# Statistical Inference and Mathematical Properties of Burr X Logistic-Exponential Distribution with Applications to Engineering Data

Authors: Mashail M. AL Sobhi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The Burr X logistic-exponential distribution is introduced in this study as a novel logistic-exponential distribution extension that may be utilized to efficiently describe engineering data. There are J-shape, symmetrical, left-skewed, reversed-J shape, and right-skewed densities available, as well as decreasing, rising, bathtub, unimodal, J-shape, and reversed-J shape hazard rates. The fundamental mathematical features of the proposed model were obtained. The new model’s parameters were estimated using seven different approaches, including maximum likelihood, Anderson–Darling, maximum product of spacing, least-squares, Cramér–von Mises, percentiles, and weighted least squares. To evaluate the performance of the recommended estimation methods, a full simulation study was carried out. Finally, the adaptability of the provided distribution was tested using two real datasets from engineering science, revealing that the new model can yield a close match when compared to competing models.


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## Body

## 1. Introduction

Survival and reliability analysis is an important field of statistics with several applications in engineering, actuarial science, biomedical investigations, demography, and dependability. In these applicable disciplines, several writers have constructed generalized distributions to model various data. The exponential distribution is a popular data modeling model because it is analytically tractable and has a low memory need. However, due to its declining probability density function (PDF) and constant hazard rate function, its use was limited (HRF). As a result, various academics have developed generalized variants of the exponential distribution in order to improve its capacity to represent data in applicable domains and increase its flexibility in terms of PDF and HRF. Some important generlizations of the exponential (E) distribution are proposed as follows: the exponentiated-E [1], beta-exponential [2], logoistic-E [15], beta generalized-E [3], Nadarajah–Haghighi [4], transmuted generalized-E [5], Harris extended-E [6], Marshall–Olkin Nadarajah–Haghighi [7], transmuted Topp–Leone E [8], alpha power-E [9], Marshall–Olkin logistic E [10], extended-E [11], odd inverse power generalized Weibull-E [12, 13] studied the heavy-tailed E and Type I half logistic Burr E [14] distributions.The logistic-E (LE) distribution [15] is a significant generalization of the E distribution. We suggested a more flexible variant of the LE model termed the Burr X logistic-E (BXLE) distribution in this study. The BXLE allows for greater flexibility and application when modeling engineering data. The BXLE distribution was constructed using the Burr X family (Yousof et al. [16]).The Burr X family’s cumulative distribution function (CDF) looks like this:(1)Fx;κ,φ=1−e−Gx;φ/1−Gx;φ2κ,κ>0,x∈ℜ,where Gx;φ is the baseline model CDF and φ is the baseline parameter vector. The Burr X family’s PDF shrinks to(2)fx;κ,φ=2κgx;φGx;φ1−Gx;φ3e−Gx;φ/1−Gx;φ21−e−Gx;φ/1−Gx;φ2κ−1,κ>0,x∈ℜ.Some qualities can inspire the suggested BXLE distribution, for example, the BXLE model includes the Burr X E distribution as a special submodel; the BXLE distribution provides J-shape, symmetrical, left-skewed, reversed-J shape, and right-skewed densities, as well as decreasing, increasing, bathtub, then unimodal, J-shape, and reversed-J shape hazard rates; its PDF and CDF have simple closed forms and can thus be used effectively in analyzing censored data, and it has also been employed to model engineering datasets, where it outperforms other competing distributions in terms of fit.The key goal of this article is to investigate and deduce some of the basic distributional features of a new extension of the LE model based on the Burr X family. We are also interested in investigating the estimation of BXLE parameters using seven classical estimation methods, including maximum likelihood estimators (MLEs), Anderson–Darling estimators (ADEs), maximum product of spacing estimators (MPSEs), least-squares estimators (LSEs), Cramér–von Mises estimators (CVMEs), percentile estimators (PCEs), and weighted least-squares estimators (WLSEs). Extensive simulations were used to examine and analyze the performance of the suggested estimate approaches.The CDF of the BXLE model is obtained by substituting the CDF of the LE model in (1), as follows:(3)Fx=1−e−eρx−12δκ,ρ,δ,κ>0,x>0.The corresponding PDF of the BXLE distribution follows, by inserting the PDF and CDF of the LE model in (2), as(4)fx=2δκρeρx−eρx−12δeρx−12δ−11−e−eρx−12δκ−1,ρ,δ,κ>0,x>0.The BXLE distribution’s survival function (SF) and HRF take the required forms:(5)Sx=1−1−e−eρx−12δκ,hx=2δκρeρxeρx−12δ−11−e−eρx−12δκ−eeρx−12δ+11−e−eρx−12δκ−1.Figures1 and 2 provide plots of the PDF and HRF of the BXLE distribution, respectively. The BXLE model produces J-shape, symmetrical, left-skewed, reversed-J shape, and right-skewed densities, as well as decreasing, increasing, bathtub, then unimodal, J-shape, and reversed-J shape hazard rates.Figure 1
PDF plots of BXLE distribution.Figure 2
HRF plots of BXLE distribution.The remainder of this work is structured as follows. Section2 determined some basic distributional features of the BXLE distribution. In Section 3, the BXLE parameters were calculated using seven different approaches. Section 4 investigated the performance of these estimators using numerical simulations. In Section 5, two engineering actual datasets were evaluated to demonstrate the relevance and adaptability of the BLLE distribution. Finally, Section 6 gave the conclusions.


## 2. Mathematical Properties

In this section, we will introduce some important statistical properties such as linear representation, quantile function, moments, and order statistics.


### 2.1. Linear Representation

An expansion for (3) can be derived using the power series(6)1−zb=∑j=0∞−1jbjzj,where Z<1 and b>0.Then, the BXLE CDF could be written as(7)Fx=∑j=0∞−1jκje−jeρx−12δ.Applying the E series,(8)Fx=∑j,k=0∞−1j+kjkk!κje2δkρx1−e−ρx2δk,and then(9)Fx=∑j,k,m=0∞−1j+k+mjkk!κj2δkme−ρm−2δkx.By differentiating the previous equation, we have(10)fx=∑k,m=0∞ϕk,mgρm−2δkx,where gA is the PDF of E model with scale parameter ρm−2δk and ϕk,m=∑j=0∞−1j+k+m+1jk/k!κj2δkm.


### 2.2. Quantile Function

Obtaining the inverse CDF yields the quantile function (QF) of the BXLE distribution (3) as(11)Qp=log−log1−p1/κ1/2δ+1ρ,0<p<1.The QF may be used to generate random data from the BXLE distribution:(12)xi=log−log1−pi1/κ1/2δ+1ρ,i=1,2,…,n,where p∈0,1 follows the uniform distribution.


