# Properties and Applications of the Modified Kies–Lomax Distribution with Estimation Methods

Authors: Abdelaziz Alsubie
Journal: Journal of Mathematics
 (2021)

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## Abstract

The present study introduces a new three-parameter model called the modified Kies–Lomax (MKL) distribution to extend the Lomax distribution and increase its flexibility in modeling real-life data. The MKL distribution, due to its flexibility, provides left-skewed, symmetrical, right-skewed, and reversed-J shaped densities and increasing, unimodal, decreasing, and bathtub hazard rate shapes. The MKF density can be expressed as a linear mixture of Lomax densities. Some basic mathematical properties of the MKF model are derived. Its parameters are estimated via six estimation algorithms. We explore their performances using detailed simulation results, and the partial and overall ranks are provided for the measures of absolute biases, mean square errors, and mean relative errors to determine the best estimation method. The results show that the maximum product of spacings and maximum likelihood approaches are recommended to estimate the MKL parameters. Finally, the flexibility of the MKL distribution is checked using two real datasets, showing that it can provide close fit to both datasets as compared with other competing Lomax models. The three-parameter MKL model outperforms some four-parameter and five-parameter rival models.


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## Body

## 1. Introduction

The Lomax distribution has several applications in different applied fields such as biological sciences, income and wealth inequality, engineering, reliability, and actuarial sciences. Chahkandi and Ganjali [1] showed that the Lomax model belongs to decreasing failure rate family. Detailed information about the Lomax distribution can be explored in [2, 3].The procedure of expanding classical distributions by adding new shape parameters is well-known technique in statistical literature. Recently, various extensions of the Lomax distribution have been constructed using well-known generators to increase its flexibility in modeling various types of data. The new added shape parameters are important to provide skewness and to increase tail weights as well as to enhance the flexibility to model monotonic and nonmonotonic hazard rates particularly if the baseline hazard rate is only monotonic.Some of the most notable extensions of the Lomax model are the exponentiated-Lomax [4], Marshall–Olkin Lomax [5], Kumaraswamy–Lomax [6], Weibull–Lomax [7], exponentiated half logistic-Lomax [8], and Fréchet Topp–Leone Lomax distributions [9]. On the contrary, some authors studied the estimation of the Lomax distribution using the Bayesian approach. For example, Okasha [10] studied the E-Bayesian estimation for the Lomax distribution based on type-II censored data and Hassan and Zaky [11] studied the entropy Bayesian estimation for the Lomax distribution based on records. For more details about Bayesian inference for probability distributions, the interested reader can refer to [12–19].Recently, there have been a great interest among statisticians to develop new families or generators of distributions by adding extra shape parameter(s) to the well-known classical distributions. One of the most recent generators is called modified Kies-G (MK-G) family due to Al-Babtain et al. [20]. Consider any baseline cumulative distribution function (CDF) with a vector of parameters υ, denoted by Gx;υ; then, the CDF of the MK-G class with additional shape parameter a takes the form(1)Fx;a,υ=1−e−Gx;υ/1−Gx;υa,x>0,a>0.The corresponding probability density function (PDF) of (1) reduces to(2)fx;a,υ=agx;υGx;υa−11−Gx;υ−a−1e−Gx;υ/1−Gx;υa.The hazard rate function (HRF) of the MK-G class has the form(3)hx;a,υ=agx;υGx;υa−11−Gx;υ−a−1.In this paper, a new three-parameter model called the modified Kies–Lomax (MKL) distribution is introduced to extend the Lomax model and improve its flexibility in fitting real-life data in various applied areas. The MKL distribution is generated by replacing the baseline Lomax distribution in the MK-G family. Shafiq et al. [21] adopted the MK-G family to define the MK-Fréchet distribution.The MKL distribution gives a better fit than some existing rival extensions of the Lomax model which have three, four, and five parameters. The MKL distribution, with three parameters, provides decreasing, unimodal, increasing, bathtub, J shape, and reversed-J shaped hazard functions, as well as right-skewed, symmetrical, left-skewed, and reversed-J shaped densities. Further important aim of the current paper is to explore the estimation of the MKL parameters using classical methods such as maximum likelihood, Anderson–Darling, Cramér–von Mises, least-squares, weighted least-squares, and maximum product of spacings. The performance of these methods are explored via simulation results based on the average values of absolute biases (AVBs), mean square error (MSEs), and mean relative errors of the estimates (MREs). Besides, these measures are ordered using the partial and overall ranks to compare the performances of the proposed estimators and to determine the best estimation method for the MKL parameters. The findings show that the maximum product of spacings and maximum likelihood methods are recommended to estimate the model parameters.The rest of the article is structured as follows. The MKL distribution is introduced in Section2. Its basic distributional properties are determined in Section 3. Six estimation approaches are introduced in Section 4. Simulation results for the six estimation methods are given in Section 5. The empirical importance MKL model is checked using two real datasets in Section 6. Some concluding remarks are presented in Section 7.


## 2. The MKL Distribution

The Lomax distribution with shape parameter,λ, and scale parameter, β, is specified by the CDF(4)Gx;λ,β=1−xβ+1−λ,x>0,λ,β>0.The PDF of the Lomax distribution reduces to(5)gx;λ,β=λβxβ+1−λ−1,x>0,λ,β>0.The CDF of the MKL distribution follows, by inserting (4) in (1), as(6)Fx;a,λ,β=1−e−β+x/βλ−1a,x>0,a,λ,β>0.Its PDF is determined by inserting (4) and (5) in equation (2) as follows:(7)fx;a,λ,β=aλβ−λβ+xλ−1e−β+x/βλ−1aβ+xβλ−1a−1,x>0,a,λ,β>0.The survival function (SF) and HRF of the MKL distribution take the forms(8)Sx;a,λ,β=e−β+x/βλ−1a,hx;a,λ,β=aλβ−λβ+xλ−1β+xβλ−1a−1.Some possible shapes for the PDF and HRF of the MKL distribution are displayed in Figures1 and 2, respectively. These plots show that the MKL distribution provides right-skewed, symmetrical, left-skewed, and reversed-J shaped densities, as well as decreasing, unimodal, increasing, bathtub, J shape, and reversed-J shaped hazard functions.Figure 1
Some possible density curves of the MKL distribution for different values of its parameters.Figure 2
Some possible hazard rate shapes of the MKL distribution for different values of its parameters.


## 3. Mathematical Properties

### 3.1. Quantile Function

The quantile function (QF) of the MKL distribution is obtained by determining the inverse function of the MKL CDF (6) as(9)Qp=β−log1−p1/a+11/λ−1,0<p<1.The three quartiles of the MKL distribution can be obtained by settingp=0.25,0.5, and 0.75, respectively, in (9).Letp∼U0,1; then, the QF can be used directly in generating random data from the MKL distribution as follows:(10)xi=β−log1−pi1/a+11/λ−1,i=1,2,…,n.


