
In this paper, we consider the following periodic discrete nonlinear Schrödinger equation
Lun−ωun=gn(un),n=(n1,n2,...,nm)∈Zm,
where ω∉σ(L)(the spectrum of L) and gn(s) is super or asymptotically linear as |s|→∞. Under weaker conditions on gn, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.
Citation: Xionghui Xu, Jijiang Sun. Ground state solutions for periodic Discrete nonlinear Schrödinger equations[J]. AIMS Mathematics, 2021, 6(12): 13057-13071. doi: 10.3934/math.2021755
[1] | M. G. M. Ghazal . Modified Chen distribution: Properties, estimation, and applications in reliability analysis. AIMS Mathematics, 2024, 9(12): 34906-34946. doi: 10.3934/math.20241662 |
[2] | Refah Alotaibi, Hassan Okasha, Hoda Rezk, Abdullah M. Almarashi, Mazen Nassar . On a new flexible Lomax distribution: statistical properties and estimation procedures with applications to engineering and medical data. AIMS Mathematics, 2021, 6(12): 13976-13999. doi: 10.3934/math.2021808 |
[3] | Hisham Mahran, Mahmoud M. Mansour, Enayat M. Abd Elrazik, Ahmed Z. Afify . A new one-parameter flexible family with variable failure rate shapes: Properties, inference, and real-life applications. AIMS Mathematics, 2024, 9(5): 11910-11940. doi: 10.3934/math.2024582 |
[4] | M. G. M. Ghazal, Yusra A. Tashkandy, Oluwafemi Samson Balogun, M. E. Bakr . Exponentiated extended extreme value distribution: Properties, estimation, and applications in applied fields. AIMS Mathematics, 2024, 9(7): 17634-17656. doi: 10.3934/math.2024857 |
[5] | M. Nagy, H. M. Barakat, M. A. Alawady, I. A. Husseiny, A. F. Alrasheedi, T. S. Taher, A. H. Mansi, M. O. Mohamed . Inference and other aspects for q−Weibull distribution via generalized order statistics with applications to medical datasets. AIMS Mathematics, 2024, 9(4): 8311-8338. doi: 10.3934/math.2024404 |
[6] | A. M. Abd El-Raheem, Ehab M. Almetwally, M. S. Mohamed, E. H. Hafez . Accelerated life tests for modified Kies exponential lifetime distribution: binomial removal, transformers turn insulation application and numerical results. AIMS Mathematics, 2021, 6(5): 5222-5255. doi: 10.3934/math.2021310 |
[7] | Qasim Ramzan, Muhammad Amin, Ahmed Elhassanein, Muhammad Ikram . The extended generalized inverted Kumaraswamy Weibull distribution: Properties and applications. AIMS Mathematics, 2021, 6(9): 9955-9980. doi: 10.3934/math.2021579 |
[8] | Ayed. R. A. Alanzi, M. Qaisar Rafique, M. H. Tahir, Farrukh Jamal, M. Adnan Hussain, Waqas Sami . A novel Muth generalized family of distributions: Properties and applications to quality control. AIMS Mathematics, 2023, 8(3): 6559-6580. doi: 10.3934/math.2023331 |
[9] | Ekramy A. Hussein, Hassan M. Aljohani, Ahmed Z. Afify . The extended Weibull–Fréchet distribution: properties, inference, and applications in medicine and engineering. AIMS Mathematics, 2022, 7(1): 225-246. doi: 10.3934/math.2022014 |
[10] | Jumanah Ahmed Darwish, Saman Hanif Shahbaz, Lutfiah Ismail Al-Turk, Muhammad Qaiser Shahbaz . Some bivariate and multivariate families of distributions: Theory, inference and application. AIMS Mathematics, 2022, 7(8): 15584-15611. doi: 10.3934/math.2022854 |
In this paper, we consider the following periodic discrete nonlinear Schrödinger equation
Lun−ωun=gn(un),n=(n1,n2,...,nm)∈Zm,
where ω∉σ(L)(the spectrum of L) and gn(s) is super or asymptotically linear as |s|→∞. Under weaker conditions on gn, the existence of ground state solitons is proved via the generalized linking theorem developed by Li and Szulkin and concentration-compactness principle. Our result sharply extends and improves some existing ones in the literature.
The classical probability distributions were generalized through the induction of location, scale and shape parameters. Recently, appreciable attempts have been made in the development of the new probability distributions which have big privilege of more flexibility, fitting specific and several real world sequence of events.
The improvement in the G-Classes revolution began with the fundamental article of Alzaatreh et al. [9] in which they proposed transformed (T)-transformer (X) (T-X) family.
Consider the random variable (rv) T∈[c,d] for −∞≤c<d≤∞ with probability density function (pdf) r(t), and consider a link function, W(⋅):[0,1]→R, that satisfy the conditions: For any baseline cumulative distribution function (cdf) G(x), W[G(x)]∈[c,d], is monotonically non-decreasing and differentiable, for x→−∞, W[G(x)]→c, and for x→∞, W[G(x)]→d. Hence, the cdf of the T-X class has the form
F(x)=∫W[G(x)]ar(t)dt. | (1.1) |
Many authors constructed extended generalized families by using T-X approach. Some examples of generalized classes are, beta-G [19], Kw-G type-1 [17], log-gamma-G type-2 [10], gamma-X [40], exponentiated T-X [9], Weibull-G [11], exponentiated-Weibull-H [13] and generalized odd Lindley-G [3].
Motivated by the new prospect in term of accuracy and exibility of a new distribution, we propose a new propose a new flexible family called, log-logistic tan generalized (LLT-G) family which provides greater accuracy and flexibility in fitting real-life data. Some general properties of the LLT-G class will be provided here. We provide two applications for one special sub-model of the proposed class, called log-logistic tan-Weibull (LLT-W) distribution which has decreasing, increasing, bathtub and unimodal hazard rate functions.
Let r(t)=cs(ts)c−1[1+(ts)C]−2 be the pdf of a rv0<t<∞ and W[G(x)]=tan[π2Gα(x)] which satisfies the conditions of T-X family. The cdf and pdf of the new LLT-G family take the forms
F(x)=∫tan[π2Gα(x)]0r(t)dt=1−(1+s−c{tan[π2Gα(x)]}c)−1,x>0,α,c,s>0 | (1.2) |
and
f(x)=παc2scg(x)[G(x)]α−1{sec2[π2Gα(x)]}{tan[π2Gα(x)]}c−1×(1+s−c{tan[π2Gα(x)]}c)−2,x>0,α,c,s>0. | (1.3) |
The LLT-G hazard rate function (hrf) has the form
h(x)=παc2scg(x)[G(x)]α−1[sec2(π2Gα(x))]{tan[π2Gα(x)]}c−1×(1+s−c{tan[π2Gα(x)]}c)−1,x>0,α,c,s>0. | (1.4) |
Henceforth, the rv with PDF (1.3) is denoted by X∼LLT-G(α,c,s).
The LLT-G class has some desirable properties including the following.
(ⅰ) The special sub-models of the LLT-G class can provide left skewed, symmetrical, right skewed, unimodal, bimodal and reversed-J densities, and increasing, modified bathtub, decreasing, bathtub, upside-down bathtub, reversed-J shaped, and J-shaped hazard rates. Hence, their special sub-models are capable of fitting different shapes of failure criteria.
(ⅱ) The LLT-G class provides extended versions of baseline distributions with closed forms for the cdf and hrf. Hence the special sub-models of this family can be used in modeling and analyzing censored data.
(ⅲ) The LLT-Weibull as a special sub-model of LLT-G class provides adequate fits than other modified models generated by other existing families under same baseline model.
Recently, several authors have proposed many generalized classes based on the logistic and log logistic distributions. For example, Afify et al. [4], Alizadeh et al. [5], Alizadeh et al. [6], Alizadeh et al. [7], Altun et al. [8], Cordeiro et al. [14], Cordeiro et al. [15], Cordeiro et al. [16], Gleaton and Lynch [20], Mehdi et al. [21], Hassan et al [26], Haghbin et al. [27], Korkmaz et al. [28], Mahadevi et al. [29] and Torabi and Montazeri [40].
This paper is outlined as follows. Five special models of the LLT-G family are provided in Section 2. In Section 3, we derived some mathematical properties of LLT-G family. Estimation of the LLT-G parameters using maximum likelihood is discussed in Section 4. Further, we present simulations for the LLT-W model to address the performance of the proposed estimators in Section 4. In Section 5, we show the importance and applicability of the LLT-G family via two real-life data applications. In Section 6, the paper is summarized.