### 2.3. Moments

Therth moment of the BXLE distribution has the form(13)μr′=EXr=∫0∞xrfxdx=∑k,m=0∞ϕk,mρm−2δk−rΓr+1.One can obtain the first four original moments of the BXLE distribution by settingr=1,2,3, and 4 in the last formula.The BXLE distribution’s moment generating function has the following form:(14)Mt=∑k,m=0∞ϕk,mρm−2δkρm−2δk−t.Characteristic function of the BXLE distribution follows from the last formula by replacingt with it.


### 2.4. Order Statistics

The PDF and CDF of the BXLE distribution’sith order statistic (OS) are provided by(15)fi:nx=n!i−1!n−i!Fxi−11−Fxn−ifx=2δκρn!eρxeρx−12δ−11−e−eρx−12δiκ1−1−e−eρx−12δκn−iΓiΓ−i+n+1eeρx−12δ−1,Fi:nx=∑r=innrFxr1−Fxn−r=Γn+11−e−eρx−12δiκ1−1−e−eρx−12δκn−iΩΓi+1Γ−i+n+1,where Ω=2F11,i−n;i+1;1+1/1−e−−1+exρ2δκ−1 is a hyper-geometric function.(16)fi:nx=fxBi,n−i+1∑h=0∞−1hn−ihFh+i−1x=∑k,m=0∞ϕk,mgρm−2δkxBi,n−i+1∑h=0∞−1hn−ih1−e−eρx−12δκh+i−1,where B⋅,⋅ is beta function.


## 2.1. Linear Representation

An expansion for (3) can be derived using the power series(6)1−zb=∑j=0∞−1jbjzj,where Z<1 and b>0.Then, the BXLE CDF could be written as(7)Fx=∑j=0∞−1jκje−jeρx−12δ.Applying the E series,(8)Fx=∑j,k=0∞−1j+kjkk!κje2δkρx1−e−ρx2δk,and then(9)Fx=∑j,k,m=0∞−1j+k+mjkk!κj2δkme−ρm−2δkx.By differentiating the previous equation, we have(10)fx=∑k,m=0∞ϕk,mgρm−2δkx,where gA is the PDF of E model with scale parameter ρm−2δk and ϕk,m=∑j=0∞−1j+k+m+1jk/k!κj2δkm.


## 2.2. Quantile Function

Obtaining the inverse CDF yields the quantile function (QF) of the BXLE distribution (3) as(11)Qp=log−log1−p1/κ1/2δ+1ρ,0<p<1.The QF may be used to generate random data from the BXLE distribution:(12)xi=log−log1−pi1/κ1/2δ+1ρ,i=1,2,…,n,where p∈0,1 follows the uniform distribution.


## 2.3. Moments

Therth moment of the BXLE distribution has the form(13)μr′=EXr=∫0∞xrfxdx=∑k,m=0∞ϕk,mρm−2δk−rΓr+1.One can obtain the first four original moments of the BXLE distribution by settingr=1,2,3, and 4 in the last formula.The BXLE distribution’s moment generating function has the following form:(14)Mt=∑k,m=0∞ϕk,mρm−2δkρm−2δk−t.Characteristic function of the BXLE distribution follows from the last formula by replacingt with it.


## 2.4. Order Statistics

The PDF and CDF of the BXLE distribution’sith order statistic (OS) are provided by(15)fi:nx=n!i−1!n−i!Fxi−11−Fxn−ifx=2δκρn!eρxeρx−12δ−11−e−eρx−12δiκ1−1−e−eρx−12δκn−iΓiΓ−i+n+1eeρx−12δ−1,Fi:nx=∑r=innrFxr1−Fxn−r=Γn+11−e−eρx−12δiκ1−1−e−eρx−12δκn−iΩΓi+1Γ−i+n+1,where Ω=2F11,i−n;i+1;1+1/1−e−−1+exρ2δκ−1 is a hyper-geometric function.(16)fi:nx=fxBi,n−i+1∑h=0∞−1hn−ihFh+i−1x=∑k,m=0∞ϕk,mgρm−2δkxBi,n−i+1∑h=0∞−1hn−ih1−e−eρx−12δκh+i−1,where B⋅,⋅ is beta function.


## 3. Different Estimation Methods

In this part, we will look at how to estimate the BXLE parameters using seven different approaches, including the MLE, ADE, MPSE, CVME, LSE, WLSE, and PCE.


### 3.1. Maximum Likelihood Method of Estimation

Letx1,x2,…,xn be a random sample of size n from the PDF (4); then, the log-likelihood function holds to(17)L=κ−1∑i=1nlog1−e−eρxi−12δ+∑i=1nρxi−eρxi−12δ+2δ−1∑i=1nlogeρxi−1+nlog2δκρ.By differentiating (17) with respect to ρ, δ, and κ, respectively, and equating to 0, then(18)∂L∂ρ=κ−1∑i=1n2δxieρxi−eρxi−12δeρxi−12δ−11−e−eρxi−12δ+2δ−1∑i=1nxieρxieρxi−1+∑i=1nxi−2δxieρxieρxi−12δ−1+nρ,∂L∂δ=κ−1∑i=1n2e−eρxi−12δeρxi−12δlogeρxi−11−e−eρxi−12δ+∑i=1n−2eρxi−12δlogeρxi−1+2∑i=1nlogeρxi−1+nδ,∂L∂κ=∑i=1nlog1−e−eρxi−12δ+nκ.By solving the above equations, we derive MLEs of the BXLE parameters.


### 3.2. Ordinary and Weighted Least-Squares Methods of Estimations

Letx1:n,x2:n,…,x2:n be the OS of a random sample of size n from the BXLE model. Hence, we have the OLSE of the BXLE parameters by minimizing the next equation:(19)O=∑i=1nFxi:n−in+12=∑i=1n1−e−eρxi:n−12δκ−in+12.The OLSE of the BXLE parameters may also be calculated by solving the nonlinear equations:(20)∑i=1n1−e−eρxi:n−12δκ−in+1Δsxi:n=0,s=1,2,3,where(21)Δ1xi:n=∂∂ρFxi:n=2δκxi:neρxi:n−eρxi:n−12δeρxi:n−12δ−11−e−eρxi:n−12δκ−1,(22)Δ2xi:n=∂∂δFxi:n=2κeρxi:n−12δlogeρxi:n−11−e−eρxi:n−12δκeeρxi:n−12δ−1,(23)Δ3xi:n=∂∂κFxi:n=1−e−eρxi:n−12δκlog1−e−eρxi:n−12δ.The WLSE of the BXLE parameters can be calculated by minimizing the following equation:(24)W=∑i=1nn+12n+2in−i+1Fxi:n−in+12=∑i=1nn+12n+2in−i+11−e−eρxi:n−12δκ−in+12.Furthermore, the WLSE of the BXLE parameters can be obtained by solving the following nonlinear equations:(25)∑i=1nn+12n+2in−i+1Fxi:n−in+1Δsxi:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