### 3.2. Mixture Representation

Here, we derive a useful linear representation for the PDF (7) of the MKL distribution. Al-Babtain et al. [22] derived a linear representation of the MK-G CDF (1) as follows:(11)Fx=∑j,k=0∞δj,kHj+1a+kx,where Hj+1a+kx is the CDF of the exp-G family with power parameter j+1a+k and the term δj,k is given by(12)δj,k=a−1jj!j+1a+kj+1a+kk.Then, the linear representation of the CDF of the MKL distribution takes the form(13)Fx=∑j,k=0∞δj,k1−xβ+1−λj+1a+k=∑j,k,m=0∞Φj,k,mxβ+1−mλ,where Φj,k,m=a−1j+m/j!j+1a+kj+1a+kkj+1a+km.By differentiating the previous equation, we obtain a linear representation of the MKL PDF as(14)fx=∑j,k,m=0∞Ψj,k,mhmx,where Ψj,k,m=a−1j+m+1/j!j+1a+kj+1a+kkj+1a+km and hmx is the PDF of Lomax distribution with shape parameter mλ and scale parameter β. Hence, the properties of the proposed MKL model, such as moments, follow from those of the Lomax distribution.


### 3.3. Moments

Therth moments of the MKL distribution has the form(15)μr′=EXr=∫0∞xrfxdx=∑j,k,m=0∞Ψj,k,m∫0∞xrhmxdx.Then, therth moments of the MKL distribution follows, from the moments of Lomax model, as(16)μr′=∑j,k,m=0∞Ψj,k,mλmβrΓr+1Γmλ−rΓmλ+1.Settingr=1,2,3, and 4, respectively, we obtain the first four-moments of the MKL distribution.The moment generating function of the MKL distribution follows directly from the previous formula as(17)Mt=∑j,k,m,r=0∞Ψj,k,mtrr!∫0∞xrhmx=∑j,k,m,r=0∞Ψj,k,mtrr!λmβrΓr+1Γmλ−rΓmλ+1.The characteristic function of the MKL distribution follows from the last equation by settingt=it.


### 3.4. Order Statistics

The PDF and CDF of theith order statistic of the MKL distribution are(18)fi:nx=n!i−1!n−i!Fxi−11−Fxn−ifx=aλn!β−λβ+xλ−1β+x/βλ−1a−11−e−β+x/βλ−1ai−1e−β+x/βλ−1a−i+n+1ΓiΓ−i+n+1,Fi:nx=∑r=innrFxr1−Fxn−r=ni1−e−β+x/βλ−1aie−β+x/βλ−1an−iF211,i−n;i+1;1−ex+β/βλ−1a,where F211,i−n;i+1;1−ex+β/βλ−1a is a hypergeometric function.The joint PDF of therth and ith order d (for 1≤r<i≤n) by(19)fr,i:nx,y=n!r−1!n−i!i−r−1!Fxr−11−Fyn−iFy−Fxi−r−1fxfy.Hence, the joint PDF of two order statistics from the MKL distribution takes the form(20)fr,i:nx,y=n!a2λ2β−2λβ+xλ−1r−1!n−i!i−r−1!β+yλ−1e−β+x/βλ−1ae−β+y/βλ−1a×β+xβλ−1a−1β+yβλ−1a−1e−β+y/βλ−1an−i×1−e−β+x/βλ−1ar−1e−β+x/βλ−1a−e−β+y/βλ−1ai−r−1.


### 3.5. Incomplete Moments

Thesth incomplete moment of the MKL distribution is given by(21)Ψst=∫0txsfxdx=∑j,k,m=0∞Ψj,k,m∫0txshmxdx=∑j,k,m=0∞Ψj,k,mλm−11−sβsB−t/βs+1,−mλ,where Bza,b=∫0zta−11−tb−1dt.The important application of the first incomplete moment is related to the Bonferroni and Lorenz curves defined byLp=Ψ1t/μ1′ and Bp=Ψ1xp/pμ1′, respectively, where xp can be evaluated numerically by equation (22) for a given probability p. These curves are very useful in economics, demography, insurance, engineering, and medicine. Another application of the first incomplete moment refers to the mean residual life (MRL) and the mean waiting time given by m1t=1−Ψ1t/St−t and M1t=t−Ψ1t/Ft, respectively.


## 3.1. Quantile Function

The quantile function (QF) of the MKL distribution is obtained by determining the inverse function of the MKL CDF (6) as(9)Qp=β−log1−p1/a+11/λ−1,0<p<1.The three quartiles of the MKL distribution can be obtained by settingp=0.25,0.5, and 0.75, respectively, in (9).Letp∼U0,1; then, the QF can be used directly in generating random data from the MKL distribution as follows:(10)xi=β−log1−pi1/a+11/λ−1,i=1,2,…,n.


## 3.2. Mixture Representation

Here, we derive a useful linear representation for the PDF (7) of the MKL distribution. Al-Babtain et al. [22] derived a linear representation of the MK-G CDF (1) as follows:(11)Fx=∑j,k=0∞δj,kHj+1a+kx,where Hj+1a+kx is the CDF of the exp-G family with power parameter j+1a+k and the term δj,k is given by(12)δj,k=a−1jj!j+1a+kj+1a+kk.Then, the linear representation of the CDF of the MKL distribution takes the form(13)Fx=∑j,k=0∞δj,k1−xβ+1−λj+1a+k=∑j,k,m=0∞Φj,k,mxβ+1−mλ,where Φj,k,m=a−1j+m/j!j+1a+kj+1a+kkj+1a+km.By differentiating the previous equation, we obtain a linear representation of the MKL PDF as(14)fx=∑j,k,m=0∞Ψj,k,mhmx,where Ψj,k,m=a−1j+m+1/j!j+1a+kj+1a+kkj+1a+km and hmx is the PDF of Lomax distribution with shape parameter mλ and scale parameter β. Hence, the properties of the proposed MKL model, such as moments, follow from those of the Lomax distribution.


## 3.3. Moments

Therth moments of the MKL distribution has the form(15)μr′=EXr=∫0∞xrfxdx=∑j,k,m=0∞Ψj,k,m∫0∞xrhmxdx.Then, therth moments of the MKL distribution follows, from the moments of Lomax model, as(16)μr′=∑j,k,m=0∞Ψj,k,mλmβrΓr+1Γmλ−rΓmλ+1.Settingr=1,2,3, and 4, respectively, we obtain the first four-moments of the MKL distribution.The moment generating function of the MKL distribution follows directly from the previous formula as(17)Mt=∑j,k,m,r=0∞Ψj,k,mtrr!∫0∞xrhmx=∑j,k,m,r=0∞Ψj,k,mtrr!λmβrΓr+1Γmλ−rΓmλ+1.The characteristic function of the MKL distribution follows from the last equation by settingt=it.


## 3.4. Order Statistics

The PDF and CDF of theith order statistic of the MKL distribution are(18)fi:nx=n!i−1!n−i!Fxi−11−Fxn−ifx=aλn!β−λβ+xλ−1β+x/βλ−1a−11−e−β+x/βλ−1ai−1e−β+x/βλ−1a−i+n+1ΓiΓ−i+n+1,Fi:nx=∑r=innrFxr1−Fxn−r=ni1−e−β+x/βλ−1aie−β+x/βλ−1an−iF211,i−n;i+1;1−ex+β/βλ−1a,where F211,i−n;i+1;1−ex+β/βλ−1a is a hypergeometric function.The joint PDF of therth and ith order d (for 1≤r<i≤n) by(19)fr,i:nx,y=n!r−1!n−i!i−r−1!Fxr−11−Fyn−iFy−Fxi−r−1fxfy.Hence, the joint PDF of two order statistics from the MKL distribution takes the form(20)fr,i:nx,y=n!a2λ2β−2λβ+xλ−1r−1!n−i!i−r−1!β+yλ−1e−β+x/βλ−1ae−β+y/βλ−1a×β+xβλ−1a−1β+yβλ−1a−1e−β+y/βλ−1an−i×1−e−β+x/βλ−1ar−1e−β+x/βλ−1a−e−β+y/βλ−1ai−r−1.