We study five sub-models of the LLT-G class using the baseline Weibull, normal, Rayleigh, exponential and Burr XII distributions (1.3) and provide some plots for their pdfs and hrfs. Figures 1–5 reveals that the special sub-models of the LLT-G class can provide left skewed, symmetrical, right skewed, unimodal, bimodal and reversed-J densities, and increasing, modified bathtub, decreasing, bathtub, unimodal, reversed-J shaped, and J-shaped failure rates.
Using the Weibull pdf, g(x)=bxb−1e−xb, x>0,b>0, we obtain the cdf of the LLT-W distribution
F(x)=1−(1+s−c{tan[π2(1−e−xb)α]}c)−1. | (2.1) |
The pdf associated with (2.1) reduces to
f(x)=παbc2scxb−1e−xb(1−e−xb)α−1{sec2[π2(1−e−xb)α]}×{tan[π2(1−e−xb)α]}c−1(1+s−c{tan[π2(1−e−xb)α]}c)−2. | (2.2) |
The plots in Figure 1 depict pdf and hrf shapes of the LLT-W distribution for different parametric values.
The pdf of normal distribution is g(x)=1σ√2πe−x22σ2, x∈ℜ,σ>0. The cdf of the LLT-N reduces to
F(x)=1−{1+s−c[tan(π2{12[1+erf(xσ√2)]}α)]c}−1. | (2.3) |
The LLT-N pdf takes the form
f(x)=παc2scσ√2πe−x22σ2{12[1+erf(xσ√2)]}α−1[sec2(π2{12[1+erf(xσ√2)]}α)]×[tan(π2{12[1+erf(xσ√2)]}α)]c−1{1+s−c[tan(π2{12[1+erf(xσ√2)]}α)]c}−2. | (2.4) |
Some plots of the density and hazard rate functions of the LLT-N distribution for several parametric values are shown in Figure 2.
Using the Rayleigh rv with pdf, g(x)=2axe−ax2, x>0,a>0, the cdf of the LLT-R distribution becomes
F(x)=1−(1+s−c{tan[π2(1−e−ax2)α]}c)−1. | (2.5) |
The pdf of the LLT-R reduces to
f(x)=παcascxe−ax2(1−e−ax2)α−1{sec2[π2(1−e−ax2)α]}×{tan[π2(1−e−ax2)α]}c−1(1+s−c{tan[π2(1−e−ax2)α]}c)−2. | (2.6) |
Figure 3 depicts possible plots of the LLT-R pdf and hrf for various parametric values.
Consider the exponential distribution with pdf g(x)=be−bx, x>0,b>0. Hence, the cdf and pdf of LLT-E distribution are
F(x)=1−(1+s−c{tan[π2(1−e−bx)α]}c)−1 | (2.7) |
and
f(x)=παbc2sce−bx(1−e−bx)α−1{sec2[π2(1−e−bx)α]}×{tan[π2(1−e−bx)α]}c−1(1+s−c{tan[π2(1−e−bx)α]}c)−2. | (2.8) |
Figure 4 depicts pdf and hrf plots of the LLT-E distribution for different parametric values.
Let X be a Burr XII rv with pdf g(x;a,b)=abxa−1(1+xa)−b−1, x>0,a,b>0.
The cdf of the LLT-BXII distribution is
F(x)=1−[1+s−c(tan{π2[1−(1+xa)−b]α})c]−1. | (2.9) |
The pdf of the LLT-BXII reduces to
f(x)=παabc2scxa−1(1+xa)−b−1[1−(1+xa)−b]α−1(sec2{π2[1−(1+xa)−b]α})×(tan{π2[1−(1+xa)−b]α})c−1[1+s−c(tan{π2[1−(1+xa)−b]α})c]−2. | (2.10) |
Figure 5 displays density and hazard rate plots of the LLT-BXII distribution.
In this section, the LLT-G properties such as quantile function (qf), useful expansion, moments, generating function, and order statistics are derived. The expressions derived for the LLT-G family can be handled using symbolic computation software, such as, Maple, Matlab, Mathematica, Mathcad, and R because of their ability to deal with complex and formidable size mathematical expressions. Established explicit formulae to evaluate statistical and mathematical measures can be more efficient than computing them directly by numerical integration. It is noted that the infinity limit in the sums of these expressions can be substituted by a large positive integer such as 40 or 50 for most practical purposes.
The qf of X is calculated directly by inverting (2.1) as
Q(u)=G−1{2πtan−1[s(u1−u)1c]}1α, | (3.1) |
Eq (3.1) can be used to simulate any baseline model and to obtain the median = Q(1/2), Bowley's skewness and Moors kurtosis.
The formation of the pdf and cdf in power series is an important concept.
Some essential properties of the exponentiated-G (exp-G) distributions are studied by [23,24,25,31,32,33,34,35].
The cdf of the LLT-G family can be written as
F(x)=1−(1+s−c{tan[π2Gα(x)]}c)−1,x>0,α,c,s>0. | (3.2) |
Using the binomial expansion
(1+x)−1=∞∑i=1(−1)ixi,x≤1 |
and the convergent series of tan function which can be calculated using the Mathematica software.
[tan(πG(x)2)]ic=∞∑j=0[bj(ic)](πG(x)2)α(2j+ic),G(x)≤1,x>0. |
Applying the last two equations to (3.2), the cdf of LLT-G reduces to
F(x)=∞∑i=1∞∑j=0(−1)(i+1)sic[bj(ic)](π2)(2j+ic)[G(x)]α(2j+ic),x>0. | (3.3) |
We can rewrite the Eq (3.2) as
F(x)=∞∑i=1∞∑j=0Wi,jHα(2j+ic)(x), | (3.4) |
where
Wi,j=(−1)(i+1)sicbj(ic)(π2)(2j+ic). | (3.5) |
The below power series can be calculated using the Mathematica software
[tan(πG(x)2)]ic=∞∑j=0bj(ic)(πG(x)2)α(2j+ic), | (3.6) |
where b0(ic)=1,b1(ic)=ic3,b2(ic)= , etc. and Hα(2j+ic)(x)=[G(x)]α(2j+ic) denotes the cdf of exp-G model with parameter α(2j+ic). So, the LLT-G pdf reduces to
f(x)=∞∑i=1∞∑j=0wi,jhα(2j+ic)(x), | (3.7) |
where hα(2j+ic)(x) denotes the exp-G pdf with parameter α(2j+ic). Hereafter, a rv having the pdf hα(2j+ic)(x) is denoted by Yα(2j+i)∼exp-Gα(2j+ic). Eq (3.7) delivers up that the LLT-G pdf is a linear combination of exp-G densities. Thus, a few properties of LLT-G class can be calculated someplace precisely from exp-G properties.
Assuming that Z is a rv with a baseline G(x). The moments of X can be derived from the (r,k)th probability weighted moment (PWM) of Z defined by [22] as
τr,k=E[G(Z)kZr]=∫∞−∞xrg(x)G(x)kdx. | (3.8) |
Using Eq (3.7), one can write
μ′r=E(Xr)=∞∑i=1∞∑j=0wi,jhα(2j+ic)(x)τr,α(2j+ic), | (3.9) |
where τr,α(2j+ic)=∫10QG(u)ruα(2j+ic)du which is computed numerically for any baseline qf.
The measures of skewness, β1, and kurtosis, β2, can be defined by the following two formulae
β1=μ′3+2μ′13−3μ′2μ′1(μ′2−μ′12)32 | (3.10) |
and
β2=μ′4−4μ′1μ′3+6μ′12μ′3−3μ′14μ′2−μ′12 | (3.11) |
The plots of skewness and kurtosis for the LLT-W distribution are visualized in Figure 6.
We present two formulae for the mgf, M(s)=E(esX), of the rvX.
The first one comes from Eq (3.7) as
M(s)=∞∑i=1∞∑j=0wi,jMα(2j+ic)(s), | (3.12) |
where Mα(2j+ic)(s) is the exp-G mgf with power parameter α(2j+ic).
Further, Eq (3.12) can also take the form
M(s)=∞∑i=1∞∑j=0α(2j+ic)wi,jρα(2j+ic)(s), | (3.13) |
where the quantity ρ2j+iβ(s)=∫10exp[sQG(u)]uα(2j+ic)du is calculated numerically.
Incomplete moments are beneficial in calculating some inequality measures and mean deviations. The nth incomplete moments of the LLT-G family has the form
mn(y)=∞∑i=1∞∑j=0wi,jα(2j+ic)∫G(y;ξξ)0QG(u)nuα(2j+ic)du. | (3.14) |
For most baseline distributions, the above (3.14) can be obtained numerically.