### 3.3. Anderson–Darling Estimation

The ADEs of the BXLE parameters are obtained by minimizing the following equation:(26)A=−n−1n∑i=1n2i−1logFxi:n+logSxi:n.The ADE can also be calculated by solving the following nonlinear equations:(27)∑i=1n2i−1Δsxi:nFxi:n−Δsxn+1−i:nSxn+1−i:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


### 3.4. Cramér–von Mises Estimators

The CVMEs of BXLE parameters are obtained by minimizing the following equation:(28)CV112n∑i=1nFxi:n−2i−12n2=212n+∑i=1n1−e−eρxi:n−12δκ−2i−12n2,or by solving the following nonlinear equations:(29)∑i=1n1−e−eρxi:n−12δκ−2i−12nΔsxi:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


### 3.5. Maximum Product of Spacing Method of Estimation

As an alternative to the ML approach, the maximum product of spacing (MPS) method is used to estimate the parameters of continuous univariate models. The uniform spacings of a random sample of sizen drawn from the BXLE distribution may be defined as follows:(30)Di=Fxi−Fxi−1,where Di denotes the uniform spacings, Fx0=0, Fxn+1=1, and ∑i=1n+1Di=1. MPS estimators (MPSEs) of the BXLE parameters can be obtained by maximizing(31)G=1n+1∑i=1n+1logDi,with respect to ρ, δ, and κ. Further, the MPSE of the BXLE parameters can also be obtained by solving(32)1n+1∑i=1n+11DiΔsxi:n−Δsxi−1:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


### 3.6. Percentile Method of Estimation

If we considerpi=i/n+1 to be an estimate of Fxi:n, then the PCE of the BXLE parameters is derived by minimizing the following expression:(33)PCE=∑i=1nxi:n−Qpi2=∑i=1nxi:n−log−log1−pi1/κ1/2δ+1ρ2.Alternatively, solve the associated nonlinear equations:(34)∑i=1nxi:n−log−log1−pi1/κ1/2δ+1ρΦsxi:n=0,s=1,2,3,where(35)Φ1xi:n=∂∂ρQpi=−log−log1−pi1/κ1/2δ+1ρ2,Φ2xi:n=∂∂ρQpi=−log−log1−pi1/κ2δ2ρ−log1−pi1/κ−1/2δ+1,Φ3xi:n=∂∂ρQpi=−pi1/κlogpi−log1−pi1/κ1/2δ−12δκ2ρpi1/κ−1−log1−pi1/κ1/2δ+1.


## 3.1. Maximum Likelihood Method of Estimation

Letx1,x2,…,xn be a random sample of size n from the PDF (4); then, the log-likelihood function holds to(17)L=κ−1∑i=1nlog1−e−eρxi−12δ+∑i=1nρxi−eρxi−12δ+2δ−1∑i=1nlogeρxi−1+nlog2δκρ.By differentiating (17) with respect to ρ, δ, and κ, respectively, and equating to 0, then(18)∂L∂ρ=κ−1∑i=1n2δxieρxi−eρxi−12δeρxi−12δ−11−e−eρxi−12δ+2δ−1∑i=1nxieρxieρxi−1+∑i=1nxi−2δxieρxieρxi−12δ−1+nρ,∂L∂δ=κ−1∑i=1n2e−eρxi−12δeρxi−12δlogeρxi−11−e−eρxi−12δ+∑i=1n−2eρxi−12δlogeρxi−1+2∑i=1nlogeρxi−1+nδ,∂L∂κ=∑i=1nlog1−e−eρxi−12δ+nκ.By solving the above equations, we derive MLEs of the BXLE parameters.


## 3.2. Ordinary and Weighted Least-Squares Methods of Estimations

Letx1:n,x2:n,…,x2:n be the OS of a random sample of size n from the BXLE model. Hence, we have the OLSE of the BXLE parameters by minimizing the next equation:(19)O=∑i=1nFxi:n−in+12=∑i=1n1−e−eρxi:n−12δκ−in+12.The OLSE of the BXLE parameters may also be calculated by solving the nonlinear equations:(20)∑i=1n1−e−eρxi:n−12δκ−in+1Δsxi:n=0,s=1,2,3,where(21)Δ1xi:n=∂∂ρFxi:n=2δκxi:neρxi:n−eρxi:n−12δeρxi:n−12δ−11−e−eρxi:n−12δκ−1,(22)Δ2xi:n=∂∂δFxi:n=2κeρxi:n−12δlogeρxi:n−11−e−eρxi:n−12δκeeρxi:n−12δ−1,(23)Δ3xi:n=∂∂κFxi:n=1−e−eρxi:n−12δκlog1−e−eρxi:n−12δ.The WLSE of the BXLE parameters can be calculated by minimizing the following equation:(24)W=∑i=1nn+12n+2in−i+1Fxi:n−in+12=∑i=1nn+12n+2in−i+11−e−eρxi:n−12δκ−in+12.Furthermore, the WLSE of the BXLE parameters can be obtained by solving the following nonlinear equations:(25)∑i=1nn+12n+2in−i+1Fxi:n−in+1Δsxi:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


## 3.3. Anderson–Darling Estimation

The ADEs of the BXLE parameters are obtained by minimizing the following equation:(26)A=−n−1n∑i=1n2i−1logFxi:n+logSxi:n.The ADE can also be calculated by solving the following nonlinear equations:(27)∑i=1n2i−1Δsxi:nFxi:n−Δsxn+1−i:nSxn+1−i:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


## 3.4. Cramér–von Mises Estimators

The CVMEs of BXLE parameters are obtained by minimizing the following equation:(28)CV112n∑i=1nFxi:n−2i−12n2=212n+∑i=1n1−e−eρxi:n−12δκ−2i−12n2,or by solving the following nonlinear equations:(29)∑i=1n1−e−eρxi:n−12δκ−2i−12nΔsxi:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


## 3.5. Maximum Product of Spacing Method of Estimation

As an alternative to the ML approach, the maximum product of spacing (MPS) method is used to estimate the parameters of continuous univariate models. The uniform spacings of a random sample of sizen drawn from the BXLE distribution may be defined as follows:(30)Di=Fxi−Fxi−1,where Di denotes the uniform spacings, Fx0=0, Fxn+1=1, and ∑i=1n+1Di=1. MPS estimators (MPSEs) of the BXLE parameters can be obtained by maximizing(31)G=1n+1∑i=1n+1logDi,with respect to ρ, δ, and κ. Further, the MPSE of the BXLE parameters can also be obtained by solving(32)1n+1∑i=1n+11DiΔsxi:n−Δsxi−1:n=0,where Δsxi:n,s=1,2,3, were defined in (21), (22), and (23), respectively.


