.46465 | -0.34499 | 0.45684 | -0.60354 | -0.97991 | -0.76199 |
BIAS | 0.07837 | 0.04604 | 0.07246 | 0.04604 | 0.06003 | 0.05431 | 0.07916 | 0.08133 | 0.08408 | 0.08889 | |
MSE | 0.00993 | 0.00330 | 0.00831 | 0.00330 | 0.00570 | 0.00859 | 0.01919 | 0.01077 | 0.01236 | 0.01358 | |
MRE | 0.15788 | 0.07564 | 0.13656 | 0.11767 | 0.13656 | 0.18274 | 0.20465 | 0.14272 | 0.08938 | 0.12019 | |
60 | \hat{E} | -0.51151 | 0.60061 | -0.54401 | -0.39939 | -0.45067 | -0.33185 | 0.43748 | -0.58617 | -0.96029 | -0.75437 |
BIAS | 0.04763 | 0.02850 | 0.04447 | 0.02850 | 0.03684 | 0.04169 | 0.06052 | 0.04871 | 0.04965 | 0.05278 | |
MSE | 0.00366 | 0.00128 | 0.00315 | 0.00128 | 0.00216 | 0.00597 | 0.01312 | 0.00382 | 0.00409 | 0.00461 | |
MRE | 0.09595 | 0.04683 | 0.08379 | 0.07285 | 0.08379 | 0.14026 | 0.15646 | 0.08548 | 0.05278 | 0.07137 | |
100 | \hat{E} | -0.50365 | 0.60492 | -0.53699 | -0.39508 | -0.44486 | -0.32447 | 0.42653 | -0.57848 | -0.95201 | -0.74854 |
BIAS | 0.03577 | 0.02159 | 0.03354 | 0.02159 | 0.02778 | 0.03496 | 0.05049 | 0.03662 | 0.03749 | 0.04155 | |
MSE | 0.00203 | 0.00073 | 0.00178 | 0.00073 | 0.00122 | 0.00432 | 0.00937 | 0.00212 | 0.00225 | 0.00277 | |
MRE | 0.07206 | 0.03546 | 0.06320 | 0.05517 | 0.06320 | 0.11763 | 0.13054 | 0.06426 | 0.03985 | 0.05618 | |
150 | \hat{E} | -0.50062 | 0.60654 | -0.53431 | -0.39346 | -0.44263 | -0.31955 | 0.41918 | -0.57587 | -0.94935 | -0.74777 |
BIAS | 0.02875 | 0.01742 | 0.02701 | 0.01742 | 0.02237 | 0.02987 | 0.04291 | 0.02945 | 0.03022 | 0.03410 | |
MSE | 0.00131 | 0.00048 | 0.00115 | 0.00048 | 0.00079 | 0.00307 | 0.00658 | 0.00136 | 0.00147 | 0.00187 | |
MRE | 0.05791 | 0.02861 | 0.05090 | 0.04451 | 0.05090 | 0.10051 | 0.11094 | 0.05167 | 0.03213 | 0.04610 | |
200 | \hat{E} | -0.49922 | 0.60731 | -0.53305 | -0.39269 | -0.44159 | -0.31498 | 0.41241 | -0.57418 | -0.94720 | -0.74586 |
BIAS | 0.02534 | 0.01537 | 0.02382 | 0.01537 | 0.01973 | 0.02508 | 0.03583 | 0.02571 | 0.02619 | 0.02975 | |
MSE | 0.00100 | 0.00037 | 0.00088 | 0.00037 | 0.00061 | 0.00205 | 0.00433 | 0.00104 | 0.00110 | 0.00142 | |
MRE | 0.05105 | 0.02525 | 0.04489 | 0.03928 | 0.04489 | 0.08439 | 0.09264 | 0.04512 | 0.02784 | 0.04023 | |
250 | \hat{E} | -0.49872 | 0.60755 | -0.53263 | -0.39245 | -0.44125 | -0.31227 | 0.40843 | -0.57363 | -0.94642 | -0.74544 |
BIAS | 0.02237 | 0.01358 | 0.02104 | 0.01358 | 0.01743 | 0.02194 | 0.03125 | 0.02259 | 0.02298 | 0.02633 | |