The first incomplete moment, m1(z), can be calculated from Eq (3.7) as
m1(z)=∞∑i=1∞∑j=0wi,jJα(2j+ic)(z), | (3.15) |
where
Jα(2j+ic)(z)=∫z−∞xhα(2j+ic)(x)dx. | (3.16) |
Eq (3.15) is the primary quantity to obtain the mean deviations. We apply (3.15) to the LLT-W model. The LLT-W pdf with power parameter α(2j+ic), reduces to
hα(2j+ic)(x)=α(2j+ic)g(x)[G(x)]α(2j+ic)−1, | (3.17) |
and then
Jα(2j+ic)(x)=∞∑i=1∞∑j=0(−1)(i+1)sicbj(ic)(π2)α(2j+ic)∫z0xα(2j+ic)g(x)[G(x)]α(2j+ic)−1dx. | (3.18) |
Another formula for m1(z) follows, Eq (3.7) with u=G(x), as
m1(z)=∞∑i=1∞∑j=0α(2j+ic)wi,jTα(2j+ic)(z), | (3.19) |
where Tα(2j+ic)(z)=∫G(z)0QG(u)uα(2j+ic)du.
Consider a random sample from the LLT-G class, X1,…,Xn. The pdf of the ith order statistic, Xi:n, has the form
fi:n(x)=Dn−i∑p=0(−1)p(n−ip)F(x)p+i−1f(x), | (3.20) |
where D=n!/[(i−1)!(n−i)!].
After some algebra, the pdf of Xi:n takes the form
fi:n(x)=∞∑m,r,t=0dm,r,thα(2(r+t)+c(m+1))(x), | (3.21) |
where the exp-G density, hα(2(r+t)+c(m+1))(x), has a power parameter α(2(r+t)+c(m+1))
dm,r,t=∞∑j,l=0k(−1)j+lsc(m+1)(n−ij)(i+j−1l)(−(l+2)m)br[c(m+1)−1]ct(2)(π2)2(r+t)+c(m+1) |
where br[c(m+1)−1] and ct(2) are the coefficients of the power series expansion of tan and sec functions by using Mathematica.
Eq (3.21) can be used to calculate some quantities of the LLT-G order statistics from the exp-G quantities.
We will discuss the estimation of the LLT-G parameters using the maximum likelihood (MLE) and study the performance of these estimators via simulations.
Consider a random sample from the LLT-G class, x1,…,xn, with Θ=(α,s,c,ξ)⊤. Then, the log-likelihood of Θ reduces to
ℓn(Θ)=nlog(π2)+nlog(αc)−nlog(s)+n∑i=1logG(xi;ξ)+n∑i=1log[sec2(Ai)]+(α−1)n∑i=1log[G(xi;ξ)]+(c−1)n∑i=1log[1stan(Ai)]−2n∑i=1log{1+[1stan(Ai)]c}, | (4.1) |
where Ai=π2Gα(xi;ξ).
The above log-likelihood (4.1) is maximized simply by statistical programs namely, R, Mathcad, SAS, Mathematica and Ox programs. It also is maximized by solving the following likelihood equations.
The score vector elements, Un(Θ)=(∂ℓn/∂α,∂ℓn/∂s,∂ℓn/∂c,∂ℓn/∂ξ)⊤, take the following formulae
∂ℓn∂α=∞∑i=1log[G(xi)]−πcs∞∑i=1[sec2(Ai)][1stan(Ai)]c−1{Gα(xi)log[G(xi)]}{[1stan(Ai)]c+1}+(c−1)∞∑i=1π2[csc(Ai)][sec(Ai)]{Gα(xi)log[G(xi)]}+∞∑i=1π[sec2(Ai)][tan(Ai)]{Gα(xi)log[G(xi)]}, | (4.2) |
∂ℓn∂s=(1−c)∞∑i=11s−ns−2cs2∞∑i=1−[tan(Ai)][1stan(Ai)]c−1{[1stan(Ai)]c+1}, | (4.3) |
∂ℓn∂c=nc+∞∑i=1log[1stan(Ai)]−2∞∑i=1[1stan(Ai)]clog[1stan(Ai)]{[1stan(Ai)]c+1} | (4.4) |
and
∂ℓn∂ξ=n∑i=1g(xi;ξ)(ξ)g(xi;ξ)+(α−1)n∑i=1G(xi;ξ)(ξ)G(xi;ξ)+πn∑i=1[1stan(Ai)]G(xi;ξ)(αξ)−πs∞∑i=1[cstan(Ai)]c−1[sec2(Ai)]{[1stan(Ai)]c+1}G(xi;ξ)(αξ)+π2(c−1)∞∑i=1sec2(Ai)tan(Ai)G(xi;ξ)(αξ), | (4.5) |
where g(ξ)(⋅)=∂g/∂ξ.
This section deals with checking the MLEs performance in estimating the LLT-W parameters using a simulation study. For n=50,100,300 and 500, we generated 1,000 samples from the LLT-W model for various parametric values using the inversion method via the qf of the LLT-W given by
Q(u)=[−log(1−{2πtan−1[s(u1−u)1c]}1α)]1b. | (4.6) |
The numerical results of the mean square error (MSE), bias, coverage probability (C.P) and average width (AW) of the MLEs of the model parameters are obtained using R software. Further, the graphical representation of bias, MSE and C.P on several parametric values are also provided. The numerical results for bias, MSE, C.P and AW are listed in Tables 1 and 2. The bias, MSE and C.P are depicted in Figures 7–9. The values in Tables 1 and 2, and shapes of Figures 7–9 show quite stable estimates for the considered sample sizes. Moreover, the MSEs decrease as n increases, proving the consistency of maximum likelihood estimators.
MLE(α=0.20,s=0.50,c=1.30,b=1.50) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.02632 | 0.00861 | 0.99023 | 2.36578 |
s | 0.18709 | 0.28943 | 0.94067 | 2.19272 | |
c | 0.17656 | 0.20302 | 0.94821 | 0.44550 | |
b | -0.16017 | 0.18987 | 0.93041 | 1.56365 | |
100 | α | 0.01978 | 0.00444 | 1.00022 | 1.25564 |
s | 0.09531 | 0.07086 | 0.96764 | 1.40323 | |
c | 0.11487 | 0.08334 | 0.93042 | 0.29029 | |
b | -0.15492 | 0.09865 | 0.98562 | 1.49110 | |
300 | α | 0.01298 | 0.00175 | 1.00276 | 0.65864 |
s | 0.03895 | 0.01812 | 0.98054 | 0.758178 | |
c | 0.05497 | 0.02252 | 0.97097 | 0.15874 | |
b | -0.10343 | 0.03767 | 0.9819 | 1.47684 | |
500 | α | 0.00787 | 0.00156 | 1.00876 | 0.48943 |
s | 0.01798 | 0.01028 | 0.99098 | 0.56132 | |
c | 0.02509 | 0.01161 | 0.98987 | 0.11954 | |
b | -0.05684 | 0.01249 | 0.99076 | 1.50318 | |
MLE(α=0.80,s=1.10,c=1.60,b=1.80) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.17012 | 0.34589 | 0.88320 | 5.73426 |
s | 0.04296 | 1.17838 | 0.82209 | 3.51921 | |
c | -0.19793 | 0.70917 | 0.84190 | 2.01749 | |
b | 0.70237 | 1.92221 | 0.84084 | 3.51065 | |
100 | α | 0.06832 | 0.10920 | 0.94890 | 3.29742 |
s | 0.04132 | 0.54720 | 0.90908 | 2.76689 | |
c | -0.05829 | 0.45318 | 0.9593 | 1.25046 | |
b | 0.29772 | 0.66458 | 0.9032 | 2.72295 | |
300 | α | 0.01620 | 0.02809 | 0.99870 | 1.46536 |
s | 0.03836 | 0.16018 | 0.95430 | 1.62928 | |
c | -0.01011 | 0.19239 | 0.9673 | 0.65048 | |
b | 0.11101 | 0.18638 | 0.95650 | 2.23627 | |
500 | α | 0.00503 | 0.01737 | 0.99201 | 1.02895 |
s | -0.00776 | 0.07429 | 0.96807 | 1.24003 | |
c | -0.03110 | 0.10385 | 0.97956 | 0.49509 | |
b | 0.07738 | 0.09929 | 0.9982 | 2.12406 |
MLE(α=1.20,s=1.50,c=2.00,b=2.50) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.55202 | 1.38710 | 0.76780 | 7.32437 |
s | 0.05138 | 1.54372 | 0.8452 | 6.13638 | |
c | -0.40001 | 1.29038 | 0.96980 | 4.15307 | |
b | 0.96352 | 2.67448 | 0.76980 | 5.539872 | |
100 | α | 0.21052 | 0.45489 | 0.85098 | 5.69928 |
s | 0.04592 | 1.25029 | 0.92290 | 4.14301 | |
c | -0.12428 | 0.85789 | 0.91428 | 2.57578 | |
b | 0.41103 | 1.00012 | 0.80098 | 4.19802 | |
300 | α | 0.06776 | 0.13128 | 0.90097 | 2.44256 |
s | 0.03576 | 0.34829 | 0.96980 | 2.50462 | |
c | -0.02872 | 0.42232 | 0.98098 | 1.40965 | |
b | 0.15902 | 0.34752 | 0.81980 | 3.36362 | |
500 | α | 0.03962 | 0.07652 | 0.92620 | 1.48882 |
s | 0.03172 | 0.15372 | 0.98095 | 1.99142 | |
c | 0.02030 | 0.27289 | 0.99980 | 1.07363 | |
b | 0.05729 | 0.19258 | 0.88765 | 3.09498 | |
MLE(α=1.80,s=2.20,c=2.60,b=3.20) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 2.55609 | 20.36652 | 0.63065 | 21.15316 |
s | 1.01329 | 12.664i2 | 0.73098 | 7.87152 | |
c | -0.80862 | 2.19272 | 1.00098 | 13.64625 | |
b | 1.24726 | 3.59262 | 0.97980 | 11.27043 | |
100 | α | 1.08072 | 4.70272 | 0.71980 | 16.88462 |
s | 0.79362 | 7.37992 | 0.79809 | 6.31162 | |
c | -0.45762 | 1.58666 | 1.00450 | 7.46862 | |
b | 0.73178 | 1.80152 | 0.98609 | 7.66562 | |
300 | α | 0.38762 | 1.13352 | 0.75780 | 5.98709 |
s | 0.27457 | 1.24709 | 0.91550 | 3.92154 | |
c | -0.16804 | 0.79128 | 1.00010 | 3.93678 | |
b | 0.29061 | 0.58889 | 0.99987 | 5.45865 | |
500 | α | 0.17910 | 0.58267 | 0.80087 | 4.02328 |
s | 0.19884 | 0.78817 | 0.93709 | 3.28665 | |
c | -0.03543 | 0.60569 | 1.00120 | 2.99141 | |
b | 0.12976 | 0.33093 | 0.99120 | 4.82464 |
This section deals with checking the flexibility of the LLT-W distribution using two real-life data applications.