## 3.6. Percentile Method of Estimation

If we considerpi=i/n+1 to be an estimate of Fxi:n, then the PCE of the BXLE parameters is derived by minimizing the following expression:(33)PCE=∑i=1nxi:n−Qpi2=∑i=1nxi:n−log−log1−pi1/κ1/2δ+1ρ2.Alternatively, solve the associated nonlinear equations:(34)∑i=1nxi:n−log−log1−pi1/κ1/2δ+1ρΦsxi:n=0,s=1,2,3,where(35)Φ1xi:n=∂∂ρQpi=−log−log1−pi1/κ1/2δ+1ρ2,Φ2xi:n=∂∂ρQpi=−log−log1−pi1/κ2δ2ρ−log1−pi1/κ−1/2δ+1,Φ3xi:n=∂∂ρQpi=−pi1/κlogpi−log1−pi1/κ1/2δ−12δκ2ρpi1/κ−1−log1−pi1/κ1/2δ+1.


## 4. Numerical Outcomes

Based on comprehensive simulation findings, this section investigates the performance of the seven estimate approaches in estimating the BXLE parameters. We explore several sample sizes,n=20,30,50,100,200,400, as well as some parametric values for ρ,δ, and κ, ρ=0.50,0.75,1.5,2.0,2.5,3.0, δ=0.50,0.75,1.5,2.0,3.0, and κ=0.25,0.5,0.75,1.5,2.0,2.5,3.0. We generate n=2000 random samples from the BXLE distribution using its QF and calculate the average values of the estimates (AVEs) with their associated average mean square errors (MSEs), average absolute biases (AVBs), and average mean relative estimates (MREs) for all sample sizes and parameter combinations using the R software.The MSE, AVB, and MRE were calculated by the following equations:(36)MSE=1N∑i=1Nη^−η2,AVB=1N∑i=1Nη^−η,MRE=1N∑i=1ηη^−ηη,where η=ρ,δ,κ′.Tables1–8 provide the simulation results for the BXLE parameters utilizing the seven estimate methodologies, including AVE, AVB, MSE, and MRE. The estimates of the BXLE parameters derived from all seven estimation techniques are fully good, that is, they are extremely trustworthy and very near to the real values, with negligible biases, MSE, and MRE in all parameter combinations. For all parameter combinations, all estimators exhibit the consistency property, in which the MSE, AVB, and MRE drop as sample size grows. We find that the MLE, ADE, CVME, LSE, MPSE, PCE, and WLSE approaches do an excellent job at estimating BXLE parameters.Table 1
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=0.5,δ=0.25,κ=0.75.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^0.756260.742940.771140.636160.668270.660660.68538δ^0.328700.322250.334040.288570.303680.294380.31013κ^1.289151.105081.293701.171671.290741.290881.18169AVBρ^0.575090.501040.569880.442170.486100.448570.48385δ^0.134290.122320.134500.101970.115720.105020.11975κ^0.848270.683730.885230.738310.861130.888170.76063MSEρ^0.611310.435850.555990.320510.375210.304290.38361δ^0.036840.027650.033350.018380.023840.017230.02682κ^1.345360.974561.512271.166751.421221.766701.13282MREρ^1.150181.002081.139750.884340.972200.897140.96769δ^0.537150.489270.538010.407890.462880.420080.47901κ^1.131030.911641.180310.984411.148181.184231.0141830AVEρ^0.749800.735530.707420.640320.683920.676730.70238δ^0.325900.314600.318390.286460.304570.293860.30745κ^1.113241.014451.238921.035111.173351.147381.05079AVBρ^0.531030.479710.500140.409620.468760.441520.46056δ^0.128020.113740.118640.094000.109690.101880.10946κ^0.669190.591220.811420.572340.743240.738380.62155MSEρ^0.524770.385970.403810.304990.353160.289670.35777δ^0.034430.023670.025240.017290.020850.015740.02189κ^0.797950.651691.285990.656051.085871.233440.75033MREρ^1.062070.959411.000280.819240.937520.883030.92112δ^0.512070.454960.474580.376000.438750.407530.43785κ^0.892260.788301.081890.763130.990990.984510.8287350AVEρ^0.696890.715600.752530.671670.674260.675170.67767δ^0.307950.310050.322360.296550.302380.293410.30187κ^0.974950.874180.991480.871061.060470.976710.94581AVBρ^0.439830.407350.475100.379680.432430.402280.39965δ^0.107060.100000.112350.092370.101500.092590.09683κ^0.488840.428800.584800.420590.618580.545870.48790MSEρ^0.396350.295970.362140.281290.299930.249580.27337δ^0.026740.018990.022540.017680.018360.013510.01783κ^0.413680.337580.668120.331580.723950.676870.46245MREρ^0.879650.814700.950200.759360.864860.804550.79929δ^0.428260.400020.449390.369470.405980.370380.38731κ^0.651780.571730.779740.560790.824770.727820.65053100AVEρ^0.648970.664630.662790.612510.635760.672570.66188δ^0.291990.293870.299460.279450.289800.292560.29370κ^0.836760.810120.930170.795470.936850.802830.85312AVBρ^0.321700.327250.372020.270990.354710.332640.34039δ^0.080020.077870.087940.065450.081670.078210.07992κ^0.310600.310960.450450.275460.447970.340720.36769MSEρ^0.253200.222880.247840.164430.220540.190680.22617δ^0.017760.013720.016010.011020.013060.010940.01367κ^0.162810.162000.400000.138300.372980.246990.25588MREρ^0.643410.654510.744050.541990.709420.665280.68077δ^0.320060.311500.351740.261790.326660.312840.31969κ^0.414130.414610.600610.367280.597290.454290.49026200AVEρ^0.628060.614410.638940.595070.623840.628300.58400δ^0.284730.280280.288280.274270.283410.280640.27245κ^0.775150.770740.819830.760580.836020.763060.81231AVBρ^0.256960.244220.299830.219290.292190.264790.23074δ^0.063490.057540.068680.052580.066280.060670.05306κ^0.225770.228090.310030.204760.315240.246110.24667MSEρ^0.189810.137680.178860.123690.169510.135190.12078δ^0.013030.008510.010760.008160.010010.007420.00712κ^0.083150.084130.173600.071870.189610.102570.11142MREρ^0.513910.488450.599660.438590.584380.529590.46149δ^0.253960.230170.274740.210330.265120.242670.21222κ^0.301020.304120.413380.273020.420320.328150.32889400AVEρ^0.542780.564000.590550.549460.596180.604430.55423δ^0.261350.267220.274070.261750.274550.273810.26417κ^0.767240.761950.781470.742780.769100.718540.76840AVBρ^0.139060.167680.216500.134220.219680.189950.16118δ^0.033480.038980.048610.031650.048770.042690.03623κ^0.143640.165930.220040.135480.217390.163070.17109MSEρ^0.058490.073690.104870.044100.108510.079490.05898δ^0.003980.004660.006080.002670.006000.004190.00344κ^0.034790.044810.084090.031300.077710.042290.04917MREρ^0.278120.335360.432990.268430.439370.379900.32236δ^0.133930.155930.194420.126620.195090.170770.14491κ^0.191520.221240.293390.180640.289860.217420.22812Table 2