## 3.5. Incomplete Moments

Thesth incomplete moment of the MKL distribution is given by(21)Ψst=∫0txsfxdx=∑j,k,m=0∞Ψj,k,m∫0txshmxdx=∑j,k,m=0∞Ψj,k,mλm−11−sβsB−t/βs+1,−mλ,where Bza,b=∫0zta−11−tb−1dt.The important application of the first incomplete moment is related to the Bonferroni and Lorenz curves defined byLp=Ψ1t/μ1′ and Bp=Ψ1xp/pμ1′, respectively, where xp can be evaluated numerically by equation (22) for a given probability p. These curves are very useful in economics, demography, insurance, engineering, and medicine. Another application of the first incomplete moment refers to the mean residual life (MRL) and the mean waiting time given by m1t=1−Ψ1t/St−t and M1t=t−Ψ1t/Ft, respectively.


## 4. Methods of Estimation

Here, we discuss the estimation of the MKL parameters by different approaches of estimation including the maximum likelihood estimates (MLE), Anderson–Darling estimates (ADE), maximum product of spacings estimates (MPSE), least-squares estimates (LSE), Cramér–von Mises estimates (CVME), and weighted least-squares estimates (WLSE). The parameters of several models have been estimated using classical methods. For example, quasi xgamma-geometric [23], Weibull Marshall–Olkin Lindley [24], heavy-tailed exponential [25], transmuted Burr X exponential [26], and discrete Poisson–Lindley and discrete Lindley [22], among many others.


### 4.1. Maximum Likelihood Estimation

Letx1,x2,…,xn be a random sample of size n from the PDF (7); then, the log-likelihood function reduces to(22)L=nlogaλβ−λ−∑i=1nβ+xiβλ−1a+a−1∑i=1nlogβ+xiβλ−1+λ−1∑i=1nlogβ+xi.By differentiating equation (22) with respect to a, λ, and β, respectively, and equating to zero, we obtain(23)∂L∂a=na−∑i=1nβ+xiβλ−1alogβ+xiβλ−1+∑i=1nlogβ+xiβλ−1=0,∂L∂λ=a−1∑i=1nβ+xi/βλlogβ+xi/ββ+xi/βλ−1−∑i=1naβ+xiβλlogβ+xiββ+xiβλ−1a−1+∑i=1nlogβ+xi+n−logβ+nλ=0,∂L∂β=a−1∑i=1n−λxiβ+xi/βλ−1β2β+xi/βλ−1−∑i=1n−aλxiβ+xi/βλ−1β+xi/βλ−1a−1β2+λ−1∑i=1n1β+xi−λnβ=0.Solving the previous equations, we obtain estimators of the MKL parameters by the MLEs. The previous equations cannot be solved explicitly; hence, the numerical techniques can be used maximize the log-likelihood function to get the MLEs using several programs such as the R, SAS, Mathcad.


### 4.2. Ordinary Least-Squares and Weighted Least-Squares Estimators

Letx1:n,x2:n,…,x2:n be the order statistics of a random sample of size n from the MKL distribution. Hence, we can obtain the LSE of the MKL parameters by minimizing the following equation:(24)O=∑i=1nFxi:n−in+12=∑i=1n1−e−β+xi/βλ−1a−in+12.The LSE of the MKL parameters can also be obtained by solving the following equations:(25)∑i=1n1−e−β+xi/βλ−1a−in+1ϑkxi:n=0,k=1,2,3,where(26)ϑ1xi:n=∂∂aFxi:n=e−β+x/βλ−1aβ+xβλ−1alogβ+xβλ−1,(27)ϑ2xi:n=∂∂λFxi:n=aβ+xβλlogβ+xβe−β+x/βλ−1aβ+xβλ−1a−1,(28)ϑ3xi:n=∂∂βFxi:n=−aλxβ+x/βλ−1e−β+x/βλ−1aβ+x/βλ−1a−1β2.The WLSE of the MKL parameters are obtained by minimizing the following equation:(29)W=∑i=1nn+12n+2in−i+1Fxi:n−in+12=∑i=1nn+12n+2in−i+11−e−β+xi/βλ−1a−in+12.Moreover, the WLSE of the MKL parameters are also obtained by solving the following equations:(30)∑i=1nn+12n+2in−i+11−e−β+xi/βλ−1a−in+1ϑkxi:n=0,where ϑkxi:n for k=1,2,3 are defined in (26)–(28).


### 4.3. Anderson–Darling Estimation

The ADE of the MKL parameters can be obtained by minimizing(31)A=−n−1n∑i=1n2i−1logFxi:n+logSxi:n.The ADE are also be calculated by solving the following equations:(32)∑i=1n2i−1ϑkxi:nFxi:n−ϑkxn+1−i:nSxn+1−i:n=0,where ϑkxi:n, for k=1,2,3, are defined in (26)–(28).


### 4.4. Cramér–von Mises Estimation

The CVME of MKL parameters are obtained by minimizing the following equation:(33)CV=112n+∑i=1nFxi:n−2i−12n2=112n+∑i=1n1−e−β+xi/βλ−1a−2i−12n2,or by solving the following equations(34)∑i=1n1−e−β+xi/βλ−1a−2i−12nϑkxi:n=0,where ϑkxi:n, for k=1,2,3, were defined in (26)–(28).


### 4.5. Maximum Product of Spacings’ Estimation

The maximum product of spacings (MPS) method is used to estimate the parameters of continuous models as a good alternative to the maximum likelihood method. The uniform spacings of a random sample of sizen from the MKL distribution is defined by(35)Di=Fxi−Fxi−1,where Di are the uniform spacings, Fx0=0, Fxn+1=1, and ∑i=1n+1Di=1. The MPSE of the MKL parameters can be obtained by maximizing(36)G=1n+1∑i=1n+1logDi,with respect to a, λ, and β. Furthermore, the MPSE of the MKL parameters can also be calculated by solving(37)1n+1∑i=1n+11Diϑkxi:n−ϑkxi−1:n=0,where ϑkxi:n, for k=1,2,3, were defined in (26)–(28).


## 4.1. Maximum Likelihood Estimation

Letx1,x2,…,xn be a random sample of size n from the PDF (7); then, the log-likelihood function reduces to(22)L=nlogaλβ−λ−∑i=1nβ+xiβλ−1a+a−1∑i=1nlogβ+xiβλ−1+λ−1∑i=1nlogβ+xi.By differentiating equation (22) with respect to a, λ, and β, respectively, and equating to zero, we obtain(23)∂L∂a=na−∑i=1nβ+xiβλ−1alogβ+xiβλ−1+∑i=1nlogβ+xiβλ−1=0,∂L∂λ=a−1∑i=1nβ+xi/βλlogβ+xi/ββ+xi/βλ−1−∑i=1naβ+xiβλlogβ+xiββ+xiβλ−1a−1+∑i=1nlogβ+xi+n−logβ+nλ=0,∂L∂β=a−1∑i=1n−λxiβ+xi/βλ−1β2β+xi/βλ−1−∑i=1n−aλxiβ+xi/βλ−1β+xi/βλ−1a−1β2+λ−1∑i=1n1β+xi−λnβ=0.Solving the previous equations, we obtain estimators of the MKL parameters by the MLEs. The previous equations cannot be solved explicitly; hence, the numerical techniques can be used maximize the log-likelihood function to get the MLEs using several programs such as the R, SAS, Mathcad.