MSE | 0.00078 | 0.00029 | 0.00069 | 0.00029 | 0.00047 | 0.00152 | 0.00317 | 0.00080 | 0.00085 | 0.00111 | |
MRE | 0.04507 | 0.02231 | 0.03965 | 0.03471 | 0.03965 | 0.07382 | 0.08078 | 0.03964 | 0.02443 | 0.03560 | |
300 | \hat{E} | -0.49867 | 0.60754 | -0.53262 | -0.39246 | -0.44124 | -0.30972 | 0.40472 | -0.57329 | -0.94570 | -0.74456 |
BIAS | 0.02057 | 0.01249 | 0.01934 | 0.01249 | 0.01602 | 0.01937 | 0.02751 | 0.02078 | 0.02112 | 0.02426 | |
MSE | 0.00066 | 0.00024 | 0.00059 | 0.00024 | 0.00040 | 0.00114 | 0.00237 | 0.00068 | 0.00071 | 0.00094 | |
MRE | 0.04143 | 0.02051 | 0.03645 | 0.03191 | 0.03645 | 0.06516 | 0.07111 | 0.03647 | 0.02246 | 0.03281 | |
400 | \hat{E} | -0.49783 | 0.60800 | -0.53187 | -0.39200 | -0.44061 | -0.30769 | 0.40175 | -0.57248 | -0.94480 | -0.74409 |
BIAS | 0.01785 | 0.01084 | 0.01679 | 0.01084 | 0.01391 | 0.01677 | 0.02374 | 0.01791 | 0.01813 | 0.02087 | |
MSE | 0.00050 | 0.00019 | 0.00045 | 0.00019 | 0.00031 | 0.00078 | 0.00159 | 0.00051 | 0.00053 | 0.00070 | |
MRE | 0.03595 | 0.01782 | 0.03165 | 0.02772 | 0.03165 | 0.05643 | 0.06138 | 0.03143 | 0.01927 | 0.02822 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.06783) | (0.93442) | (-0.08051) | (-0.06558) | (-0.06670) | (-0.55698) | (0.77529) | (-0.11870) | (-0.61333) | (-0.19500) | ||
20 | \hat{E} | -0.08600 | 0.91826 | -0.10119 | -0.08174 | -0.08383 | -0.57359 | 0.81240 | -0.14688 | -0.64206 | -0.20407 |
BIAS | 0.03295 | 0.03015 | 0.03804 | 0.03015 | 0.03152 | 0.13286 | 0.21409 | 0.05684 | 0.06063 | 0.02178 | |
MSE | 0.00181 | 0.00148 | 0.00238 | 0.00148 | 0.00163 | 0.02637 | 0.06957 | 0.00541 | 0.00640 | 0.00083 | |
MRE | 0.48578 | 0.03227 | 0.47257 | 0.45978 | 0.47257 | 0.23853 | 0.27615 | 0.47882 | 0.09886 | 0.11170 | |
60 | \hat{E} | -0.07058 | 0.93218 | -0.08350 | -0.06782 | -0.06918 | -0.54615 | 0.76306 | -0.12237 | -0.61731 | -0.19918 |
BIAS | 0.01987 | 0.01844 | 0.02311 | 0.01844 | 0.01914 | 0.08949 | 0.14217 | 0.03542 | 0.03742 | 0.01242 | |
MSE | 0.00071 | 0.00060 | 0.00095 | 0.00060 | 0.00065 | 0.01272 | 0.03216 | 0.00224 | 0.00257 | 0.00026 | |
MRE | 0.29296 | 0.01973 | 0.28701 | 0.28119 | 0.28701 | 0.16067 | 0.18337 | 0.29838 | 0.06101 | 0.06367 | |
100 | \hat{E} | -0.07079 | 0.93192 | -0.08380 | -0.06808 | -0.06942 | -0.55144 | 0.77021 | -0.12307 | -0.61799 | -0.19801 |
BIAS | 0.01830 | 0.01700 | 0.02129 | 0.01700 | 0.01764 | 0.07792 | 0.12400 | 0.03260 | 0.03419 | 0.00970 | |