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=0.5,δ=0.25,κ=2.5.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^1.798381.789703.727111.266412.562951.377142.13722δ^0.293340.306270.343610.284450.332280.288740.31289κ^4.183763.560654.147253.591923.735903.679033.50675AVBρ^1.618791.550973.515631.047342.344171.183841.90137δ^0.064300.073860.109950.055170.100560.061010.08100κ^2.610402.178153.019012.133792.612372.202802.27485MSEρ^27.2405218.29810203.498927.9052941.284739.9218528.59804δ^0.020770.025460.044690.014340.038170.018500.02930κ^13.021159.1914717.7074510.2925213.328108.4202810.35202MREρ^3.237573.101947.031272.094684.688352.367683.80274δ^0.257200.295440.439790.220680.402260.244050.32402κ^1.044160.871261.207610.853521.044950.881120.9099430AVEρ^0.990991.329212.079300.954201.533930.983881.35775δ^0.271250.286880.312260.270550.298550.271380.28670κ^3.618313.125263.731723.205623.652273.296903.28674AVBρ^0.758891.045411.845040.697771.314900.748021.10170δ^0.039110.051780.077870.039440.066900.041320.05350κ^1.888201.594612.408941.642912.252271.718621.84758MSEρ^4.301568.5998224.552733.6076611.388604.8081310.04901δ^0.007760.015010.027690.007630.021580.008860.01474κ^7.315985.2107311.466805.8761910.046475.366006.74280MREρ^1.517792.090833.690081.395532.629801.496042.20341δ^0.156450.207140.311500.157780.267590.165260.21402κ^0.755280.637840.963580.657160.900910.687450.7390350AVEρ^0.728970.878091.307930.671471.145200.687260.94686δ^0.260940.269770.288270.257660.280020.258690.27062κ^3.195622.770123.223262.888873.255502.992372.97697AVBρ^0.453190.545371.022660.368140.873460.406810.64862δ^0.026710.030930.051030.022940.044510.025540.03453κ^1.338541.163171.786931.122051.756871.253661.37743MSEρ^1.884712.093997.428020.816095.563870.909853.96788δ^0.003720.004810.013090.002010.010610.002580.00648κ^3.690032.594086.492052.690526.211492.753503.75536MREρ^0.906381.090732.045330.736271.746910.813621.29724δ^0.106830.123720.204130.091770.178060.102180.13811κ^0.535420.465270.714770.448820.702750.501460.55097100AVEρ^0.555220.595650.736940.548670.659980.544060.62726δ^0.251370.255500.262450.252060.258040.251930.25698κ^2.735972.701732.926812.627282.893902.796492.70926AVBρ^0.203720.243860.415850.190540.339220.211850.27539δ^0.015540.015420.023340.013340.019820.014050.01699κ^0.735420.787571.217450.697861.117030.838070.89749MSEρ^0.096250.255771.057030.069160.488620.100100.23843δ^0.000920.000750.002170.000310.001210.000390.00079κ^1.024681.121263.108650.936642.684271.308291.54429MREρ^0.407430.487730.831690.381080.678430.423710.55078δ^0.062180.061680.093370.053340.079280.056210.06797κ^0.294170.315030.486980.279140.446810.335230.35900200AVEρ^0.516720.541330.545520.537280.567980.511740.54949δ^0.249800.252290.252680.251830.253550.250360.25282κ^2.633832.616982.799882.497762.699182.679962.59225AVBρ^0.133230.155200.198920.130560.212630.140860.16293δ^0.010360.010150.012160.009000.012810.009580.01067κ^0.500720.557700.778070.435590.761430.565050.57875MSEρ^0.031350.050730.095650.031160.113960.037490.04950δ^0.000510.000190.000330.000140.000360.000160.00020κ^0.443810.550241.181160.348151.032250.576670.60758MREρ^0.266450.310400.397840.261120.425250.281730.32585δ^0.041430.040620.048640.036000.051250.038310.04270κ^0.200290.223080.311230.174230.304570.226020.23150400AVEρ^0.501000.522360.528230.517830.522870.512850.51279δ^0.248980.251190.251800.250860.251180.250600.25073κ^2.576222.544492.614662.493002.613372.550732.57562AVBρ^0.088620.107040.140430.086190.129790.097370.10665δ^0.007290.007220.008690.006060.008250.006720.00705κ^0.330830.394490.530500.288690.504120.372700.41309MSEρ^0.012980.019030.036370.013520.030090.016170.01763δ^0.000370.000080.000140.000060.000120.000070.00008κ^0.181160.254010.461450.163520.449000.241110.28127MREρ^0.177250.214070.280860.172380.259580.194740.21331δ^0.029150.028890.034760.024240.033000.026900.02820κ^0.132330.157790.212200.115480.201650.149080.16523Table 3
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=0.75,δ=1.50,κ=0.25.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^0.993600.957110.962700.869260.981100.864420.92370δ^2.425432.422822.662892.083192.564473.535352.35381κ^0.342950.306900.342300.311740.314610.368560.34015AVBρ^0.505820.480370.451630.451220.486770.507860.47104δ^1.290661.317341.451961.187371.454932.706801.30148κ^0.196130.158410.186840.157980.176490.251790.19231MSEρ^0.452140.463170.454740.383170.491330.419890.43086δ^4.062095.564857.767504.119206.5738748.763754.70223κ^0.120300.070430.142990.072640.103550.392860.12948MREρ^0.674430.640490.602170.601630.649020.677150.62806δ^0.860440.878230.967970.791580.969951.804530.86765κ^0.784500.633640.747350.631940.705961.007140.7692430AVEρ^0.959380.876010.922870.842520.929270.873340.89492δ^2.227332.077212.385101.939282.326282.825602.18518κ^0.302170.298950.312090.298690.288240.326250.30640AVBρ^0.448340.390200.409420.408100.420970.468670.42713δ^1.084040.965311.174021.004781.171111.914511.11519κ^0.152290.134390.153370.135900.139040.201830.15072MSEρ^0.340950.285130.343540.297570.354320.347980.34303δ^2.669242.497284.107402.688634.1169815.165343.64459κ^0.062680.048