## 4.2. Ordinary Least-Squares and Weighted Least-Squares Estimators

Letx1:n,x2:n,…,x2:n be the order statistics of a random sample of size n from the MKL distribution. Hence, we can obtain the LSE of the MKL parameters by minimizing the following equation:(24)O=∑i=1nFxi:n−in+12=∑i=1n1−e−β+xi/βλ−1a−in+12.The LSE of the MKL parameters can also be obtained by solving the following equations:(25)∑i=1n1−e−β+xi/βλ−1a−in+1ϑkxi:n=0,k=1,2,3,where(26)ϑ1xi:n=∂∂aFxi:n=e−β+x/βλ−1aβ+xβλ−1alogβ+xβλ−1,(27)ϑ2xi:n=∂∂λFxi:n=aβ+xβλlogβ+xβe−β+x/βλ−1aβ+xβλ−1a−1,(28)ϑ3xi:n=∂∂βFxi:n=−aλxβ+x/βλ−1e−β+x/βλ−1aβ+x/βλ−1a−1β2.The WLSE of the MKL parameters are obtained by minimizing the following equation:(29)W=∑i=1nn+12n+2in−i+1Fxi:n−in+12=∑i=1nn+12n+2in−i+11−e−β+xi/βλ−1a−in+12.Moreover, the WLSE of the MKL parameters are also obtained by solving the following equations:(30)∑i=1nn+12n+2in−i+11−e−β+xi/βλ−1a−in+1ϑkxi:n=0,where ϑkxi:n for k=1,2,3 are defined in (26)–(28).


## 4.3. Anderson–Darling Estimation

The ADE of the MKL parameters can be obtained by minimizing(31)A=−n−1n∑i=1n2i−1logFxi:n+logSxi:n.The ADE are also be calculated by solving the following equations:(32)∑i=1n2i−1ϑkxi:nFxi:n−ϑkxn+1−i:nSxn+1−i:n=0,where ϑkxi:n, for k=1,2,3, are defined in (26)–(28).


## 4.4. Cramér–von Mises Estimation

The CVME of MKL parameters are obtained by minimizing the following equation:(33)CV=112n+∑i=1nFxi:n−2i−12n2=112n+∑i=1n1−e−β+xi/βλ−1a−2i−12n2,or by solving the following equations(34)∑i=1n1−e−β+xi/βλ−1a−2i−12nϑkxi:n=0,where ϑkxi:n, for k=1,2,3, were defined in (26)–(28).


## 4.5. Maximum Product of Spacings’ Estimation

The maximum product of spacings (MPS) method is used to estimate the parameters of continuous models as a good alternative to the maximum likelihood method. The uniform spacings of a random sample of sizen from the MKL distribution is defined by(35)Di=Fxi−Fxi−1,where Di are the uniform spacings, Fx0=0, Fxn+1=1, and ∑i=1n+1Di=1. The MPSE of the MKL parameters can be obtained by maximizing(36)G=1n+1∑i=1n+1logDi,with respect to a, λ, and β. Furthermore, the MPSE of the MKL parameters can also be calculated by solving(37)1n+1∑i=1n+11Diϑkxi:n−ϑkxi−1:n=0,where ϑkxi:n, for k=1,2,3, were defined in (26)–(28).