The discrimination measures namely, the Kolmogrov-Smirnov (K-S) (with its p-value), Anderson-Darling (A∗) and Cramér–von Mises (W∗) are calculated to compare the competing models selected, odd log-logistic modified-Weibull (OLLMW) [37], odd Dagum-Weibull (ODW) [1], odd log logistic exponentiated-Weibull (OLLEW) [2], Kumaraswamy-Weibull (KW) [18], Kumaraswamy-exponentiated Burr XII (KEBXII) [30], Weibull-Lomax (WL) [39], and beta-Burr XII (BBXII) [36] distributions.
The first real data contain 63 observations about strengths of 1.5 cm glass fibers which are reported in Smith and Naylor [38]. The second data set from Choulakian and Stephens (2001)[12], is the exceedances of flood peaks (in m3/s) of Wheaton River, Yukon Territory, Canada. The data consists of 72 exceedances for the years 1958–1984, rounded to one decimal place.
Tables 3 and 4 report the MLEs (with their associated (standard errors)) of the parameters of competing models, and the statistics K-S, p-value, A∗ and W∗.
Model | α | c | s | b | λ | p-value | K-S | W∗ | A∗ |
LLT-W | 2.53601 | 0.55698 | 38.13266 | 3.59346 | - | 0.70310 | 0.08880 | 0.06492 | 0.38337 |
(1.35585) | (0.29409) | (54.65571) | (0.70897) | - | |||||
OLLMW | 2.62115 | 0.30243 | 0.01341 | 6.27241 | - | 0.62310 | 0.09479 | 0.08044 | 0.47130 |
(0.78255) | (0.08134) | (0.01179) | (1.09531) | - | |||||
ODW | 5.29998 | 0.29810 | 9.78229 | 0.25124 | 2.32808 | 0.37470 | 0.11506 | 0.09164 | 0.51433 |
(7.86358) | (0.16143) | (55.66654) | (0.30203) | (2.49274) | |||||
OLLEW | 0.30232 | 1.68593 | 1.99160 | 8.74967 | - | 0.22260 | 0.13196 | 0.18638 | 1.03125 |
(0.26788) | (0.74479) | (0.29822) | (3.94948) | - | |||||
KW | 0.55200 | 0.22850 | 0.10530 | 7.18550 | - | 0.27860 | 0.12499 | 0.15316 | 0.86834 |
(0.14662) | (0.05697) | (0.02394) | (0.01724) | - | |||||
KEBXII | 4.27374 | 658.39551 | 0.85420 | 1.31415 | 4.26494 | 0.02819 | 0.18391 | 0.36750 | 2.01129 |
(36.70618) | (745.20326) | (0.65802) | (1.48982) | (36.63053) | |||||
WL | 0.01519 | 3.39585 | 7.04854 | 6.96628 | - | 0.16120 | 0.14136 | 0.18795 | 1.04676 |
(0.02116) | (0.93100) | (12.01524) | (12.82023) | - | |||||
BBXII | 103.51855 | 174.50750 | 0.55986 | 0.57767 | - | 0.00230 | 0.23172 | 0.70303 | 3.84266 |
(245.89797) | (401.09856) | (0.69152) | (1.16196) | - | |||||
WBXII | 0.01210 | 2.98508 | 1.57632 | 1.46315 | - | 0.11440 | 0.15069 | 0.21760 | 1.19955 |
(0.01215) | (2.60747) | (1.02824) | (0.59959) | - |
Model | α | c | s | b | λ | p-value | K-S | W∗ | A∗ |
LLT-W | 0.22515 | 11.81810 | 0.80678 | 13.88789 | - | 0.99580 | 0.04853 | 0.02924 | 0.22831 |
(0.07329) | (14.49115) | (0.08631) | (4.14192) | - | |||||
OLLMW | 0.00425 | 0.56057 | 0.07717 | 7.29698 | - | 0.43390 | 0.10266 | 0.13676 | 0.80928 |
(0.00289) | (0.07774) | (0.05263) | (4.47235) | - | |||||
ODW | 0.31643 | 13.49715 | 0.05973 | 0.03914 | 1.50574 | 0.89770 | 0.06754 | 0.04044 | 0.25765 |
(0.07961) | (28.57142) | (0.12519) | (0.04984) | (0.36632) | |||||
OLLEW | 4.83143 | 1.38144 | 4.23678 | 0.20378 | - | 0.93890 | 0.06280 | 0.03495 | 0.22261 |
(0.00432) | (0.00431) | (0.80929) | (0.02616) | - | |||||
KW | 0.54109 | 1.36525 | 0.01059 | 1.39132 | - | 0.39510 | 0.10587 | 0.10785 | 0.64912 |
(0.21553) | (1.19205) | (0.01303) | (0.33498) | - | |||||
KEBXII | -6.33954 | 63.90982 | 0.16781 | 1.72935 | -2.82907 | 0.32580 | 0.11212 | 0.19729 | 1.08811 |
(28.76141) | (69.93447) | (0.16987) | (2.40254) | (12.82347) | |||||
WL | 1.09893 | 0.76225 | 2.66001 | 46.73481 | - | 0.44240 | 0.10198 | 0.09932 | 0.60466 |
(1.24287) | (0.13144) | (2.91198) | (98.66865) | - | |||||
BBXII | 59.58061 | 9.47 | 67.54568 | 0.13740 | - | 0.11800 | 0.14020 | 0.26630 | 1.49169 |
(118.95689) | (139.21525) | (0.14196) | (1.43564) | - | |||||
WBXII | 0.12549 | 0.25801 | 5.19935 | 0.63785 | - | 0.62690 | 0.08840 | 0.12470 | 0.69009 |
(0.03716) | (0.22083) | (4.60782) | (0.21590) | - |
The LLT-W model is compared with the ODW, OLLEW, KW, WBXII, KEBXII, BBXII, WL distributions in Tables 3 and 4. We note that the LLT-W model gives the lowest values for all discrimination measures and largest p-value among all fitted models. Hence, the LLT-W model could be chosen as a good alternative to explain glass fibers and Wheaton river data. The results in Tables 3 and 4 indicate that LLT-W provides better fits for glass fibers and Wheaton river data sets as compared to other competing models. More visual comparison of the four best competing distributions are provided in Figures 10 and 11. The fitted densities of the four best models are shown in Figure 10 for glass fibers and Wheaton river data, whereas the estimated distribution functions for four best models are depicted in Figure 11. Further, the hrf plots of the LLT-W distribution for glass fibers and Wheaton river data are illustrated in Figure 12. Based on visual comparison, we can conclude that the LLT-W distribution provides a close fit for glass fibers and Wheaton river data and it can be utilized in fitting data with increasing and modified bathtub hazard rates.