MSE | 0.00057 | 0.00049 | 0.00077 | 0.00049 | 0.00053 | 0.00945 | 0.02392 | 0.00179 | 0.00197 | 0.00015 | |
MRE | 0.26984 | 0.01819 | 0.26448 | 0.25923 | 0.26448 | 0.13990 | 0.15994 | 0.27467 | 0.05574 | 0.04975 | |
150 | \hat{E} | -0.06964 | 0.93291 | -0.08251 | -0.06709 | -0.06835 | -0.55277 | 0.77125 | -0.12140 | -0.61630 | -0.19696 |
BIAS | 0.01545 | 0.01438 | 0.01799 | 0.01438 | 0.01490 | 0.06590 | 0.10492 | 0.02758 | 0.02884 | 0.00791 | |
MSE | 0.00039 | 0.00034 | 0.00053 | 0.00034 | 0.00036 | 0.00675 | 0.01708 | 0.00124 | 0.00136 | 0.00010 | |
MRE | 0.22776 | 0.01539 | 0.22349 | 0.21929 | 0.22349 | 0.11832 | 0.13533 | 0.23232 | 0.04703 | 0.04055 | |
200 | \hat{E} | -0.06936 | 0.93313 | -0.08220 | -0.06687 | -0.06810 | -0.55416 | 0.77283 | -0.12100 | -0.61582 | -0.19650 |
BIAS | 0.01350 | 0.01258 | 0.01573 | 0.01258 | 0.01303 | 0.05742 | 0.09143 | 0.02413 | 0.02521 | 0.00679 | |
MSE | 0.00029 | 0.00025 | 0.00039 | 0.00025 | 0.00027 | 0.00513 | 0.01298 | 0.00092 | 0.00101 | 0.00007 | |
MRE | 0.19902 | 0.01346 | 0.19539 | 0.19182 | 0.19539 | 0.10308 | 0.11793 | 0.20332 | 0.04110 | 0.03481 | |
250 | \hat{E} | -0.06880 | 0.93362 | -0.08157 | -0.06638 | -0.06758 | -0.55348 | 0.77135 | -0.12007 | -0.61481 | -0.19632 |
BIAS | 0.01209 | 0.01127 | 0.01409 | 0.01127 | 0.01167 | 0.05144 | 0.08187 | 0.02162 | 0.02254 | 0.00606 | |
MSE | 0.00023 | 0.00020 | 0.00031 | 0.00020 | 0.00021 | 0.00412 | 0.01041 | 0.00073 | 0.00079 | 0.00006 | |
MRE | 0.17817 | 0.01206 | 0.17501 | 0.17189 | 0.17501 | 0.09235 | 0.10559 | 0.18216 | 0.03675 | 0.03107 | |
300 | \hat{E} | -0.06849 | 0.93389 | -0.08122 | -0.06611 | -0.06728 | -0.55377 | 0.77148 | -0.11958 | -0.61429 | -0.19608 |
BIAS | 0.01080 | 0.01008 | 0.01259 | 0.01008 | 0.01043 | 0.04597 | 0.07317 | 0.01930 | 0.02008 | 0.00549 | |
MSE | 0.00018 | 0.00016 | 0.00024 | 0.00016 | 0.00017 | 0.00331 | 0.00836 | 0.00057 | 0.00062 | 0.00005 | |
MRE | 0.15919 | 0.01079 | 0.15641 | 0.15367 | 0.15641 | 0.08253 | 0.09437 | 0.16256 | 0.03273 | 0.02814 | |
400 | \hat{E} | -0.06779 | 0.93451 | -0.08043 | -0.06549 | -0.06663 | -0.55280 | 0.76960 | -0.11845 | -0.61313 | -0.19585 |
BIAS | 0.00922 | 0.00861 | 0.01075 | 0.00861 | 0.00891 | 0.03952 | 0.06287 | 0.01649 | 0.01714 | 0.00468 | |
MSE | 0.00013 | 0.00011 | 0.00018 | 0.00011 | 0.00012 | 0.00246 | 0.00620 | 0.00042 | 0.00045 | 0.00003 | |
