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Research article

Power unit inverse Lindley distribution with different measures of uncertainty, estimation and applications

  • This paper introduced and investigated the power unit inverse Lindley distribution (PUILD), a novel two-parameter generalization of the famous unit inverse Lindley distribution. Among its notable functional properties, the corresponding probability density function can be unimodal, decreasing, increasing, or right-skewed. In addition, the hazard rate function can be increasing, U-shaped, or N-shaped. The PUILD thus takes advantage of these characteristics to gain flexibility in the analysis of unit data compared to the former unit inverse Lindley distribution, among others. From a theoretical point of view, many key measures were determined under closed-form expressions, including mode, quantiles, median, Bowley's skewness, Moor's kurtosis, coefficient of variation, index of dispersion, moments of various types, and Lorenz and Bonferroni curves. Some important measures of uncertainty were also calculated, mainly through the incomplete gamma function. In the statistical part, the estimation of the parameters involved was studied using fifteen different methods, including the maximum likelihood method. The invariant property of this approach was then used to efficiently estimate different uncertainty measures. Some simulation results were presented to support this claim. The significance of the PUILD underlying model compared to several current statistical models, including the unit inverse Lindley, exponentiated Topp-Leone, Kumaraswamy, and beta and transformed gamma models, was illustrated by two applications using real datasets.

    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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  • This paper introduced and investigated the power unit inverse Lindley distribution (PUILD), a novel two-parameter generalization of the famous unit inverse Lindley distribution. Among its notable functional properties, the corresponding probability density function can be unimodal, decreasing, increasing, or right-skewed. In addition, the hazard rate function can be increasing, U-shaped, or N-shaped. The PUILD thus takes advantage of these characteristics to gain flexibility in the analysis of unit data compared to the former unit inverse Lindley distribution, among others. From a theoretical point of view, many key measures were determined under closed-form expressions, including mode, quantiles, median, Bowley's skewness, Moor's kurtosis, coefficient of variation, index of dispersion, moments of various types, and Lorenz and Bonferroni curves. Some important measures of uncertainty were also calculated, mainly through the incomplete gamma function. In the statistical part, the estimation of the parameters involved was studied using fifteen different methods, including the maximum likelihood method. The invariant property of this approach was then used to efficiently estimate different uncertainty measures. Some simulation results were presented to support this claim. The significance of the PUILD underlying model compared to several current statistical models, including the unit inverse Lindley, exponentiated Topp-Leone, Kumaraswamy, and beta and transformed gamma models, was illustrated by two applications using real datasets.


    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)eβw,w>0,β>0, (1.1)

    with G(w;β)=0 for w0, and

    g(w;β)=β21+β(1+w)eβw,w>0,β>0, (1.2)

    with g(w;β)=0 for w0, 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)eβy,y>0,β>0, (1.3)

    with G(y;β)=0 for y0, and

    g(y;β)=β21+βy3(1+y)eβy,y>0,β>0, (1.4)

    with g(y;β)=0 for y0, 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)eβx,0<x1,β>0, (1.5)

    with G(x;β)=0 for x0 and G(x;β)=1 for x>1, and

    g(x;β)=β2eβ1+βx3eβx,0<x1,β>0, (1.6)

    with g(x;β)=0 for x0 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)eβzδ,0<z1,β,δ>0, (2.1)

    with G(z;β,δ)=0 for z0 and G(z;β,δ)=1 for z>1, and

    g(z;β,δ)=δβ2eβ1+βz2δ1eβzδ,0<z1,β,δ>0, (2.2)

    with g(z;β,δ)=0 for z0 or z>1, respectively. We can check that limz0g(z;β,δ)=0 and limz1g(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)=1eβ1+β(1+βzδ)eβzδ,
    h(z;β,δ)=δβ2eβz2δ1eβzδ1+βeβ(1+βzδ)eβzδ,
    τ(z;β,δ)=δβ2z2δ11+βzδ,

    and

    H(z;β,δ)=log[1eβ(1+β)(1+βzδ)eβzδ],

    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.

    Figure 1.  Plots of the PDF and HRF for the PUILD.