This paper provides a new log-logistic tan generalized (LLT-G) class with three additional shape parameters to capture skewness and kurtosis behavior. Five special models of the LLT-G family are presented by choosing Weibull, normal, Rayleigh, exponential and Burr XII, as baseline distributions in the proposed family, to obtain the LLT-Weibull, LLT-normal, LLT-Rayleigh, LLT-exponential and LLT-Burr XII. The general mathematical properties are obtained for the LLT-G class. The LLT-G parameters estimation is discussed by maximum-likelihood approach and simulation results are obtained to check the performance of these estimators. The importance and flexibility of the LLT-Weibull are checked empirically using two sets of real-life data, proving that it can provide better fit as compared with other competing models, such as odd log-logistic modified-Weibull, odd Dagum-Weibull, odd log logistic exponentiated-Weibull, Weibull-Lomax, Kumaraswamy-Weibull, Kumaraswamy-exponentiated-Burr XII, and beta-Burr XII distributions.
This publication was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University, Alkharj, Saudi Arabia.
There is no conflict of interest declared by the authors.
[1] |
S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Phys. D, 103 (1997), 201–250. doi: 10.1016/S0167-2789(96)00261-8
![]() |
[2] | S. Aubry, G. André, Analyticity breaking and Anderson localization in incommensurate lattices, Ann. Israel Phys. Soc., 3 (1980), 133–164. |
[3] |
S. Aubry, G. Kopidakis, V. Kadelburg, Variational proof for hard discrete breathers in some classes of Hamiltonian dynamical systems, Discrete Contin. Dyn. Syst. B, 1 (2001), 271–298. doi: 10.3934/dcdsb.2001.1.271
![]() |
[4] | M.Ya. Azbel, Energy spectrum of a conduction electron in a magnetic field, Sov. Phys. JETP, 19 (1964), 634–645. |
[5] | G. W. Chen, S. W. Ma, Discrete nonlinear Schrödinger equations with superlinear nonlinearities, Appl. Math. Comput., 218 (2012), 5496–5507. |
[6] |
G. W. Chen, S. W. Ma, Ground state and geometrically distinct solitons of discrete nonlinear Schrödinger equations with saturable nonlinearities, Stud. Appl. Math., 131 (2013), 389–413. doi: 10.1111/sapm.12016
![]() |
[7] |
G. W. Chen, S. W. Ma, Z. Q. Wang, Standing waves for discrete Schrödinger equations in infinite lattices with saturable nonlinearities, J. Differ. Equ., 261 (2016), 3493–3518. doi: 10.1016/j.jde.2016.05.030
![]() |
[8] |
D. N. Christodoulides, F. Lederer, Y. Silberberg, Discretizing light behaviour in linear and nonlinear waveguide lattices, Nature, 424 (2003), 817–823. doi: 10.1038/nature01936
![]() |
[9] |
J. Cuevas, P. G. Kevrekidis, D. J. Frantzeskakis, B. A. Malomed, Discrete solitons in nonlinear Schrodinger lattices with a power-law nonlinearity, Phys. D, 238 (2009), 67–76. doi: 10.1016/j.physd.2008.08.013
![]() |
[10] |
S. Flach, C. R. Willis, Discrete breathers, Phys Rep., 295 (1998), 181–264. doi: 10.1016/S0370-1573(97)00068-9
![]() |
[11] | S. Flach, K. Kladko, Moving discrete breathers Phys. D, 127 (1999), 61–72. |
[12] |
S. Flach, A. V. Gorbach, Discrete breathers-advances in theory and applications, Phys Rep., 467 (2008), 1–116. doi: 10.1016/j.physrep.2008.05.002
![]() |
[13] |
J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, D. N. Christodoulides, Observation of discrete solitons in optically induced real time waveguide arrays, Phys. Rev. Lett., 90 (2003), 023902. doi: 10.1103/PhysRevLett.90.023902
![]() |
[14] |
J. W. Fleischer, M. Segev, N. K. Efremidis, D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature, 422 (2003), 147–150. doi: 10.1038/nature01452
![]() |
[15] |
P. G. Harper, Single Band Motion of Conduction Electrons in a Uniform Magnetic Field, Proc. Phys. Soc. Sect. A, 68 (1955), 874–878. doi: 10.1088/0370-1298/68/10/304
![]() |
[16] |
D. Hennig, G. P. Tsironis, Wave transmission in nonlinear lattices, Physics Reports, 307 (1999), 333–432. doi: 10.1016/S0370-1573(98)00025-8
![]() |
[17] |
S. Iubini, A. Politi, Chaos and localization in the discrete nonlinear Schrödinger equation, Chaos, Solitons and Fractals, 147 (2021), 110954. doi: 10.1016/j.chaos.2021.110954
![]() |
[18] |
L. Jeanjean, K. Tanaka, A positive solution for an asymptotically linear elliptic problem on RN autonomous at infinity, ESAIM Control Optim. Calc. Var., 7 (2002), 597–614. doi: 10.1051/cocv:2002068
![]() |
[19] | P. G. Kevrekidis, K. Rasmussen, A. R. Bishop, The discrete nonlinear Schrödinger equation: a survey of recent results, Int. J. Mod. Phys. B, 15 (2001), 2883–2900. |
[20] |
G. Kopidakis, S. Aubry, G. P. Tsironis, Targeted energy transfer through discrete breathers in nonlinear systems, Phys. Rev. Lett., 87 (2001), 165501. doi: 10.1103/PhysRevLett.87.165501
![]() |
[21] |
G. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763–776. doi: 10.1142/S0219199702000853
![]() |
[22] |
S. Liu, On superlinear Schrdinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1–9. doi: 10.1007/s00526-011-0447-2
![]() |
[23] |
R. Livi, R. Franzosi, G. L. Oppo, Self-localization of Bose Einstein condensates in optical lattices via boundary dissipation, Phys. Rev. Lett., 97 (2006), 060401. doi: 10.1103/PhysRevLett.97.060401
![]() |
[24] | D. Ma, Z. Zhou, Existence and multiplicity results of homoclinic solutions for the DNLS equations with unbounded potentials, Abstr. Appl. Anal., 2012 (2012), 703596. |
[25] | A. Mai, Z. Zhou, Ground state solutions for the periodic discrete nonlinear Schrödinger equations with superlinear nonlinearities, Abstr. Appl. Anal., 2013 (2013), 317139. |
[26] | A. Mai, Z. Zhou, Discrete solitons for periodic discrete nonlinear Schrödinger equations, Appl. Math. Comput., 222 (2013), 34–41. |
[27] |
D. V. Makarov, M. Yu. Uleysky, Chaos-assisted formation of immiscible matter-wave solitons and self-stabilization in the binary discrete nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul., 43 (2017), 227–238. doi: 10.1016/j.cnsns.2016.07.006
![]() |
[28] |