MRE | 0.13587 | 0.00921 | 0.13356 | 0.13128 | 0.13356 | 0.07096 | 0.08109 | 0.13893 | 0.02795 | 0.02402 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.50778) | (0.60183) | (-0.54131) | (-0.39817) | (-0.44844) | (-0.41510) | (0.55685) | (-0.56943) | (-0.94546) | (-0.64094) | ||
20 | \hat{E} | -0.53963 | 0.58474 | -0.56950 | -0.41526 | -0.47179 | -0.42231 | 0.56992 | -0.60692 | -0.99220 | -0.68195 |
BIAS | 0.06486 | 0.03768 | 0.05965 | 0.03768 | 0.04941 | 0.06592 | 0.09848 | 0.07228 | 0.08395 | 0.06627 | |
MSE | 0.00716 | 0.00232 | 0.00593 | 0.00232 | 0.00407 | 0.00626 | 0.01412 | 0.00905 | 0.01324 | 0.00793 | |
MRE | 0.12773 | 0.06261 | 0.11019 | 0.09464 | 0.11019 | 0.15882 | 0.17685 | 0.12693 | 0.08879 | 0.10339 | |
60 | \hat{E} | -0.51075 | 0.60061 | -0.54366 | -0.39939 | -0.45038 | -0.39987 | 0.53536 | -0.57260 | -0.94830 | -0.65192 |
BIAS | 0.03416 | 0.02045 | 0.03190 | 0.02045 | 0.02642 | 0.04356 | 0.06430 | 0.03772 | 0.04263 | 0.03160 | |
MSE | 0.00191 | 0.00068 | 0.00165 | 0.00068 | 0.00114 | 0.00303 | 0.00661 | 0.00240 | 0.00323 | 0.00169 | |
MRE | 0.06728 | 0.03398 | 0.05893 | 0.05135 | 0.05893 | 0.10495 | 0.11547 | 0.06624 | 0.04509 | 0.04930 | |
100 | \hat{E} | -0.50817 | 0.60195 | -0.54141 | -0.39805 | -0.44852 | -0.40209 | 0.53840 | -0.56973 | -0.94512 | -0.64743 |
BIAS | 0.02656 | 0.01595 | 0.02484 | 0.01595 | 0.02058 | 0.03806 | 0.05627 | 0.02990 | 0.03474 | 0.02354 | |
MSE | 0.00118 | 0.00042 | 0.00102 | 0.00042 | 0.00070 | 0.00235 | 0.00513 | 0.00154 | 0.00218 | 0.00093 | |
MRE | 0.05231 | 0.02650 | 0.04589 | 0.04005 | 0.04589 | 0.09169 | 0.10105 | 0.05251 | 0.03675 | 0.03672 | |
150 | \hat{E} | -0.50866 | 0.60156 | -0.54195 | -0.39844 | -0.44896 | -0.40642 | 0.54463 | -0.57036 | -0.94611 | -0.64604 |
BIAS | 0.02290 | 0.01375 | 0.02142 | 0.01375 | 0.01774 | 0.03357 | 0.04972 | 0.02612 | 0.03078 | 0.01965 | |
MSE | 0.00085 | 0.00031 | 0.00074 | 0.00031 | 0.00051 | 0.00177 | 0.00388 | 0.00113 | 0.00162 | 0.00064 | |
MRE | 0.04510 | 0.02284 | 0.03956 | 0.03452 | 0.03956 | 0.08087 | 0.08930 | 0.04588 | 0.03256 | 0.03065 | |
200 | \hat{E} | -0.50805 | 0.60185 | -0.54143 | -0.39815 | -0.44853 | -0.40717 | 0.54558 | -0.56956 | -0.94505 | -0.64468 |
BIAS | 0.01964 | 0.01180 | 0.01838 | 0.01180 | 0.01523 | 0.02931 | 0.04342 | 0.02235 | 0.02632 | 0.01691 | |
MSE | 0.00061 | 0.00022 | 0.00054 | 0.00022 | 0.00037 | 0.00134 | 0.00294 | 0.00081 | 0.00115 | 0.00046 | |
MRE | 0.03868 | 0.01961 | 0.03395 | 0.02965 | 0.03395 | 0.07061 | 0.07797 | 0.03925 | 0.02783 | 0.02639 | |