    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)eβ(1+βzδ)1eβzδ1]1,
    FRA(z;β,δ)=log[1eβ1+β(1+βzδ)eβzδ]z,

    and

    MR(z;β,δ)=1+βeβ(1+βzδ)eβzδδβ2eβz2δ1eβzδ,

    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 βδ2δ+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)=F1(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)eβ(Q(u;β,δ))δ=u,

    that provides

    (1+β(Q(u;β,δ))δ)eβ(Q(u;β,δ))δ=(1+β)eβu.

    Multiplying each side of the previous equation by e1 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 W1(.), we find that

    Q(u;β,δ)={1β1βW1[(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

    μr=E(Zr)=10zrg(z;β,δ)dz=δβ2eβ1+β10zr2δ1eβzδdz.

    If we apply the change of the variable v=βzδ, we get

    μr=βrδeβ1+ββv1rδevdv.

    Then, by introducing the upper incomplete gamma function Γ(u,t)=tzu1ezdz, with u>0 and t0, we get

    μr=βrδeβ1+βΓ(2rδ,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)=μ1=β1δeβ1+βΓ(21δ,β),

    and

    var(Z)=μ2μ12=β2δeβ1+βΓ(22δ,β)β2δe2β(1+β)2[Γ(21δ,β)]2.

    Similarly, the corresponding skewness is given as E{[ZE(Z)]3/[var(Z)]3/2}, the kurtosis is specified as E{[ZE(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.

    Figure 2.  3D plots of the most important moment measures for the PUILD, namely, from left to right, the mean, the variance, the skewness, the kurtosis, the CV, and the ID.

    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+βt0zω2δ1eβzδdz=βωδeβ1+βΓ(2ωδ,βtδ).

    By taking t=1 and ω=r, as expected, we refind ϱω(t)=μr.

    For any nonnegative integer r, the inverse rth moment of Z can be calculated as follows:

    μr=E(Zr)=10zrg(z;β,δ)dz=δβ2eβ1+β10zr2δ1eβzδdz.

    Again, by applying v=βzδ, we obtain

    μr=βrδeβ1+ββv1+rδevdv.

    Then, using the incomplete gamma function, we find that

    μr=βrδeβ1+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=β1δeβ1+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(21δ,βtδ)Γ(21δ,β),

    and

    BON=LORG(t;β,δ)=(1+β)Γ(21δ,βtδ)eβtδΓ(21δ,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)eβlog(δβ2eβ1+β)(2+1δ)[(β+1)eβlog(β)Γ(2,β)2]Γ(3,β)],

    where Γn(.,.) denotes the nth derivative of the incomplete gamma function, that is, Γn(u,t)=tzu1(log(z))nezdz, with u>0 and t0.

    Proof. Thanks to Eq (2.2), we have

    Q(β,δ)=10g(z;β,δ)log[g(z;β,δ)]dz=δβ2eβ1+β10z2δ1eβzδlog[δβ2eβ1+βz2δ1eβzδ]dz.

    Then, we have

    Q(β,δ)=δβ2eβ1+β[I1(2δ+1)I2βI3], (4.1)

    where

    I1=log(δβ2eβ1+β)10z2δ1eβzδdz,I2=10z2δ1log(z)eβzδdz,

    and

    I3=10z3δ1eβzδdz.

    For I1, by the change of variables v=βzδ, we have

    I1=1δβ2log(δβ2eβ1+β)βvevdv=(1+β)eβδβ2log(δβ2eβ1+β)

    and

    I2=1δβ2βvlog(β1δv1δ)evdv=1δ2β2βv[log(β)log(v)]evdv,

    which implies that

    I2=1δ2β2[(β+1)eβlog(β)Γ(2,β)2].

    Also, with the same technique, we get

    I3=1β3δβv2evdv=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)eβlog(δβ2eβ1+β)(2+1δ)[(β+1)eβlog(β)Γ(2,β)2]Γ(3,β)].

    This ends the proof of Proposition 1.

    Proposition 2. Let κ>0, κ1, 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+1κδΓ(2κ+κ1δ,κβ).

    Proof. Owing to Eq (2.2), we have

    Iκ(β,δ)=10g(z;β,δ)κdz=(δβ2eβ1+β)κ10z2κδκeκβzδdz.