990.085410.042800.065110.253340.06424MREρ^0.597790.520260.545900.544130.561300.624890.56950δ^0.722690.643540.782680.669850.780741.276340.74346κ^0.609150.537570.613470.543580.556160.807320.6028950AVEρ^0.911630.867780.928520.767030.900270.845280.87249δ^2.033221.991312.229801.659482.122202.319592.00310κ^0.291400.279620.281090.297440.279580.273040.28505AVBρ^0.370580.363210.388860.325280.385550.417030.37056δ^0.869390.870811.014920.726220.980871.367920.89554κ^0.131160.113500.122460.112790.122620.133450.11958MSEρ^0.233600.231920.293730.184270.272720.257890.25123δ^1.561491.813612.692811.256852.507045.786692.05061κ^0.048400.026960.045900.026490.040630.076760.03561MREρ^0.494110.484280.518480.433710.514070.556040.49408δ^0.579590.580540.676620.484150.653920.911950.59703κ^0.524660.453990.489840.451150.490500.533810.47833100AVEρ^0.883980.831830.864780.756350.866410.814950.88168δ^1.901331.812691.957391.573531.968061.941501.91637κ^0.273960.273600.268710.280690.267510.258850.26220AVBρ^0.322230.302170.313930.262180.328130.358250.33039δ^0.736750.662490.747210.543750.789390.966080.73887κ^0.106530.095100.098140.086240.099250.103750.09643MSEρ^0.165530.150500.171770.118580.191270.179360.17703δ^1.035170.885631.258410.647491.482421.983621.04959κ^0.030140.019130.022730.014420.021740.020170.01795MREρ^0.429640.402890.418570.349570.437510.477670.44052δ^0.491170.441660.498140.362500.526260.644050.49258κ^0.426140.380390.392560.344980.396980.415010.38571200AVEρ^0.805150.806910.821730.735130.825070.747420.82156δ^1.671921.714971.780581.502741.795911.663701.74915κ^0.268000.260740.261660.272870.261120.272810.26201AVBρ^0.247610.243220.254200.179480.268230.292540.26068δ^0.528150.524320.558040.346590.608180.695850.56620κ^0.079140.072380.076550.058400.078500.090530.07860MSEρ^0.092360.093560.106330.063310.117240.111680.10681δ^0.452330.514970.641300.300210.765180.897610.60677κ^0.011560.009030.011460.007400.011100.009870.01097MREρ^0.330150.324300.338940.239300.357630.390050.34758δ^0.352100.349540.372030.231060.405450.463900.37747κ^0.316560.289510.306220.233600.314000.362130.31439400AVEρ^0.782160.792460.803160.725500.803970.723950.79052δ^1.618561.647911.708731.462161.694331.538571.64473κ^0.265640.259900.260920.269240.259470.276060.26031AVBρ^0.209200.213470.228810.125260.225470.240470.20776δ^0.441920.443670.494320.229550.489180.526590.43133κ^0.068050.064470.068710.042160.067580.079650.06396MSEρ^0.066000.067370.079300.035910.079750.073700.06616δ^0.316040.322930.441190.153850.438080.412150.31885κ^0.008540.006940.008540.004560.007980.008130.00710MREρ^0.278930.284620.305080.167020.300620.320620.27702δ^0.294610.295780.329550.153040.326120.351060.28755κ^0.272180.257860.274860.168640.270310.318590.25582Table 4
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=2,δ=0.75,κ=0.5.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^2.199722.108602.180551.833872.024121.895132.04918δ^0.804640.828060.818340.760800.841190.765880.83656κ^1.283760.950151.241560.963761.192950.978281.11270AVBρ^1.484641.276511.429150.846191.228181.094571.33954δ^0.320120.324440.315840.227690.337010.267180.34797κ^0.929990.600280.890610.606760.857830.635550.76614MSEρ^3.086432.616103.474361.566702.765911.781223.07681δ^0.206310.365240.208910.262420.582760.137250.36637κ^3.218041.353412.977301.807242.823491.495412.12508MREρ^0.742320.638260.714570.423100.614090.547290.66977δ^0.426830.432590.421110.303590.449340.356240.46396κ^1.859991.200571.781231.213521.715671.271091.5322830AVEρ^2.072082.098791.930541.816431.879201.809821.97688δ^0.773450.794700.754830.739110.762130.736080.79586κ^1.021100.825721.170150.815421.101330.845510.93708AVBρ^1.353151.183421.276980.780841.200011.030241.23836δ^0.290550.275080.278600.202570.285350.246670.30605κ^0.659610.467800.798910.440440.735750.479290.57549MSEρ^2.354642.208502.393161.256772.266251.507542.29529δ^0.104270.147600.104810.222130.217660.109720.31649κ^1.510820.787212.207390.856981.934550.769411.10169MREρ^0.676570.591710.638490.390420.600000.515120.61918δ^0.387390.366770.371470.270090.380470.328890.40807κ^1.319230.935611.597830.880891.471500.958591.1509850AVEρ^2.216382.019401.987351.847381.913681.980211.99521δ^0.805720.768810.771590.731320.758730.768220.77033κ^0.707920.694090.866230.677360.837300.672830.78059AVBρ^1.204551.042621.161570.697441.072140.979111.14091δ^0.264080.237460.262070.170300.247310.232060.26052κ^0.355630.320680.494990.285920.468320.316350.41074MSEρ^1.826871.426651.965390.912671.646351.305711.78899δ^0.084680.083720.100390.042970.082390.069190.14433κ^0.375120.298180.789680.289400.736920.269780.50205MREρ^0.602280.521310.580790.348720.536070.489560.57045δ^0.352100.316620.349430.227070.329750.309410.34736κ^0.711260.641360.989990.571840.936640.632700.82148100AVEρ^2.167242.142231.979141.955341.970422.119362.02067δ^0.797780.793470.760160.750130.761230.795960.76815κ^0.595430.585230.686960.566000.679150.571610.63676AVBρ^1.027910.935181.019500.516361.007020.916380.97203δ^0.231460.213150.226530.131790.230560.218540.21899κ^0.231380.218660.309260.155520.304410.213390.25835MSEρ^1.328141.114331.267800.606591.282611.129641.17428δ^0.066930.055480.059850.030100.064350.065560.05745κ^0.100690.094670.243640.063310.237050.084340.14838MREρ^0.513950.467590.509750.258180.503510.458190.48601δ^0.308610.284200.302040.175720.307420.291390.29199κ^0.462770.437320.618520.311050.608810.426780.51670200AVEρ^2.205812.156102.131581.921662.011042.228792.12827δ^0.804400.796200.794850.737740.765250.816140.78879κ^0.542410.543820.589590.545910.613000.519870.55222