250 | \hat{E} | -0.50768 | 0.60204 | -0.54111 | -0.39796 | -0.44827 | -0.40773 | 0.54631 | -0.56909 | -0.94448 | -0.64387 |
BIAS | 0.01792 | 0.01078 | 0.01678 | 0.01078 | 0.01390 | 0.02639 | 0.03910 | 0.02037 | 0.02394 | 0.01534 | |
MSE | 0.00051 | 0.00018 | 0.00044 | 0.00018 | 0.00030 | 0.00108 | 0.00237 | 0.00066 | 0.00093 | 0.00038 | |
MRE | 0.03529 | 0.01791 | 0.03099 | 0.02707 | 0.03099 | 0.06359 | 0.07022 | 0.03578 | 0.02532 | 0.02394 | |
300 | \hat{E} | -0.50732 | 0.60223 | -0.54079 | -0.39777 | -0.44801 | -0.40808 | 0.54678 | -0.56866 | -0.94398 | -0.64321 |
BIAS | 0.01623 | 0.00976 | 0.01520 | 0.00976 | 0.01259 | 0.02435 | 0.03607 | 0.01864 | 0.02221 | 0.01370 | |
MSE | 0.00042 | 0.00015 | 0.00037 | 0.00015 | 0.00025 | 0.00091 | 0.00200 | 0.00056 | 0.00078 | 0.00030 | |
MRE | 0.03196 | 0.01623 | 0.02807 | 0.02452 | 0.02807 | 0.05866 | 0.06477 | 0.03274 | 0.02349 | 0.02138 | |
400 | \hat{E} | -0.50687 | 0.60247 | -0.54040 | -0.39753 | -0.44768 | -0.40877 | 0.54769 | -0.56812 | -0.94337 | -0.64228 |
BIAS | 0.01396 | 0.00840 | 0.01307 | 0.00840 | 0.01083 | 0.02063 | 0.03056 | 0.01594 | 0.01885 | 0.01196 | |
MSE | 0.00031 | 0.00011 | 0.00027 | 0.00011 | 0.00019 | 0.00065 | 0.00142 | 0.00040 | 0.00055 | 0.00023 | |
MRE | 0.02748 | 0.01396 | 0.02415 | 0.02111 | 0.02415 | 0.04969 | 0.05487 | 0.02800 | 0.01994 | 0.01866 |
n | Measure | RE | ExE | HCE | ArE | TsE | AA1E | AA2E | ShE | DEX | WEX |
(-0.24470) | (0.78294) | (-0.27803) | (-0.21706) | (-0.23032) | (-0.93457) | (1.43805) | (-0.42447) | (-0.96243) | (-0.20000) | ||
20 | \hat{E} | -0.22848 | 0.79765 | -0.25933 | -0.20235 | -0.21483 | -0.89872 | 1.37489 | -0.39453 | -0.93073 | -0.20048 |
BIAS | 0.06023 | 0.04755 | 0.06458 | 0.04755 | 0.05350 | 0.08863 | 0.16768 | 0.10001 | 0.12505 | 0.01564 | |
MSE | 0.00513 | 0.00318 | 0.00587 | 0.00318 | 0.00403 | 0.01248 | 0.04391 | 0.01414 | 0.02254 | 0.00041 | |
MRE | 0.24614 | 0.06073 | 0.23228 | 0.21907 | 0.23228 | 0.09483 | 0.11660 | 0.23562 | 0.12993 | 0.07820 | |
60 | \hat{E} | -0.22444 | 0.79973 | -0.25575 | -0.20027 | -0.21187 | -0.90532 | 1.38427 | -0.39031 | -0.92289 | -0.19737 |
BIAS | 0.03918 | 0.03122 | 0.04221 | 0.03122 | 0.03497 | 0.05784 | 0.10973 | 0.06527 | 0.08070 | 0.00931 | |
MSE | 0.00233 | 0.00150 | 0.00272 | 0.00150 | 0.00187 | 0.00543 | 0.01921 | 0.00653 | 0.00994 | 0.00013 | |
MRE | 0.16010 | 0.03987 | 0.15182 | 0.14382 | 0.15182 | 0.06189 | 0.07631 | 0.15377 | 0.08385 | 0.04657 | |