    By performing the change of variables v=κβzδ, we get

    Iκ(β,δ)=1δ(δβ2eβ1+β)κ(κβ)2κ+1κδκβv2κ+κ1δ1evdv,

    which implies that

    Iκ(β,δ)=1δ(δβ2eβ1+β)κ(κβ)2κ+1κδΓ(2κ+κ1δ,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 κ1 are basic assumptions.

    Table 1.  Important entropy measures of a distribution with PDF g(z;β,δ) at κ.
    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)=11κlog[10g(z;β,δ)κdz]
    Exponential [46] Eκ(β,δ)=[10g(z;β,δ)κdz]11κ
    Havrda and Charvat [47] HCκ(β,δ)=121κ1[10g(z;β,δ)κdz1]
    Arimoto [48] Aκ(β,δ)=κ1κ{[10g(z;β,δ)κdz]1κ1}
    Tsallis [49] Tκ(β,δ)=1κ1[110g(z;β,δ)κdz]
    Awad and Alawneh 1 [50] AA1κ(β,δ)=1κ1log{[supzRg(z;β,δ)]1κ10g(z;β,δ)κdz}
    Awad and Alawneh 2 [50] AA2κ(β,δ)=121κ1[{[supzRg(z;β,δ)]1κ10g(z;β,δ)κdz}1]

     | Show Table
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    It is assumed that supzRg(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)eβlog(β)Γ(2,β)2]+Γ(3,β)(1+β)eβlog(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)=11κlog[Iκ(β,δ)]=11κlog{1δ(δβ2eβ1+β)κ(κβ)2κ+1κδΓ(2κ+κ1δ,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)]11κ={1δ(δβ2eβ1+β)κ(κβ)2κ+1κδΓ(2κ+κ1δ,κβ)}11κ.

    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)=121κ1[Iκ(β,δ)1]=121κ1[1δ(δβ2eβ1+β)κ(κβ)2κ+1κδΓ(2κ+κ1δ,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)=κ1κ[Iκ(β,δ)1κ1]=κ1κ{[1δ(δβ2eβ1+β)κ(κβ)2κ+1κδΓ(2κ+κ1δ,κβ)]1κ1}.

    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)=1κ1[1Iκ(β,δ)]=1κ1[11δ(δβ2eβ1+β)κ(κβ)2κ+1κδΓ(2κ+κ1δ,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)=1κ1log{[sup0<z<1g(z;β,δ)]1κIκ(β,δ)}. (4.2)

    Before going any further, we need to find sup0<z1g(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<z1g(z;β,δ)=g(z;β,δ)=(δeβ1δ+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)=1κ1log{[s(β,δ)]1κ1δ(δβ2eβ1+β)κΓ(2κ+κ1δ,κβ)(κβ)2κ1κδ}.

    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=121κ1[{[sup0<z<1g(z;β,δ)]1κIκ(β,δ)}1]=121κ1[{[s(β,δ)]1κ1δ(δβ2eβ1+β)κΓ(2κ+κ1δ,κβ)(κβ)2κ1κδ}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=1210g(z;β,δ)2dz. (4.5)

    By inserting Eq (2.2) into Eq (4.5), we obtain

    Φ=12[δ2β4e2β(1+β)210z4δ2e2βzδdz].

    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=1210zg(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)+nβnlog(1+β)(1+2δ)ni=1log(zi)ni=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

    logLβ=n+2nβn1+βni=11ziδ, (5.2)

    and

    logLδ=nδ2ni=1log(zi)+βni=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=n1nni=1(2i1){log[G(zi:n,β,δ)]+log[S(zni1:n,β,δ)]}=n1nni=1(2i1){log[eβ1+β(1+βzδi:n)eβzδi:n]+log[1eβ1+β(1+βzδni1:n)eβzδni1:n]}.

    The Cramér-von Mises estimates (CVMEs) (^β3,^δ3) are determined by minimizing the following function:

    C=112n+ni=1[G(zi:n,β,δ)2i12n]2=112n+ni=1[eβ1+β(1+βzδi:n)eβzδi:n2i12n]2.