A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations, Nonlinearity, 19 (2006), 27–40. doi: 10.1088/0951-7715/19/1/002
![]() |
[29] |
A. Pankov, Gap solitons in periodic discrete nonlinear Schrödinger equations. II. A generalized Nehari manifold approach, Discrete Contin. Dyn. Syst. Ser. A, 19 (2007), 419–430. doi: 10.3934/dcds.2007.19.419
![]() |
[30] |
H. Shi, Gap solitons in periodic discrete nonlinear Schrödinger equations with nonlinearity, Acta Appl. Math., 109 (2010), 1065–1075. doi: 10.1007/s10440-008-9360-x
![]() |
[31] | J. J. Sun, S. W. Ma, Multiple solutions for discrete periodic nonlinear Schrödinger equations, J. Math. Phys., 56 (2015), 1413–1442. |
[32] |
A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Func. Anal., 257 (2009), 3802–3822. doi: 10.1016/j.jfa.2009.09.013
![]() |
[33] |
X. Tang, New super-quadratic conditions on ground state solutions for superlinear Schrödinger equation, Adv. Nonlinear Stud., 14 (2014), 361–373. doi: 10.1515/ans-2014-0208
![]() |
[34] | X. Tang, Non-nehari-manifold method for asymptotically linear schrodinger equation, J. Math. Phys., 56 (2015), 1413–1442. |
[35] | A. Trombettoni, A. Smerzi, Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates, Phys. Rev. Lett., 16 (2001), 2353–2356. |
[36] |
Z. Yang, W. Chen, Y. Ding, Solutions for discrete periodic Schrödinger equations with spectrum 0, Acta Appl. Math., 110 (2010), 1475–1488. doi: 10.1007/s10440-009-9521-6
![]() |
[37] |
L. Zhang, S. Ma, Ground state solutions for periodic discrete nonlinear Schrödinger equations with saturable nonlinearities, Adv. Difference Equ., 2018 (2018), 1–13. doi: 10.1186/s13662-017-1452-3
![]() |
[38] |
Z. Zhou, J. Yu, Y. Chen, On the existence of gap solitons in a periodic discrete nonlinear Schrödinger equation with saturable nonlinearity, Nonlinearity, 23 (2010), 1727–1740. doi: 10.1088/0951-7715/23/7/011
![]() |
1. | Mahmoud EL-Morshedy, Fahad Sameer Alshammari, Abhishek Tyagi, Iberahim Elbatal, Yasser S. Hamed, Mohamed S. Eliwa, Bayesian and Frequentist Inferences on a Type I Half-Logistic Odd Weibull Generator with Applications in Engineering, 2021, 23, 1099-4300, 446, 10.3390/e23040446 | |
2. | Ahmed Z. Afify, Hassan M. Aljohani, Abdulaziz S. Alghamdi, Ahmed M. Gemeay, Abdullah M. Sarg, Barbara Martinucci, A New Two-Parameter Burr-Hatke Distribution: Properties and Bayesian and Non-Bayesian Inference with Applications, 2021, 2021, 2314-4785, 1, 10.1155/2021/1061083 | |
3. | Ahmed Z. Afify, Hazem Al-Mofleh, Hassan M. Aljohani, Gauss M. Cordeiro, The Marshall–Olkin–Weibull-H family: Estimation, simulations, and applications to COVID-19 data, 2022, 34, 10183647, 102115, 10.1016/j.jksus.2022.102115 | |
4. | Nikolay Kyurkchiev, Anton Iliev, Asen Rahnev, Properties and Applications of a Tan–G Family of ”Adaptive Functions”, 2021, 15, 1998-4464, 1292, 10.46300/9106.2021.15.139 | |
5. | Fei Wang, Zubair Ahmad, Faridoon Khan, Eslam Hussam, Abdal-Aziz H El-Bagoury, Huda M. Alshanbari, A new statistical distribution with applications to sports and health sciences, 2022, 61, 11100168, 9661, 10.1016/j.aej.2022.02.062 | |
6. | Thatayaone Moakofi, Broderick Oluyede, Fastel Chipepa, Type II exponentiated half-logistic Topp-Leone Marshall-Olkin-G family of distributions with applications, 2021, 7, 24058440, e08590, 10.1016/j.heliyon.2021.e08590 | |
7. | Yinghui Zhou, Zubair Ahmad, Zahra Almaspoor, Faridoon Khan, Elsayed tag-Eldin, Zahoor Iqbal, Mahmoud El-Morshedy, On the implementation of a new version of the Weibull distribution and machine learning approach to model the COVID-19 data, 2022, 20, 1551-0018, 337, 10.3934/mbe.2023016 | |
8. | Muhammad Arif, Dost Muhammad Khan, Muhammad Aamir, Mahmoud El-Morshedy, Zubair Ahmad, Zardad Khan, A New Flexible Exponentiated-X Family of Distributions: Characterizations and Applications to Lifetime Data, 2022, 0377-2063, 1, 10.1080/03772063.2022.2034537 | |
9. | Mahmoud El-Morshedy, Adel A. El-Faheem, Afrah Al-Bossly, Mohamed El-Dawoody, Exponentiated Generalized Inverted Gompertz Distribution: Properties and Estimation Methods with Applications to Symmetric and Asymmetric Data, 2021, 13, 2073-8994, 1868, 10.3390/sym13101868 | |
10. | Emrah Altun, Mustafa Ç. Korkmaz, M. El-Morshedy, M. S. Eliwa, The extended gamma distribution with regression model and applications, 2020, 6, 2473-6988, 2418, 10.3934/math.2021147 | |
11. | Adebisi A. Ogunde, Subhankar Dutta, Ehab M. Almetawally, Half Logistic Generalized Rayleigh Distribution for Modeling Hydrological Data, 2024, 2198-5804, 10.1007/s40745-024-00527-2 | |
12. | Coşkun Kuş, Kadir Karakaya, Caner Tanış, Yunus Akdoğan, Sümeyra Sert, Fahreddin Kalkan, Compound transmuted family of distributions: properties and applications, 2023, 0035-5038, 10.1007/s11587-023-00808-7 | |
13. | Huda M. Alshanbari, Zubair Ahmad, Abd Al-Aziz Hosni El-Bagoury, Omalsad Hamood Odhah, Gadde Srinivasa Rao, A New Modification of the Weibull Distribution: Model, Theory, and Analyzing Engineering Data Sets, 2024, 16, 2073-8994, 611, 10.3390/sym16050611 | |
14. | Hisham Mahran, Mahmoud M. Mansour, Enayat M. Abd Elrazik, Ahmed Z. Afify, A new one-parameter flexible family with variable failure rate shapes: Properties, inference, and real-life applications, 2024, 9, 2473-6988, 11910, 10.3934/math.2024582 | |
15. | Getachew Tekle, Rasool Roozegar, Zubair Ahmad, Alessandro De Gregorio, A New Type 1 Alpha Power Family of Distributions and Modeling Data with Correlation, Overdispersion, and Zero-Inflation in the Health Data Sets, 2023, 2023, 1687-9538, 1, 10.1155/2023/6611108 |
MLE(α=0.20,s=0.50,c=1.30,b=1.50) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.02632 | 0.00861 | 0.99023 | 2.36578 |
s | 0.18709 | 0.28943 | 0.94067 | 2.19272 | |
c | 0.17656 | 0.20302 | 0.94821 | 0.44550 | |
b | -0.16017 | 0.18987 | 0.93041 | 1.56365 | |
100 | α | 0.01978 | 0.00444 | 1.00022 | 1.25564 |
s | 0.09531 | 0.07086 | 0.96764 | 1.40323 | |
c | 0.11487 | 0.08334 | 0.93042 | 0.29029 | |
b | -0.15492 | 0.09865 | 0.98562 | 1.49110 | |