100 | \hat{E} | -0.22910 | 0.79586 | -0.26088 | -0.20414 | -0.21612 | -0.91192 | 1.39608 | -0.39810 | -0.93166 | -0.19806 |
BIAS | 0.03308 | 0.02624 | 0.03556 | 0.02624 | 0.02946 | 0.04549 | 0.08654 | 0.05470 | 0.06766 | 0.00776 | |
MSE | 0.00179 | 0.00113 | 0.00207 | 0.00113 | 0.00142 | 0.00351 | 0.01253 | 0.00486 | 0.00735 | 0.00010 | |
MRE | 0.13519 | 0.03351 | 0.12789 | 0.12087 | 0.12789 | 0.04868 | 0.06018 | 0.12887 | 0.07030 | 0.03881 | |
150 | \hat{E} | -0.23449 | 0.79145 | -0.26676 | -0.20855 | -0.22099 | -0.91935 | 1.40990 | -0.40710 | -0.94232 | -0.19880 |
BIAS | 0.02881 | 0.02270 | 0.03087 | 0.02270 | 0.02557 | 0.03868 | 0.07394 | 0.04755 | 0.05926 | 0.00679 | |
MSE | 0.00132 | 0.00082 | 0.00151 | 0.00082 | 0.00104 | 0.00245 | 0.00889 | 0.00358 | 0.00556 | 0.00007 | |
MRE | 0.11772 | 0.02900 | 0.11102 | 0.10459 | 0.11102 | 0.04138 | 0.05141 | 0.11203 | 0.06158 | 0.03394 | |
200 | \hat{E} | -0.23524 | 0.79074 | -0.26765 | -0.20926 | -0.22173 | -0.92085 | 1.41249 | -0.40845 | -0.94356 | -0.19891 |
BIAS | 0.02538 | 0.02001 | 0.02721 | 0.02001 | 0.02254 | 0.03402 | 0.06510 | 0.04188 | 0.05223 | 0.00608 | |
MSE | 0.00098 | 0.00061 | 0.00113 | 0.00061 | 0.00078 | 0.00184 | 0.00669 | 0.00268 | 0.00415 | 0.00006 | |
MRE | 0.10374 | 0.02556 | 0.09785 | 0.09220 | 0.09785 | 0.03640 | 0.04527 | 0.09866 | 0.05426 | 0.03038 | |
250 | \hat{E} | -0.23646 | 0.78972 | -0.26900 | -0.21028 | -0.22285 | -0.92295 | 1.41637 | -0.41058 | -0.94608 | -0.19896 |
BIAS | 0.02336 | 0.01841 | 0.02504 | 0.01841 | 0.02074 | 0.03051 | 0.05844 | 0.03856 | 0.04798 | 0.00550 | |
MSE | 0.00082 | 0.00051 | 0.00095 | 0.00051 | 0.00065 | 0.00149 | 0.00544 | 0.00224 | 0.00346 | 0.00005 | |
MRE | 0.09548 | 0.02352 | 0.09005 | 0.08483 | 0.09005 | 0.03265 | 0.04064 | 0.09083 | 0.04986 | 0.02752 | |
300 | \hat{E} | -0.23566 | 0.79028 | -0.26819 | -0.20972 | -0.22217 | -0.92243 | 1.41526 | -0.40936 | -0.94439 | -0.19879 |
BIAS | 0.02066 | 0.01630 | 0.02215 | 0.01630 | 0.01835 | 0.02713 | 0.05194 | 0.03399 | 0.04211 | 0.00489 | |
MSE | 0.00066 | 0.00041 | 0.00076 | 0.00041 | 0.00052 | 0.00124 | 0.00450 | 0.00181 | 0.00278 | 0.00004 | |
MRE | 0.08442 | 0.02082 | 0.07967 | 0.07511 | 0.07967 | 0.02903 | 0.03612 | 0.08008 | 0.04375 | 0.02447 | |
400 | \hat{E} | -0.23764 | 0.78865 | -0.27036 | -0.21135 | -0.22397 | -0.92502 | 1.42006 | -0.41266 | -0.94823 | -0.19911 |