## 5. Simulation Results

Now, we explore and compare the performance of the introduced methods in estimating the MKL parameters based on simulation results. Several sample sizes,n=20,70,150,250,400, and several parametric values for the parameters, a=0.25,0.50,0.75,1.50,2.0,3.0, λ=0.50,0.75,1.50,2.0,3.0, and β=0.25,0.50,0.75,1.50,2.0, are considered to generate N=10,000 random samples from the MKL distribution using its QF and to determine the AVBs’, MSEs’, and MREs’ measures using the R program. The AVBs’, MSEs’, and MREs’ measures are calculated using the following equations:(38)AVBs=1N∑i=1Nφ^−φ,MSEs=1N∑i=1Nφ^−φ2,MREs=1N∑i=1Nφ^−φφ,where φ=a,λ,β′.Tables1–6 report detailed simulation results for the six estimation methods including AVBs, MSEs, and MREs for the parameters of the MKL distribution. These tables also present the ordering of the simulation measures, AVBs, MSEs, and MREs, based on partial and overall ranks which are given for each combination and sample size. In conclusion, the estimates, from six estimation methods, of the MKL parameters are quite good and close to their true values, showing small and decreasing AVBs, MSEs, and MREs in all parameter combinations. Moreover, all methods have the consistency property, i.e., the MREs and MSEs decrease as sample size, n, increases, for all parameter combinations.Table 1
Simulation results for the AVBs, MSEs and MREs of the MKL distribution witha=0.25,λ=0.5,andβ=0.75.
nEst.Est. par.MLEADECVMEMPSELSEWLSE20AVBsa^0.0493140.0487530.0565650.0411310.05860.044762λ^0.1379740.1195430.1186620.1084910.1469660.138355β^0.3810430.3733720.4083150.3452110.4115560.383674MSEsa^0.0050940.0044530.0063260.0032510.0061250.003492λ^0.030160.0214420.022130.01910.0300150.026664β^0.1946240.1836920.2034660.1669110.2024750.191813MREsa^0.1972340.1950130.2262350.1645110.2320160.179042λ^0.2759440.2390830.2373120.2169810.2939360.276695β^0.5080630.4978320.5444250.4602710.5487460.511564∑Ranks3642323959151631370AVBsa^0.027423.50.026820.0363660.0230610.027423.50.031185λ^0.0786810.079320.0927260.0837540.0879350.082023β^0.2813750.2565420.263130.2664440.239810.292976MSEsa^0.0011620.0012640.002460.0008910.0011830.00175λ^0.0090310.0091920.0131260.0101740.012750.010093β^0.1056350.08720.0897130.1031640.0838510.10736MREsa^0.1096630.1071920.1454460.0922310.1096840.124725λ^0.1573610.1585920.1854360.1674940.1758650.164043β^0.3751750.3420520.350830.3552540.3197410.390626∑Ranks26.5220145627328.54425150AVBsa^0.0209340.0193310.0228960.0194620.0207330.021575λ^0.0598540.0564520.066660.0567930.0634950.056371β^0.1802820.203740.2344860.1429710.2170750.20233MSEsa^0.0007130.0006520.0008860.0006410.0007550.000734λ^0.005840.0049320.0064560.0053330.006450.00471β^0.050320.057940.0682760.041410.0622450.056993MREsa^0.0837240.077310.0915560.0778620.0829130.086285λ^0.1196940.1129120.133260.1135730.1269850.112741β^0.2403820.271640.3126560.1906310.2894350.269743∑Ranks294222546171415263250AVBsa^0.0142120.0155630.0174260.0160650.0158340.014021λ^0.0410620.0452630.0514650.0409210.0515760.046324β^0.1212220.1584750.1652760.0949110.1425330.156464MSEsa^3e−0410.0003930.000485.50.000485.54e−0440.000352λ^0.0027110.0033930.0041750.0032420.0042360.003414β^0.0254420.0349360.0341750.0210.0285430.032854MREsa^0.0568320.0622430.0696960.0642350.0633340.056091λ^0.0821220.0905230.1029250.0818510.1031560.092654β^0.1616320.211350.2203760.1265510.1900530.208624∑Ranks20133447.5620.52415273400AVBsa^0.009320.0088210.0116460.0093330.0108940.011075λ^0.0233510.0281630.0352850.0265920.0359460.031814β^0.0503520.0781130.0897360.0418210.0890950.083674MSEsa^0.0001520.0001412e−0450.0001630.0001940.000216λ^0.0010710.0013930.00260.0012220.0018750.001534β^0.005920.0090330.0104750.0052610.0105460.009724MREsa^0.037220.0352710.0465460.0373130.0435740.044275λ^0.046710.0563230.0705650.0531920.0718960.063614β^0.0671320.1041430.1196460.0557710.1187950.111564∑Ranks151213516182455394Table 2
Simulation results for the AVBs, MSEs, and MREs of the MKL distribution witha=1.5,λ=2,andβ=0.5.
nEst.Est. par.MLEADECVMEMPSELSEWLSE20AVBsa^0.3309230.3658960.3495540.2500710.3223420.365875λ^0.9002840.9713450.8090220.5328710.8501931.034246β^0.2686940.2870550.2449220.1758810.2567630.301196MSEsa^0.1805320.2276660.1877830.1083810.1966740.208795λ^1.0219331.1680650.8987120.6462611.0250841.229896β^0.088540.1004950.0789420.0551910.0867830.104326MREsa^0.2206130.243925.50.2330340.1667110.214920.243925.5λ^0.4501440.4856750.4045120.2664410.425130.517126β^0.5373840.574150.4898420.3517710.5135230.602376∑Ranks31447.552329127351.5670AVBsa^0.2199740.1986620.2279350.1347510.2547460.210253λ^0.6446920.7297450.7973860.3030310.6653230.701664β^0.1923720.212640.2358360.1068310.1967930.21585MSEsa^0.0835150.0650520.0832940.0292910.1160260.073553λ^0.5509620.6253540.7805860.2967110.6114430.633465β^0.0482320.0538740.0665960.026210.0523530.05765MREsa^0.1466540.1324420.1519550.0898310.1698360.140173λ^0.3223420.3648750.3986960.1515110.3326630.350834β^0.3847520.4252140.4716760.2136610.3935830.43165∑Ranks25232350691364375150AVBsa^0.1254950.118230.1208140.0821410.1288460.117972λ^0.4474720.4700530.4710840.1078710.492650.493156β^0.1326720.1390130.1399340.0466610.1520560.148175MSEsa^0.0255760.0222120.0242240.0108110.0253650.023133λ^0.2966520.3085730.3095840.0851410.3356460.327135β^0.0255820.0265230.0269740.0080810.0303960.029915MREsa^0.0836650.078830.0805440.0547610.0858960.078652λ^0.2237420.2350330.2355440.0539410.246350.246586β^0.2653420.2780230.2798540.0933110.304160.296345∑Ranks28326236491516395250AVBsa^0.0846830.0905740.1035360.060510.1017350.080932λ^0.3579220.3893650.3959360.0137510.3830940.3813β^0.1073520.1140240.1146250.0162310.1121430.118816MSEsa^0.0109820.0126540.0175460.0054310.0161450.011143λ^0.1723420.203560.2021250.0058210.1938540.192013β^0.0151820.0173850.0170348e−0410.0168130.018196MREsa^0.0564630.0603840.0690260.0403310.0678250.053952λ^0.1789620.1946850.1979760.0068810.1915440.19053β^0.214720.2280440.2292550.0324710.2242930.237626∑Ranks192405486141354333400AVBsa^0.0566230.049920.0690660.0460710.0614340.061695λ^0.2067340.2224160.1933420.0059610.2028130.207185β^0.063450.0678660.0587920.0107610.0603530.062864MSEsa^0.0044630.0040420.0073960.0032710.005650.005474λ^0.0579750.0643660.0522626e−0510.0549930.057314β^0.0052850.0057360.0046322e−0410.0048730.005134MREsa^0.0377530.0332720.0460460.0307110.0409540.041135λ^0.1033740.111260.0966720.0029810.1014130.103595β^0.1267950.1357260.1175720.0215210.120730.125724∑Ranks354406282191293385Table 3