    The method of maximum product of spacings estimates (MPSEs) (^β4,^δ4) are obtained by maximizing the following function:

    MPS=1n+1n+1i=1log(ξi,n),

    where

    ξi,n=G(zi:n,β,δ)G(zi1:n,β,δ)=eβ1+β(1+βzδi:n)eβzδi:neβ1+β(1+βzδi1:n)eβzδi1: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=ni=1[G(zi:n,β,δ)in+1]2=ni=1[eβ1+β(1+βzδi:n)eβzδi:nin+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 26. Furthermore, the partial and total ranks for the estimates are given in Table 7.

    Table 2.  Numerical values of simulation measures for \delta = 0.7 and \beta = 2.5 .
    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 \}}

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    Table 3.  Numerical values of simulation measures for \delta = 0.25 and \beta = 0.75 .
    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 \}}

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    Table 4.  Numerical values of simulation measures for \delta = 1.5 and \beta = 1.5 .
    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 \}}

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    Table 5.  Numerical values of simulation measures for \delta = 0.5 and \beta = 2.0 .
    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 \}}

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    Table 6.  Numerical values of simulation measures for \delta = 2.5 and \beta = 0.4 .
    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 \}}

     | Show Table
    DownLoad: CSV
    Table 7.  Partial and overall ranks of all the methods of estimation for various values of the parameters.
    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

     | Show Table
    DownLoad: CSV

    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 311.

    Figure 3.  Graphical representations for the BIAS values of \hat{\delta} presented in Table 2.
    Figure 4.  Graphical representations for the BIAS values of \hat{\beta} presented in Table 2.
    Figure 5.  Graphical representations for the MSE values of \hat{\delta} presented in Table 2.
    Figure 6.  Graphical representations for the MSE values of \hat{\beta} presented in Table 2.
    Figure 7.  Graphical representations for the MRE values of \hat{\delta} presented in Table 2.
    Figure 8.  Graphical representations for the MRE values of \hat{\beta} presented in Table 2.
    Figure 9.  Graphical representations for the D_{abs} values presented in Table 2.
    Figure 10.  Graphical representations for the D_{max} values presented in Table 2.
    Figure 11.  Comparison between the D_{abs} and D_{max} values presented in Table 2.

    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 811. 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:

    Table 8.  Numerical values of the different entropy measures and their BIAS, MSE, and MRE for \delta = 1.5 and \beta = 2.0 .
    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

     | Show Table
    DownLoad: CSV
    Table 9.  Numerical values of the different entropy measures and their BIAS, MSE, and MRE for \delta = 0.5 and \beta = 1.5 .
    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

     | Show Table
    DownLoad: CSV
    Table 10.  Numerical values of the different entropy measures and their BIAS, MSE, and MRE for \delta = 2.5 and \beta = 0.75 .
    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

     | Show Table
    DownLoad: CSV
    Table 11.  Numerical values of the different entropy measures and their BIAS, MSE, and MRE for \delta = 0.9 and \beta = 0.5 .
    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

     | Show Table
    DownLoad: CSV

    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 811 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 1214.

    Figure 12.  Graphical representation for the BIAS values presented in Table 8.
    Figure 13.  Graphical representation for the MSE values presented in Table 8.
    Figure 14.  Graphical representation for the MRE values presented in Table 8.

    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.

    Figure 15.  Some nonparametric plots for the first dataset.
    Figure 16.  Some nonparametric plots for the second dataset.

    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.

    Table 12.  Estimated parameters with the SEs of all the compared models for the first dataset.
    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

     | Show Table
    DownLoad: CSV
    Table 13.  Estimated parameters with the SEs of all the compared models for the second dataset.
    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

     | Show Table
    DownLoad: CSV

    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.

    Table 14.  Numerical values for analyzing the first dataset.
    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

     | Show Table
    DownLoad: CSV
    Table 15.  Numerical values for analyzing the second dataset.
    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

     | Show Table
    DownLoad: CSV

    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.

    Figure 17.  Estimated CDFs and PDFs for the first dataset.
    Figure 18.  Estimated CDFs and PDFs for the second dataset.

    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.

    Figure 19.  Profile plots of the L-LF for the first dataset.
    Figure 20.  Profile plots of the L-LF for the second dataset.

    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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