AVBρ^0.896380.830000.953990.340230.912640.847940.84106δ^0.204320.190640.216940.088640.205230.199650.19182κ^0.170650.170540.223670.100930.231820.162100.17306MSEρ^1.062580.869821.120360.366021.013620.930020.89204δ^0.056310.045480.057750.017840.049740.052040.04582κ^0.044950.046930.093510.030080.109020.041360.04966MREρ^0.448190.415000.477000.170120.456320.423970.42053δ^0.272420.254180.289260.118190.273640.266200.25576κ^0.341310.341070.447330.201860.463650.324210.34612400AVEρ^2.199282.233512.124341.986562.104872.227012.12101δ^0.802810.812100.787770.749600.784190.813090.78822κ^0.521400.509380.542360.512220.549960.506160.52888AVBρ^0.782120.768310.801150.161290.817740.771480.74657δ^0.179150.176290.180840.048120.184780.180590.17113κ^0.137030.136000.159750.048710.168070.137280.14135MSEρ^0.871250.745390.788990.162530.815620.764140.69966δ^0.047200.039580.040380.008230.041670.042560.03742κ^0.026710.026240.039590.009440.044420.026000.02909MREρ^0.391060.384160.400570.080640.408870.385740.37329δ^0.238870.235050.241120.064150.246370.240790.22817κ^0.274060.272010.319500.097420.336130.274550.28269Table 5
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=1.5,δ=2,κ=3.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^1.484261.501431.539191.397271.497291.400271.51141δ^1.995851.998402.000211.981681.994761.978841.99683κ^3.359613.253443.309623.158303.211903.236873.19768AVBρ^0.450240.465690.473070.499420.497060.523600.49561δ^0.071240.076660.080520.082120.083430.087900.08176κ^0.703690.684210.700030.722080.737400.708540.70656MSEρ^0.237410.255370.257620.294100.284500.319370.28166δ^0.007160.008590.009150.009380.010120.010650.00953κ^0.614380.582700.615100.643820.664240.627500.61700MREρ^0.300160.310460.315380.332950.331370.349070.33041δ^0.035620.038330.040260.041060.041720.043950.04088κ^0.234560.228070.233340.240690.245800.236180.2355230AVEρ^1.493491.527581.500131.408731.489821.403881.48786δ^1.994402.000951.996981.982371.993781.979271.99263κ^3.327883.183103.291923.167893.199373.230343.21322AVBρ^0.443560.451610.452180.477390.490110.498570.45999δ^0.064030.069940.073720.075410.078390.080680.07318κ^0.619830.613540.682970.658740.690340.663900.65282MSEρ^0.230180.233370.238550.269560.272630.289040.24734δ^0.005810.006700.007510.007920.008390.008840.00743κ^0.510570.496470.585010.554310.595660.557940.54755MREρ^0.295710.301070.301460.318260.326740.332380.30666δ^0.032020.034970.036860.037710.039190.040340.03659κ^0.206610.204510.227660.219580.230110.221300.2176150AVEρ^1.522721.488161.507161.446091.502001.426131.48624δ^1.998261.993511.997751.989391.995681.984191.99475κ^3.231163.208373.244153.111423.176263.189043.17970AVBρ^0.430700.430990.449920.443340.455810.460980.43779δ^0.061900.065720.067650.068120.068350.071240.06679κ^0.572890.584290.624870.585660.627220.603440.59943MSEρ^0.213950.218830.229880.236180.239000.250580.22438δ^0.005090.005740.006030.006130.006230.006840.00597κ^0.444400.457490.509460.460320.514620.481270.47750MREρ^0.287130.287330.299940.295560.303870.307320.29186δ^0.030950.032860.033820.034060.034180.035620.03340κ^0.190960.194760.208290.195220.209070.201150.19981100AVEρ^1.498611.498541.489771.494531.482201.450971.52141δ^1.996591.995201.996001.996761.993321.988991.99940κ^3.181753.172243.203543.072783.198643.163003.13747AVBρ^0.377600.398280.428410.378440.437410.400960.40783δ^0.053900.058170.062130.055870.063800.058920.05851κ^0.494060.515210.573050.473510.572080.512390.52015MSEρ^0.173400.190140.212420.183430.220340.196690.19731δ^0.003840.004420.004860.004230.005190.004590.00445κ^0.346080.373800.443340.329610.439200.368420.37435MREρ^0.251730.265520.285600.252300.291610.267300.27189δ^0.164690.171740.191020.157840.190690.170800.17338κ^0.026950.029080.031070.027930.031900.029460.02926200AVEρ^1.521121.519051.502441.507401.510741.473111.51276δ^1.999751.999851.997051.998901.998081.992551.99789κ^3.113703.098693.153363.039613.132563.125423.11391AVBρ^0.336640.361350.393230.305830.396430.353010.36308δ^0.047110.051900.055300.045390.055350.051080.05125κ^0.419550.447320.502820.371720.499590.445720.44781MSEρ^0.145640.162920.184320.135450.186760.159760.16363δ^0.002960.003500.003860.003040.003840.003480.00340κ^0.262020.282630.356060.230010.347790.289640.28877MREρ^0.224430.240900.262150.203890.264290.235340.24205δ^0.023550.025950.027650.022700.027680.025540.02563κ^0.139850.149110.167610.123910.166530.148570.14927400AVEρ^1.509521.518721.488521.507351.513291.468001.50446δ^1.999772.000381.996091.999821.999351.992881.99783κ^3.072393.078183.141573.028343.104993.112623.08511AVBρ^0.271990.300580.342990.232730.347380.297090.29563δ^0.038840.041810.047370.033480.048640.042390.04145κ^0.333490.357230.437720.275380.430710.371780.35158MSEρ^0.102120.120740.149130.094130.153490.118520.11873δ^0.002110.002380.002940.001920.003110.002470.00239κ^0.167750.191160.277500.148990.271920.207600.19001MREρ^0.181320.200390.228660.155150.231590.198060.19709δ^0.019420.020910.023690.016740.024320.021190.02072κ^0.111160.119080.145910.091790.143570.123930.11719Table 6