Simulation results for the AVBs, MSEs, and MREs of the MKL distribution witha=0.5,λ=1.5,andβ=0.25.
nEst.Est. par.MLEADECVMEMPSELSEWLSE20AVBsa^0.0900520.0971330.1352360.0803410.1047750.097614λ^0.4874410.4898920.5986860.5230940.5920350.503033β^0.1170410.1188120.1238140.1310460.1299850.123023MSEsa^0.0136820.0165230.0323360.0128210.0195750.017634λ^0.3452320.3433910.4592260.384540.4452550.356083β^0.0180110.0180820.0194740.0209650.0218860.019453MREsa^0.180120.1942730.2704760.1606810.2095350.195234λ^0.3249610.3265920.3991260.3487340.3946950.335353β^0.4681410.4752620.4952440.5241560.5199150.492083∑Ranks13120248632446530370AVBsa^0.061940.0670250.0713660.0540210.059720.060773λ^0.357210.4158850.3767330.3650220.4251860.415044β^0.082510.0948550.0931330.0856220.1010660.094524MSEsa^0.0069540.0076950.0090660.0045110.0059720.006363λ^0.19520.2382950.1932210.1981530.2395960.228634β^0.0101120.011950.0114630.0099610.0126960.011544MREsa^0.123840.1340350.1427360.1080410.119420.121543λ^0.2381310.2772550.2511530.2433520.2834660.27674β^0.3300110.3794250.372530.3424820.4042560.378084∑Ranks202456344151425333150AVBsa^0.0399130.0393420.0372410.0400140.0420160.040735λ^0.2691740.257510.3028750.2596530.3063260.258882β^0.0638540.0573110.0677260.0607920.0676850.06283MSEsa^0.0024840.0023720.002210.0024530.0031160.002735λ^0.104840.0924510.1215850.1001620.1285360.102773β^0.0054930.0046610.0059850.0053820.0064560.005544MREsa^0.0798130.0786820.0744910.0800340.0840360.081465λ^0.1794540.1716710.2019150.173130.2042160.172592β^0.255440.2292510.2708960.2431820.2707350.251213∑Ranks334121355252526323250AVBsa^0.0294140.0270420.0354260.0241210.028530.029515λ^0.1891620.2287150.2540960.1772610.2134630.216994β^0.0471520.0545750.0596560.0467610.0497330.050944MSEsa^0.0015250.001162.50.001969e−0410.001162.50.001354λ^0.0513620.0706650.0869560.0491910.0638340.063663β^0.0031320.0038950.0043960.002910.0033630.003444MREsa^0.0588240.0540720.0708360.0482310.05730.059025λ^0.1261120.1524750.1693960.1181810.1423130.144664β^0.188620.2182850.2385960.1870310.198930.203754∑Ranks24235.5453614126.53365400AVBsa^0.020330.0209950.0220560.0192910.0201420.020764λ^0.1042510.1208740.1402660.108320.1375850.118823β^0.0269110.0304450.0310560.0279830.028140.026982MSEsa^0.0006330.0006540.0007460.0005410.0006127e−045λ^0.0171810.0226740.0273360.0221930.0255850.020882β^0.000991.50.0011850.0013360.0011240.001130.000991.5MREsa^0.0405930.0419950.044160.0385910.0402820.041524λ^0.069510.0805840.0935160.072220.0917250.079213β^0.1076310.1217550.1242160.1119330.1124240.107922∑Ranks15.5141553620232427.53Table 4
Simulation results for the AVBs, MSEs, and MREs of the MKL distribution witha=0.75,λ=1.5,andβ=2.
nEst.Est. par.MLEADECVMEMPSELSEWLSE20AVBsa^0.1741140.1957560.1908550.1291620.1238610.161583λ^0.5417330.6175760.599440.423510.5086720.605365β^0.9718431.0691660.9792540.6951410.8483621.01955MSEsa^0.0530540.0680660.0609950.0299120.0278310.046953λ^0.3998730.4849260.4775540.2784810.3454620.482275β^1.2349131.408761.3088540.8490110.9748921.344145MREsa^0.2321540.2610160.2544750.1722120.1651410.215443λ^0.3611630.4117260.399640.2823310.3391120.403575β^0.4859230.5345860.4896340.3475710.4241820.509755∑Ranks303546394.5121152394.570AVBsa^0.1024250.0965830.1111360.0871410.090420.098374λ^0.4278130.4750160.4675940.3649610.4716950.405412β^0.7513530.8576160.8335450.5949810.8000140.714552MSEsa^0.0177350.0160430.0215560.0126610.0149920.017684λ^0.2507230.2990160.29140.2259410.2927650.232142β^0.7375430.9234460.8767850.6532110.8277340.696142MREsa^0.1365650.1287830.1481760.1161910.1205420.131164λ^0.285230.3166760.3117340.2433110.3144650.270272β^0.3756830.428860.4167750.2974910.440.357282∑Ranks333.5455.5455.591333.5242150AVBsa^0.0603920.0514210.0640850.0607640.0607430.068786λ^0.295430.2831520.3367460.2672810.3091740.310075β^0.5210120.5338630.5888760.4255110.5467140.562295MSEsa^0.0061630.0041310.0068550.00620.0062140.00746λ^0.1261430.1197110.1522160.1253520.1356540.136325β^0.3857920.3858630.4516760.370110.4095440.427885MREsa^0.0805220.0685610.0854450.0810140.0809830.091716λ^0.1969430.1887720.224560.1781810.2061140.206715β^0.260520.2669330.2944360.2127510.2733540.281155∑Ranks223171.5516171.5344485250AVBsa^0.0424630.0471560.0405820.0388910.0467750.045384λ^0.214520.2294640.2280930.1383510.255960.230935β^0.399530.3984320.399940.1966110.4671760.429765MSEsa^0.0029330.0034960.0025920.0025810.0033850.003354λ^0.0627820.0759340.0732930.0434310.0889160.07715β^0.2202430.2211840.2175220.1220410.2809860.248065MREsa^0.0566130.0628760.054120.0518510.0623750.060514λ^0.14320.1529840.1520630.0922310.170660.153965β^0.1997530.1992120.1999540.0983110.2335860.214885∑Ranks24238425391516425400AVBsa^0.0314530.0273520.0339250.0249410.0332640.034266λ^0.1205120.1239530.1284440.0379610.144160.138275β^0.2174420.224330.2370340.0218110.2420250.244946MSEsa^0.0014430.0011620.0018350.0009310.0017840.001976λ^0.0212320.0214630.0243740.0022710.0279560.026365β^0.0663320.0667130.0764240.0007210.0778650.079736MREsa^0.0419330.0364720.0452250.0332610.0443440.045686λ^0.0803420.0826430.0856240.0253110.0960760.092185β^0.1087220.1121530.1185140.010910.1210150.122476∑Ranks21224339491455516Table 5
Simulation results for the AVBs, MSEs and MREs of the MKL distribution witha=2,λ=3,andβ=1.5.