300 | α | 0.01298 | 0.00175 | 1.00276 | 0.65864 |
s | 0.03895 | 0.01812 | 0.98054 | 0.758178 | |
c | 0.05497 | 0.02252 | 0.97097 | 0.15874 | |
b | -0.10343 | 0.03767 | 0.9819 | 1.47684 | |
500 | α | 0.00787 | 0.00156 | 1.00876 | 0.48943 |
s | 0.01798 | 0.01028 | 0.99098 | 0.56132 | |
c | 0.02509 | 0.01161 | 0.98987 | 0.11954 | |
b | -0.05684 | 0.01249 | 0.99076 | 1.50318 | |
MLE(α=0.80,s=1.10,c=1.60,b=1.80) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.17012 | 0.34589 | 0.88320 | 5.73426 |
s | 0.04296 | 1.17838 | 0.82209 | 3.51921 | |
c | -0.19793 | 0.70917 | 0.84190 | 2.01749 | |
b | 0.70237 | 1.92221 | 0.84084 | 3.51065 | |
100 | α | 0.06832 | 0.10920 | 0.94890 | 3.29742 |
s | 0.04132 | 0.54720 | 0.90908 | 2.76689 | |
c | -0.05829 | 0.45318 | 0.9593 | 1.25046 | |
b | 0.29772 | 0.66458 | 0.9032 | 2.72295 | |
300 | α | 0.01620 | 0.02809 | 0.99870 | 1.46536 |
s | 0.03836 | 0.16018 | 0.95430 | 1.62928 | |
c | -0.01011 | 0.19239 | 0.9673 | 0.65048 | |
b | 0.11101 | 0.18638 | 0.95650 | 2.23627 | |
500 | α | 0.00503 | 0.01737 | 0.99201 | 1.02895 |
s | -0.00776 | 0.07429 | 0.96807 | 1.24003 | |
c | -0.03110 | 0.10385 | 0.97956 | 0.49509 | |
b | 0.07738 | 0.09929 | 0.9982 | 2.12406 |
MLE(α=1.20,s=1.50,c=2.00,b=2.50) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.55202 | 1.38710 | 0.76780 | 7.32437 |
s | 0.05138 | 1.54372 | 0.8452 | 6.13638 | |
c | -0.40001 | 1.29038 | 0.96980 | 4.15307 | |
b | 0.96352 | 2.67448 | 0.76980 | 5.539872 | |
100 | α | 0.21052 | 0.45489 | 0.85098 | 5.69928 |
s | 0.04592 | 1.25029 | 0.92290 | 4.14301 | |
c | -0.12428 | 0.85789 | 0.91428 | 2.57578 | |
b | 0.41103 | 1.00012 | 0.80098 | 4.19802 | |
300 | α | 0.06776 | 0.13128 | 0.90097 | 2.44256 |
s | 0.03576 | 0.34829 | 0.96980 | 2.50462 | |
c | -0.02872 | 0.42232 | 0.98098 | 1.40965 | |
b | 0.15902 | 0.34752 | 0.81980 | 3.36362 | |
500 | α | 0.03962 | 0.07652 | 0.92620 | 1.48882 |
s | 0.03172 | 0.15372 | 0.98095 | 1.99142 | |
c | 0.02030 | 0.27289 | 0.99980 | 1.07363 | |
b | 0.05729 | 0.19258 | 0.88765 | 3.09498 | |
MLE(α=1.80,s=2.20,c=2.60,b=3.20) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 2.55609 | 20.36652 | 0.63065 | 21.15316 |
s | 1.01329 | 12.664i2 | 0.73098 | 7.87152 | |
c | -0.80862 | 2.19272 | 1.00098 | 13.64625 | |
b | 1.24726 | 3.59262 | 0.97980 | 11.27043 | |
100 | α | 1.08072 | 4.70272 | 0.71980 | 16.88462 |
s | 0.79362 | 7.37992 | 0.79809 | 6.31162 | |
c | -0.45762 | 1.58666 | 1.00450 | 7.46862 | |
b | 0.73178 | 1.80152 | 0.98609 | 7.66562 | |
300 | α | 0.38762 | 1.13352 | 0.75780 | 5.98709 |
s | 0.27457 | 1.24709 | 0.91550 | 3.92154 | |
c | -0.16804 | 0.79128 | 1.00010 | 3.93678 | |
b | 0.29061 | 0.58889 | 0.99987 | 5.45865 | |
500 | α | 0.17910 | 0.58267 | 0.80087 | 4.02328 |
s | 0.19884 | 0.78817 | 0.93709 | 3.28665 | |
c | -0.03543 | 0.60569 | 1.00120 | 2.99141 | |
b | 0.12976 | 0.33093 | 0.99120 | 4.82464 |
Model | α | c | s | b | λ | p-value | K-S | W∗ | A∗ |
LLT-W | 2.53601 | 0.55698 | 38.13266 | 3.59346 | - | 0.70310 | 0.08880 | 0.06492 | 0.38337 |
(1.35585) | (0.29409) | (54.65571) | (0.70897) | - | |||||
OLLMW | 2.62115 | 0.30243 | 0.01341 | 6.27241 | - | 0.62310 | 0.09479 | 0.08044 | 0.47130 |
(0.78255) | (0.08134) | (0.01179) | (1.09531) | - | |||||
ODW | 5.29998 | 0.29810 | 9.78229 | 0.25124 | 2.32808 | 0.37470 | 0.11506 | 0.09164 | 0.51433 |
(7.86358) | (0.16143) | (55.66654) | (0.30203) | (2.49274) | |||||
OLLEW | 0.30232 | 1.68593 | 1.99160 | 8.74967 | - | 0.22260 | 0.13196 | 0.18638 | 1.03125 |
(0.26788) | (0.74479) | (0.29822) | (3.94948) | - | |||||
KW | 0.55200 | 0.22850 | 0.10530 | 7.18550 | - | 0.27860 | 0.12499 | 0.15316 | 0.86834 |
(0.14662) | (0.05697) | (0.02394) | (0.01724) | - | |||||
KEBXII | 4.27374 | 658.39551 | 0.85420 | 1.31415 | 4.26494 | 0.02819 | 0.18391 | 0.36750 | 2.01129 |
(36.70618) | (745.20326) | (0.65802) | (1.48982) | (36.63053) | |||||
WL | 0.01519 | 3.39585 | 7.04854 | 6.96628 | - | 0.16120 | 0.14136 | 0.18795 | 1.04676 |
(0.02116) | (0.93100) | (12.01524) | (12.82023) | - | |||||
BBXII | 103.51855 | 174.50750 | 0.55986 | 0.57767 | - | 0.00230 | 0.23172 | 0.70303 | 3.84266 |
(245.89797) | (401.09856) | (0.69152) | (1.16196) | - | |||||
WBXII | 0.01210 | 2.98508 | 1.57632 | 1.46315 | - | 0.11440 | 0.15069 | 0.21760 | 1.19955 |
(0.01215) | (2.60747) | (1.02824) | (0.59959) | - |
Model | α | c | s | b | λ | p-value | K-S | W∗ | A∗ |
LLT-W | 0.22515 | 11.81810 | 0.80678 | 13.88789 | - | 0.99580 | 0.04853 | 0.02924 | 0.22831 |
(0.07329) | (14.49115) | (0.08631) | (4.14192) | - | |||||
OLLMW | 0.00425 | 0.56057 | 0.07717 | 7.29698 | - | 0.43390 | 0.10266 | 0.13676 | 0.80928 |
(0.00289) | (0.07774) | (0.05263) | (4.47235) | - | |||||
ODW | 0.31643 | 13.49715 | 0.05973 | 0.03914 | 1.50574 | 0.89770 | 0.06754 | 0.04044 | 0.25765 |
(0.07961) | (28.57142) | (0.12519) | (0.04984) | (0.36632) | |||||
OLLEW | 4.83143 | 1.38144 | 4.23678 | 0.20378 | - | 0.93890 | 0.06280 | 0.03495 | 0.22261 |
(0.00432) | (0.00431) | (0.80929) | (0.02616) | - | |||||
KW | 0.54109 | 1.36525 | 0.01059 | 1.39132 | - | 0.39510 | 0.10587 | 0.10785 | 0.64912 |
(0.21553) | (1.19205) | (0.01303) | (0.33498) | - | |||||
KEBXII | -6.33954 | 63.90982 | 0.16781 | 1.72935 | -2.82907 | 0.32580 | 0.11212 | 0.19729 | 1.08811 |
(28.76141) | (69.93447) | (0.16987) | (2.40254) | (12.82347) | |||||
WL | 1.09893 | 0.76225 | 2.66001 | 46.73481 | - | 0.44240 | 0.10198 | 0.09932 | 0.60466 |
(1.24287) | (0.13144) | (2.91198) | (98.66865) | - | |||||
BBXII | 59.58061 | 9.47 | 67.54568 | 0.13740 | - | 0.11800 | 0.14020 | 0.26630 | 1.49169 |
(118.95689) | (139.21525) | (0.14196) | (1.43564) | - | |||||
WBXII | 0.12549 | 0.25801 | 5.19935 | 0.63785 | - | 0.62690 | 0.08840 | 0.12470 | 0.69009 |
(0.03716) | (0.22083) | (4.60782) | (0.21590) | - |
MLE(α=0.20,s=0.50,c=1.30,b=1.50) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.02632 | 0.00861 | 0.99023 | 2.36578 |
s | 0.18709 | 0.28943 | 0.94067 | 2.19272 | |
c | 0.17656 | 0.20302 | 0.94821 | 0.44550 | |