BIAS | 0.01779 | 0.01402 | 0.01906 | 0.01402 | 0.01579 | 0.02303 | 0.04416 | 0.02926 | 0.03636 | 0.00412 | |
MSE | 0.00048 | 0.00030 | 0.00055 | 0.00030 | 0.00038 | 0.00085 | 0.00310 | 0.00129 | 0.00200 | 0.00003 | |
MRE | 0.07271 | 0.01790 | 0.06856 | 0.06457 | 0.06856 | 0.02464 | 0.03071 | 0.06893 | 0.03777 | 0.02061 |
Model | \hat{\delta} | SE( \hat{\delta} ) | \hat{\beta} | SE( \hat{\beta} ) |
PUIL | 2.4144 | 0.4321 | 0.0068 | 0.0072 |
UIL | 0.2045 | 0.0326 | ||
ETL | 1.7370 | 0.2896 | 9.7115 | 3.8780 |
Km | 1.5878 | 0.2444 | 21.8673 | 10.2082 |
Be | 3.1127 | 0.9368 | 21.8246 | 7.0422 |
TrG | 14.6813 | 2.3213 |
Model | \hat{\delta} | SE( \hat{\delta} ) | \hat{\beta} | SE( \hat{\beta} ) |
PUIL | 2.9709 | 0.5340 | 0.1091 | 0.0645 |
UIL | 0.9867 | 0.1666 | ||
ETL | 4.6858 | 0.9595 | 4.1306 | 1.5083 |
Km | 3.4039 | 0.6073 | 12.0731 | 5.4978 |
Be | 6.9757 | 2.1638 | 9.3522 | 2.9276 |
TrG | 3.4438 | 0.54452 |
Model | Aic | Caic | Bic | Hqic | A | W | KS | KSp | ShE | DEX | WEX |
PUIL | -71.5426 | -70.8367 | -69.5511 | -71.1538 | 0.4162 | 0.0500 | 0.1259 | 0.9092 | -1.9713 | -4.6321 | -0.4527 |
UIL | -57.9514 | -57.7292 | -56.9557 | -57.7570 | 2.6972 | 0.5309 | 0.3036 | 0.05012 | -0.9807 | -1.9026 | -0.1944 |
ETL | 48.2272 | -47.5213 | -46.2358 | -47.8385 | 2.6147 | 0.4524 | 0.2641 | 0.1229 | -1.2521 | -2.0067 | -0.2121 |
Km | -47.2969 | -46.5910 | -45.3054 | -46.9081 | 2.6889 | 0.4681 | 0.2627 | 0.1265 | -1.2290 | -1.9560 | -0.2031 |
Be | -51.7626 | -51.0567 | -49.7711 | -51.3738 | 2.2611 | 0.3727 | 0.2538 | 0.1521 | -1.3941 | -2.3467 | -0.2561 |
TrG | -51.8497 | -51.6275 | -50.8540 | -51.6553 | 2.5040 | 0.4327 | 0.2709 | 0.1062 | -1.2456 | -2.0362 | -0.2010 |
Model | Aic | Caic | Bic | Hqic | A | W | KS | KSp | ShE | DEX | WEX |
PUIL | -30.0341 | -29.3282 | -28.0427 | -29.6454 | 0.2522 | 0.0425 | 0.1226 | 0.9247 | -0.8583 | -1.4576 | -0.5632 |
UIL | -16.4854 | -16.2631 | -15.4896 | -16.2910 | 2.4384 | 0.4656 | 0.3009 | 0.0535 | -0.2021 | -0.6580 | -0.2946 |
ETL | -23.6156 | -22.9097 | -21.6241 | -23.2268 | 0.8845 | 0.1553 | 0.2110 | 0.3353 | -0.6532 | -1.0934 | -0.4659 |
Km | -22.0935 | -21.3876 | -20.1020 | -21.7047 | 1.0040 | 0.17602 | 0.2151 | 0.3132 | -0.6031 | -1.0336 | -0.4439 |
Be | -24.6329 | -23.9270 | -22.6414 | -24.2441 | 0.7991 | 0.1345 | 0.2038 | 0.3771 | -0.7158 | -1.1654 | -0.4923 |
TrG | -15.3907 | -15.1684 | -14.3949 | -15.1963 | 2.1343 | 0.3959 | 0.2905 | 0.0684 | -0.2401 | -0.6892 | -0.2587 |