nEst.Est. par.MLEADECVMEMPSELSEWLSE20AVBsa^0.4573350.4605860.3732820.2501110.4072430.410644λ^1.3079441.4189450.5926420.1283110.8680331.597736β^0.7441140.7977950.3700120.1234410.5224630.874776MSEsa^0.3652450.378560.2464620.0891410.2818830.322634λ^2.4401442.7076751.2027620.2275111.7689433.010496β^0.7623940.8318750.3768620.0761610.5520730.9116MREsa^0.2286750.2302960.1866420.1250610.2036230.205324λ^0.4359840.4729850.1975520.0427710.2893430.532586β^0.4960740.5318650.2466720.082310.3483130.583186∑Ranks394485.518291273485.570AVBsa^0.2714760.2038520.216330.1635210.2645350.217384λ^1.1162251.0915840.7218120.0242710.8407731.16466β^0.6245340.6313250.4232120.0458110.482430.652286MSEsa^0.1220560.0638720.0842240.0388610.1146850.082413λ^1.6203651.5811241.1505129e−0411.3839731.710566β^0.5039740.5090250.3650120.0033910.4297430.538396MREsa^0.1357460.1019220.1081530.0817610.1322650.108694λ^0.3720750.3638640.240620.0080910.2802630.38826β^0.4163540.4208850.2821420.0305410.321630.434866∑Ranks445323.5212141323.5466150AVBsa^0.1585330.1721250.1765160.1172510.1426120.163514λ^0.8378840.9112360.8065730.014410.7105520.883375β^0.4703440.5129160.4626530.0306210.4062820.49775MSEsa^0.0385830.0460350.0502360.0208610.0302620.042354λ^0.9732131.0451761.0207940.0003310.8472421.026045β^0.3032930.3266560.3188140.0015410.2661220.322075MREsa^0.0792730.0860650.0882660.0586210.0713120.081764λ^0.2792940.3037460.2688630.004810.2368520.294465β^0.3135640.3419460.3084330.0204210.2708520.33185∑Ranks31351638491182425250AVBsa^0.0772410.090730.1067940.0877420.1095960.107185λ^0.5546830.5611340.5446520.0147710.5745650.59376β^0.3186140.3175130.3103620.0289410.3242550.331326MSEsa^0.0093810.0141130.0180640.0114420.0182250.01866λ^0.4285120.4314430.4714240.0003110.484660.482595β^0.1402230.1369220.14940.0012510.1491450.149786MREsa^0.0386210.0453530.053440.0438720.0547960.053595λ^0.1848930.1870440.1815520.0049210.1915250.19796β^0.2124140.2116730.2069120.019310.2161750.220886∑Ranks222283.5283.5121485516400AVBsa^0.0694920.0801850.0744940.0687610.0902460.071283λ^0.2671930.3465260.2344320.0086210.2915940.322125β^0.1560530.1942660.1347520.0188610.1660940.177845MSEsa^0.0079720.0098750.0093240.007310.0129460.007983λ^0.1117430.1578860.1102220.0001310.1416450.134314β^0.0367130.0498360.0336920.0005810.0437450.041734MREsa^0.0347520.0400950.0372540.0343810.0451260.035643λ^0.0890630.1155160.0781420.0028710.097240.107375β^0.1040430.1295160.0898320.0125810.1107340.118565∑Ranks242.5516242.591445374Table 6
Simulation results for the AVBs, MSEs, and MREs of the MKL distribution witha=3,λ=0.75,andβ=2.
nEst.Est. par.MLEADECVMEMPSELSEWLSE20AVBsa^1.2921621.4276931.5299750.8536911.5587861.459664λ^0.281120.3702860.3357330.1724910.3613950.357824β^1.0479921.3923461.2395130.6300611.3365241.34635MSEsa^2.839522.8693233.2336961.4916213.207852.952474λ^0.1187220.1583750.150730.0728910.1624660.154574β^1.6153222.2015952.0366530.9631512.2071762.140134MREsa^0.4307220.475930.5099950.2845610.5195960.486554λ^0.3748120.493760.4476430.2299910.4818530.477094β^0.52420.6961760.6197630.3150310.6682640.673155∑Ranks1824353439147638470AVBsa^0.5038160.4335720.4758640.2184710.4448630.493995λ^0.1824230.2028560.1660620.0180310.1910340.199585β^0.7207830.8105360.6602920.0771610.7574440.801715MSEsa^0.4573260.3400730.3922840.0694210.3225220.402685λ^0.04830.0527560.0421620.0005110.0526450.051164β^0.7447830.8371860.6569920.0086410.8365950.816264MREsa^0.1679460.1445220.1586240.0728210.1482930.164665λ^0.2432230.2704760.2214120.0240310.2547140.266115β^0.3603930.4052660.3301420.0385810.3787240.400865∑Ranks364435.524291343435.5150AVBsa^0.2953920.4359260.313930.1592110.3446350.344344λ^0.1359930.1543260.120420.0158610.1492440.151545β^0.5461230.6199760.49520.0547710.6072540.616445MSEsa^0.1495330.2767460.1466720.0384410.1709340.188135λ^0.0254630.0311360.0224920.0003710.0298540.030235β^0.408730.4968660.3768720.0046510.4931840.495955MREsa^0.0984620.1453160.1046330.0530710.1148850.114784λ^0.1813230.2057560.1605420.0211510.1989840.202065β^0.2730630.3099860.247520.0273910.3036240.308225∑Ranks25354620291384435250AVBsa^0.2273450.2145230.2620360.116210.2242640.182382λ^0.111350.1031440.112560.0100110.1014830.09162β^0.4588350.4151730.4617360.0416510.4173840.367812MSEsa^0.0817250.0742730.1080560.0204710.0800840.052682λ^0.0163660.0141540.0163550.0001610.0141130.011772β^0.2720350.2303630.2757860.0026510.2352440.192722MREsa^0.0757850.0715130.0873460.0387310.0747540.060792λ^0.148450.1375240.1560.0133410.1353130.122142β^0.2294150.2075930.2308760.0208310.2086940.183912∑Ranks46530353691334182400AVBsa^0.1318140.1497360.1405750.0862610.1275630.118932λ^0.0514320.0561560.054950.0084110.0526530.054534β^0.2111720.2291360.2202340.0316510.2154130.224155MSEsa^0.0264630.035460.0295850.011110.0278540.023432λ^0.0037720.0042860.0041150.0001110.0039130.003934β^0.0628720.0728460.0666250.0015210.0653130.065924MREsa^0.0439440.0499160.0468650.0287510.0425230.039642λ^0.0685720.0748660.0732150.0112110.070230.072714β^0.1055920.1145660.1101240.0158210.1077130.112085∑Ranks23254643591283324Table7 illustrates the partial and overall ranks of the six estimators. Table 7 shows that the performance ordering of the estimators are MPSE, MLE, ADE, LSE, WLSE, and CVME, respectively. Based on simulation results and the ranks of the estimators, we conclude that the MPSE outperforms other estimators with an overall score of 39.5 and can confirm the superiority of MPSE and MLE for estimating the parameters of the MKL distribution.Table 7
Partial and overall ranks of the estimation methods for various values ofa, λ, and β.
ParametersnMLEADECVMEMPSELSEWLSEa=0.25,λ=0.5,β=0.752042516370216345150426153250146253400136254a=1.5,λ=2,β=0.52045213670236145150324165250256143400462135a=0.5,λ=1.5,β=0.252012645370264153150415263250246135400156243a=0.75,λ=1.5,β=220364.5124.5703.55.55.513.5215031.561.545250243165400234156a=2,λ=3,β=1.52045.52135.57053.5213.5615036412525023.53.51564002.562.5154a=3,λ=0.75,β=2202531647045.52135.5150362145250536142400265134∑Ranks8312113139.5127128.5Overall rank236145