b | -0.16017 | 0.18987 | 0.93041 | 1.56365 | |
100 | α | 0.01978 | 0.00444 | 1.00022 | 1.25564 |
s | 0.09531 | 0.07086 | 0.96764 | 1.40323 | |
c | 0.11487 | 0.08334 | 0.93042 | 0.29029 | |
b | -0.15492 | 0.09865 | 0.98562 | 1.49110 | |
300 | α | 0.01298 | 0.00175 | 1.00276 | 0.65864 |
s | 0.03895 | 0.01812 | 0.98054 | 0.758178 | |
c | 0.05497 | 0.02252 | 0.97097 | 0.15874 | |
b | -0.10343 | 0.03767 | 0.9819 | 1.47684 | |
500 | α | 0.00787 | 0.00156 | 1.00876 | 0.48943 |
s | 0.01798 | 0.01028 | 0.99098 | 0.56132 | |
c | 0.02509 | 0.01161 | 0.98987 | 0.11954 | |
b | -0.05684 | 0.01249 | 0.99076 | 1.50318 | |
MLE(α=0.80,s=1.10,c=1.60,b=1.80) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.17012 | 0.34589 | 0.88320 | 5.73426 |
s | 0.04296 | 1.17838 | 0.82209 | 3.51921 | |
c | -0.19793 | 0.70917 | 0.84190 | 2.01749 | |
b | 0.70237 | 1.92221 | 0.84084 | 3.51065 | |
100 | α | 0.06832 | 0.10920 | 0.94890 | 3.29742 |
s | 0.04132 | 0.54720 | 0.90908 | 2.76689 | |
c | -0.05829 | 0.45318 | 0.9593 | 1.25046 | |
b | 0.29772 | 0.66458 | 0.9032 | 2.72295 | |
300 | α | 0.01620 | 0.02809 | 0.99870 | 1.46536 |
s | 0.03836 | 0.16018 | 0.95430 | 1.62928 | |
c | -0.01011 | 0.19239 | 0.9673 | 0.65048 | |
b | 0.11101 | 0.18638 | 0.95650 | 2.23627 | |
500 | α | 0.00503 | 0.01737 | 0.99201 | 1.02895 |
s | -0.00776 | 0.07429 | 0.96807 | 1.24003 | |
c | -0.03110 | 0.10385 | 0.97956 | 0.49509 | |
b | 0.07738 | 0.09929 | 0.9982 | 2.12406 |
MLE(α=1.20,s=1.50,c=2.00,b=2.50) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 0.55202 | 1.38710 | 0.76780 | 7.32437 |
s | 0.05138 | 1.54372 | 0.8452 | 6.13638 | |
c | -0.40001 | 1.29038 | 0.96980 | 4.15307 | |
b | 0.96352 | 2.67448 | 0.76980 | 5.539872 | |
100 | α | 0.21052 | 0.45489 | 0.85098 | 5.69928 |
s | 0.04592 | 1.25029 | 0.92290 | 4.14301 | |
c | -0.12428 | 0.85789 | 0.91428 | 2.57578 | |
b | 0.41103 | 1.00012 | 0.80098 | 4.19802 | |
300 | α | 0.06776 | 0.13128 | 0.90097 | 2.44256 |
s | 0.03576 | 0.34829 | 0.96980 | 2.50462 | |
c | -0.02872 | 0.42232 | 0.98098 | 1.40965 | |
b | 0.15902 | 0.34752 | 0.81980 | 3.36362 | |
500 | α | 0.03962 | 0.07652 | 0.92620 | 1.48882 |
s | 0.03172 | 0.15372 | 0.98095 | 1.99142 | |
c | 0.02030 | 0.27289 | 0.99980 | 1.07363 | |
b | 0.05729 | 0.19258 | 0.88765 | 3.09498 | |
MLE(α=1.80,s=2.20,c=2.60,b=3.20) | |||||
n | Bias | MSE | C.P | AW | |
50 | α | 2.55609 | 20.36652 | 0.63065 | 21.15316 |
s | 1.01329 | 12.664i2 | 0.73098 | 7.87152 | |
c | -0.80862 | 2.19272 | 1.00098 | 13.64625 | |
b | 1.24726 | 3.59262 | 0.97980 | 11.27043 | |
100 | α | 1.08072 | 4.70272 | 0.71980 | 16.88462 |
s | 0.79362 | 7.37992 | 0.79809 | 6.31162 | |
c | -0.45762 | 1.58666 | 1.00450 | 7.46862 | |
b | 0.73178 | 1.80152 | 0.98609 | 7.66562 | |
300 | α | 0.38762 | 1.13352 | 0.75780 | 5.98709 |
s | 0.27457 | 1.24709 | 0.91550 | 3.92154 | |
c | -0.16804 | 0.79128 | 1.00010 | 3.93678 | |
b | 0.29061 | 0.58889 | 0.99987 | 5.45865 | |
500 | α | 0.17910 | 0.58267 | 0.80087 | 4.02328 |
s | 0.19884 | 0.78817 | 0.93709 | 3.28665 | |
c | -0.03543 | 0.60569 | 1.00120 | 2.99141 | |
b | 0.12976 | 0.33093 | 0.99120 | 4.82464 |
Model | α | c | s | b | λ | p-value | K-S | W∗ | A∗ |
LLT-W | 2.53601 | 0.55698 | 38.13266 | 3.59346 | - | 0.70310 | 0.08880 | 0.06492 | 0.38337 |
(1.35585) | (0.29409) | (54.65571) | (0.70897) | - | |||||
OLLMW | 2.62115 | 0.30243 | 0.01341 | 6.27241 | - | 0.62310 | 0.09479 | 0.08044 | 0.47130 |
(0.78255) | (0.08134) | (0.01179) | (1.09531) | - | |||||
ODW | 5.29998 | 0.29810 | 9.78229 | 0.25124 | 2.32808 | 0.37470 | 0.11506 | 0.09164 | 0.51433 |
(7.86358) | (0.16143) | (55.66654) | (0.30203) | (2.49274) | |||||
OLLEW | 0.30232 | 1.68593 | 1.99160 | 8.74967 | - | 0.22260 | 0.13196 | 0.18638 | 1.03125 |
(0.26788) | (0.74479) | (0.29822) | (3.94948) | - | |||||
KW | 0.55200 | 0.22850 | 0.10530 | 7.18550 | - | 0.27860 | 0.12499 | 0.15316 | 0.86834 |
(0.14662) | (0.05697) | (0.02394) | (0.01724) | - | |||||
KEBXII | 4.27374 | 658.39551 | 0.85420 | 1.31415 | 4.26494 | 0.02819 | 0.18391 | 0.36750 | 2.01129 |
(36.70618) | (745.20326) | (0.65802) | (1.48982) | (36.63053) | |||||
WL | 0.01519 | 3.39585 | 7.04854 | 6.96628 | - | 0.16120 | 0.14136 | 0.18795 | 1.04676 |
(0.02116) | (0.93100) | (12.01524) | (12.82023) | - | |||||
BBXII | 103.51855 | 174.50750 | 0.55986 | 0.57767 | - | 0.00230 | 0.23172 | 0.70303 | 3.84266 |
(245.89797) | (401.09856) | (0.69152) | (1.16196) | - | |||||
WBXII | 0.01210 | 2.98508 | 1.57632 | 1.46315 | - | 0.11440 | 0.15069 | 0.21760 | 1.19955 |
(0.01215) | (2.60747) | (1.02824) | (0.59959) | - |
Model | α | c | s | b | λ | p-value | K-S | W∗ | A∗ |
LLT-W | 0.22515 | 11.81810 | 0.80678 | 13.88789 | - | 0.99580 | 0.04853 | 0.02924 | 0.22831 |
(0.07329) | (14.49115) | (0.08631) | (4.14192) | - | |||||
OLLMW | 0.00425 | 0.56057 | 0.07717 | 7.29698 | - | 0.43390 | 0.10266 | 0.13676 | 0.80928 |
(0.00289) | (0.07774) | (0.05263) | (4.47235) | - | |||||
ODW | 0.31643 | 13.49715 | 0.05973 | 0.03914 | 1.50574 | 0.89770 | 0.06754 | 0.04044 | 0.25765 |
(0.07961) | (28.57142) | (0.12519) | (0.04984) | (0.36632) | |||||
OLLEW | 4.83143 | 1.38144 | 4.23678 | 0.20378 | - | 0.93890 | 0.06280 | 0.03495 | 0.22261 |
(0.00432) | (0.00431) | (0.80929) | (0.02616) | - | |||||
KW | 0.54109 | 1.36525 | 0.01059 | 1.39132 | - | 0.39510 | 0.10587 | 0.10785 | 0.64912 |
(0.21553) | (1.19205) | (0.01303) | (0.33498) | - | |||||
KEBXII | -6.33954 | 63.90982 | 0.16781 | 1.72935 | -2.82907 | 0.32580 | 0.11212 | 0.19729 | 1.08811 |
(28.76141) | (69.93447) | (0.16987) | (2.40254) | (12.82347) | |||||
WL | 1.09893 | 0.76225 | 2.66001 | 46.73481 | - | 0.44240 | 0.10198 | 0.09932 | 0.60466 |
(1.24287) | (0.13144) | (2.91198) | (98.66865) | - | |||||
BBXII | 59.58061 | 9.47 | 67.54568 | 0.13740 | - | 0.11800 | 0.14020 | 0.26630 | 1.49169 |
(118.95689) | (139.21525) | (0.14196) | (1.43564) | - | |||||
WBXII | 0.12549 | 0.25801 | 5.19935 | 0.63785 | - | 0.62690 | 0.08840 | 0.12470 | 0.69009 |
(0.03716) | (0.22083) | (4.60782) | (0.21590) | - |