## 6. Application

In this section, we consider two real-life datasets to show the applicability and flexibility of the new MKL distribution. The first dataset is studied by [9, 27], and it represents successive failures of air conditioning systems for fleet of 13 Boeing 720 jet airplanes. The data are 50, 11, 14, 130, 16, 9, 57, 102, 15, 5, 14, 11, 10, 57, 261, 12, 51, 44, 239, 18, 33, 209, 181, 37, 111, 82, 230, 71, 80, 152, 46, 386, 182, 118, 44, 120, 141, 210, 11, 14, 186, 29, 14, 102, 4, 118, 72, 270, 31, 14, 88, 130, 15, 326, 18, 130, 35, 5, 18, 1, 79, 216, 61, 41, 58, 61, 60, 48, 56, 87, 98, 54, 12, 5, 22, 43, 201, 84, 27, 20, 206, 310, 220, 34, 29, 26, 90, 163, 208, 1, 134, 3, 184, 20, 35, 29, 65, 49, 7, 104, 21, 16, 283, 5, 22, 120, 66, 57, 139, 67, 3, 27, 76, 14, 100, 32, 9, 12, 5, 18, 3, 14, 39, 18, 13, 7, 34, 36, 79, 33, 246, 25, 59, 34, 18, 44, 191, 100, 7, 156, 320, 63, 22, 23, 62, 188, 11, 104, 31, 106, 153, 59, 42, 46, 26, 54, 62, 39, 36, 30, 23, 95, 16, 101, 71, 52, 208, 24, 12, 70, 230, 97, 254, 438, 502, 493, 487, and 603. The second dataset is studied by [8], and it contains 100 observations about waiting times (in minutes) in a bank before the customers receive their services. The data are 0.8, 4.3, 0.8, 1.5, 1.3, 1.8, 2.7, 1.9, 2.1, 1.9, 2.6, 3.1, 3.5, 2.9, 3.2, 3.3, 4, 3.6, 4.2, 4.1, 4.2, 4.4, 4.3, 4.9, 4.4, 7.1, 4.6, 7.1, 4.7, 7.1, 4.7, 6.9, 4.8, 5.0, 4.9, 5.3, 8.9, 5.5, 5.7, 8.8, 5.7, 6.1, 6.2, 9.5, 6.2, 6.3, 6.2, 6.7, 7.1, 10.7, 7.4, 7.7, 7.6, 8, 8.6, 8.2, 8.6, 8.8, 11.0, 8.9, 8.6, 9.6, 11.9, 12.5, 13.0, 13.3, 13.6, 13.7, 33.1, 9.7, 9.8, 11.1, 10.9, 11.0, 17.3, 11.2, 11.5, 12.4, 13.1, 11.2, 20.6, 12.9, 27, 31.6, 23, 13.9, 21.9, 18.9, 19.0, 14.1, 18.2, 15.4, 17.3, 15.4, 18.4, 18.1, 19.9, 21.3, 38.5, and 21.4.We compare the proposed MKL model with some competing models based on some discrimination measures and criteria such as Akaike information (AIC), consistent Akaike information (CAIC), Bayesian information (BIC), and Hannan–Quinn information (HQIC). The competing distributions include odd Lomax Lomax (OLL) [28], Lomax–Weibull (LW) [29], Lomax-exponential (LE) [29], beta exponentiated-Lomax (BExL) [30], beta-Lomax (BL) [30], exponentiated-Lomax (ExL) [30], Zubair–Lomax (ZL) [31], and Lomax (L) distributions.The estimates of the model parameters and their SEs (standard errors), using the maximum likelihood, along with discrimination measures are displayed in Tables8 and 9, respectively, for the two datasets. Furthermore, the estimates of the MKL parameters are obtained using different estimators and listed, for both datasets, in Tables 10 and 11, respectively. These tables also display the values of Anderson–Darling (A), Cramér–von Mises (W), and Kolmogorov–Smirnov (KS) statistic with its p value (KS-PV) for both datasets.Table 8
Estimates, SEs, and discrimination measures for successive failures of air conditioning systems’ data.
DistributionEstimatesSEsAICCAICBICHQICMKLa^=3.038651.784231960.241960.371969.81964.11λ^=0.1443140.08508β^=0.6371951.79215Lλ^=4.905982.35221963.441963.511969.821966.03β^=350.7199.943OLLα^=0.071690.017381977.571977.81990.321982.74β^=9.897414.4404a^=17.61833.9966b^=46.379122.5866LWα^=2.924041.917161966.861967.091979.611972.03β^=11.0795115725a^=1.09480.13109b^=0.05127489.109LEβ^=2.698634663.981970.011970.151979.581973.89a^=1.400990.08629b^=0.0420651.895BExLα^=1.636777.36951964.551964.91980.491971.02β^=0.000050.000249a^=20.682462.3831b^=21.175229.5167λ^=0.128620.31728BLα^=4.949288.562061965.991966.221978.741971.16β^=0.005680.00359a^=1.226410.20528b^=0.637211.09985ExLα^=3.14731.41181964.041964.181973.611967.92β^=0.005830.00407a^=1.215910.21023ZLα^=1.617020.266651969.931970.071979.491973.81β^=32.835116.9085a^=0.683371.0219Table 9
Estimates, SEs, and discrimination measures for waiting times in a bank data.
DistributionEstimatesSEsAICCAICBICHQICMKLa^=3.402983.21709640.027640.277647.842643.19λ^=0.2188570.240281β^=0.461611.67085Lλ^=1.48985×1082.65679×1011662.042662.042667.252664.15β^=1.47152×1092.62411×1012OLLα^=0.1594040.084276643.045643.466653.466647.263β^=10.89398.03526a^=4.34048×1089.96634×1011b^=5.05811×1081.16139×1012LWα^=3.0222.37068643.016643.437653.437647.234β^=0.947521.28896a^=1.769860.25992b^=0.0550760.664065LEβ^=0.060187337.599644.82645.07652.635647.983a^=2.267120.188173b^=0.0370491.6422BExLα^=17.8535160.591643.983644.621657.009649.255β^=0.000940.00955a^=48.763276.4906b^=8.763444.56521λ^=0.077696.26012BLα^=4.696614.31371642.446642.867652.866646.663β^=0.0232260.020468a^=2.378540.47328b^=2.002881.67736ExLα^=37.8584164.125640.136640.386647.951643.299β^=0.0044260.020182a^=2.272110.52796ZLα^=62.811331.2061640.538640.788648.354643.701β^=377.76644.7447a^=0.3577080.49548Table 10
The estimates of the MKL model and goodness-of-fit measures for successive failures of air conditioning systems’ data.
Methodsa^λ^β^AWKSKS-PVMLE3.038650.1443140.6371950.252620.0336130.040880.92579ADE2.086890.209752.906080.209490.023870.034970.98086CVME1.980690.219313.378160.2178260.0228690.037820.95997MPSE2.741440.1563180.9228330.2209440.0258010.0381890.95651LSE1.949820.221113.480510.231230.023250.039430.94352WLSE2.07650.2099182.918410.2103420.0234080.035750.97615Table 11
The estimates of the MKL model and goodness-of-fit measures for waiting times in a bank data.
Methodsa^λ^β^AWKSKS-PVMLE3.402980.2188570.461610.1454150.0205920.0394330.99772ADE2.948330.260060.7809460.1418950.0194140.0405800.99655CVME3.024410.2534340.7209230.1445060.0190250.0409270.99612MPSE2.52970.3015571.178170.1445060.0190250.0409270.99612LSE2.945670.2569540.7528820.1453750.01969220.04401760.990224WLSE2.797710.2734490.9014110.1464280.02031440.04332840.991902The relative histogram with the fitted MKL density along with fitted CDF, SF, and P-P plots for the MKL model is displayed in Figures3 and 4 for both datasets. These plots support the results in Tables 8 and 9.Figure 3
The fitted MKL PDF, CDF, SF, and P-P plots for for successive failures of air conditioning systems’ data.Figure 4
The fitted MKL PDF, CDF, SF, and P-P plots for waiting times in a bank data.It is shown, from Tables8 and 9, that the MKL distribution has the lowest values for goodness-of-fit criteria for both datasets among all fitted competing models. That is, it provides close fit for the data, and hence, it could be chosen as the best distribution for the two datasets.From Tables10 and 11 and based on the values of KS statistic, we conclude that all estimation methods perform well in estimating the parameters of the MKL distribution for the two datasets.


## 7. Concluding Remarks

In this study, a new three-parameter extension of the Lomax distribution called the modified Kies–Lomax (MKL) distribution has been studied. The MKL distribution provides flexible hazard rate and density functions which can have important forms, depending on its shape parameters, including increasing, unimodal, decreasing, and bathtub hazard rate shapes, and positive-skewed, symmetrical, negative-skewed, and reversed-J shaped densities. Furthermore, its density can be viewed as a linear mixture of Lomax distribution. The MKL parameters have been estimated via six estimation approaches, called, maximum likelihood, least-squares and weighted least-squares, Cramér–von Mises, maximum product of spacings, an Anderson–Darling. The simulation study has been illustrated that maximum product of spacings is the best performing method in terms of partial and overall ranks for absolute biases, mean square errors, and mean relative errors of the estimates. Two real data applications show that all estimators perform well for both datasets. Furthermore, the MKL distribution shows its flexibility in modeling successive failures of air conditioning systems and waiting times’ datasets as compared with some rival Lomax extensions.



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*Source: 1944864-2021-11-28.xml*