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=0.5,δ=3,κ=1.5.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^1.090131.093011.599931.056211.268851.037081.36696δ^3.756593.722764.235213.657873.887743.707884.01329κ^2.497002.174622.412942.165782.488302.166852.19013AVBρ^0.897010.854431.376560.839441.080150.823231.13763δ^1.100981.064441.537751.063601.304831.121741.36835κ^1.594071.328781.745561.362121.727361.349141.46963MSEρ^2.887322.878827.287832.424814.158892.502728.32081δ^3.853253.465156.770923.197934.719995.882396.48326κ^4.907303.525436.192793.886145.631503.315934.40429MREρ^1.794031.708862.753121.678892.160301.646462.27525δ^0.366990.354810.512580.354530.434940.373910.45612κ^1.062710.885851.163710.908081.151580.899430.9797530AVEρ^0.949541.000331.303230.863141.102170.885771.11377δ^3.590343.650313.960943.422153.744203.560563.76013κ^2.260981.899102.178641.935242.173451.997362.02340AVBρ^0.741410.721041.059900.605050.869410.640250.86180δ^0.939530.945791.257240.774151.090030.926731.09069κ^1.282381.017581.441641.035141.373251.099471.21207MSEρ^2.259492.207183.669421.397812.637891.533922.77654δ^3.149154.083954.360291.958613.598005.641144.08607κ^3.053102.033744.072712.346563.618532.297712.85553MREρ^1.482821.442082.119811.210101.738821.280501.72360δ^0.313180.315260.419080.258050.363340.308910.36356κ^0.854920.678390.961100.690090.915500.732980.8080550AVEρ^0.691720.791821.008630.704840.917210.747540.82745δ^3.261233.392513.649363.251123.555903.301883.40601κ^1.988311.689502.006561.749261.971131.768221.82594AVBρ^0.434230.473550.744830.408400.653500.454310.53521δ^0.562340.634520.920030.545960.856730.605270.68306κ^0.897800.724261.142820.759301.079210.780710.87385MSEρ^0.741150.793291.979140.558561.501960.823281.15166δ^1.181601.391802.762120.991832.544171.455081.75732κ^1.549850.986472.509261.140062.366981.148111.55161MREρ^0.868460.947101.489660.816811.307000.908631.07042δ^0.187450.211510.306680.181990.285580.201760.22769κ^0.598530.482840.761880.506200.719470.520470.58257100AVEρ^0.568070.646650.713240.578650.744400.595530.65737δ^3.097933.186883.275873.084973.307203.113283.20582κ^1.697201.626761.811351.582251.735851.605841.61359AVBρ^0.231930.303420.404990.220320.429180.239570.30949δ^0.311180.398240.498460.299560.541750.322890.40949κ^0.505440.560050.789290.448300.745980.489900.54703MSEρ^0.180460.269100.552430.149650.666050.168950.34042δ^0.314020.436050.840220.252851.024440.293700.58959κ^0.492100.569941.268540.392311.103820.460660.56457MREρ^0.463870.606840.809980.440650.858360.479140.61898δ^0.103730.132750.166150.099850.180580.107630.13650κ^0.336960.373360.526190.298870.497320.326600.36468200AVEρ^0.521630.551920.587750.537570.577610.538500.56014δ^3.034693.067703.115093.041183.097813.040233.07489κ^1.572601.566491.657801.526351.652081.542601.54055AVBρ^0.130960.173480.235650.143860.231090.146100.17219δ^0.179410.231510.293930.200120.285080.197150.22459κ^0.299630.364150.519770.304600.506990.323270.35146MSEρ^0.030470.070020.156860.041580.136200.039150.06129δ^0.056480.122050.250270.076650.221070.068700.10343κ^0.157290.224930.560420.172240.509820.185050.21295MREρ^0.261910.346950.471300.287720.462180.292190.34437δ^0.059800.077170.097980.066710.095030.065720.07486κ^0.199750.242770.346510.203070.337990.215510.23431400AVEρ^0.512000.526890.536750.525400.537070.528830.51915δ^3.016243.036843.048063.029513.046373.032063.02435κ^1.535881.522571.555911.485981.557301.502311.54437AVBρ^0.090030.111490.142630.094130.149210.104980.11394δ^0.124990.150180.182670.132200.187940.141360.15208κ^0.206970.241490.320070.196070.335070.226590.26217MSEρ^0.013750.023980.043970.016410.042580.019330.02252δ^0.025700.041170.070230.031130.065970.033840.03974κ^0.070780.098430.180960.065850.194420.085450.11580MREρ^0.180060.222980.285260.188250.298430.209970.22788δ^0.041660.050060.060890.044070.062650.047120.05069κ^0.137980.160990.213380.130710.223380.151060.17478Table 7
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=2.5,δ=0.75,κ=2.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^1.934682.029491.925251.887661.932471.870491.91932δ^0.716100.724050.718850.717560.721570.717110.71864κ^4.588783.563155.024403.972014.378343.556514.10333AVBρ^1.037640.964131.066061.003921.050271.052251.04019δ^0.056030.053300.059590.060130.060460.060390.05869κ^2.736671.777333.236142.283322.677641.832432.34900MSEρ^1.606741.366111.710181.596581.667441.600031.59770δ^0.005140.004480.005460.005690.005580.005500.00527κ^35.7400616.0912047.4146526.4330133.1362814.6981127.01181MREρ^0.415060.385650.426420.401570.420110.420900.41608δ^0.074710.071060.079460.080170.080610.080520.07825κ^1.368340.888671.618071.141661.338820.916211.1745030AVEρ^2.073352.116122.002592.067201.926912.012052.00773δ^0.724140.728530.722850.727630.719520.724250.72425κ^3.314322.886863.826602.761873.841792.857583.23246AVBρ^0.896480.871940.977660.826081.034610.935780.95710δ^0.047750.046800.052960.046330.058070.051840.05307κ^1.482151.090702.027851.023662.081431.123261.47502MSEρ^1.204221.108831.451711.092461.578361.274351.34196δ^0.003790.003520.004480.003520.005120.004200.00429κ^11.760965.5912119.439845.9796819.493404.7720510.10694MREρ^0.358590.348780.391060.330430.413840.374310.38284δ^0.063670.062400.070610.061770.077420.069120.07077κ^0.741070.545351.013920.511831.040720.561630.7375150AVEρ^2.174262.205162.058712.159302.007762.039872.19373δ^0.730600.734170.726310.732240.724170.725360.73278κ^2.643432.518743.097992.408273.064242.565932.61239AVBρ^0.782670.760880.895480.721960.940800.864870.80352δ^0.791620.710541.260720.660601.278750.767110.82421κ^0.039750.039320.045290.039960.049970.045490.04171MSEρ^0.893360.845621.177620.860591.291621.060940.95492δ^0.002600.002490.003300.002670.003880.003170.00285κ^2.737911.907318.167731.588776.842341.779053.10024MREρ^0.313070.304350.358190.288780.376320.345950.32141δ^0.053000.052420.060390.053280.066630.060650.05561κ^0.395810.355270.630360.330300.639370.383550.41210100AVEρ^2.258412.265982.181052.291932.169432.201902.28508δ^0.736820.737710.732600.740430.732630.734870.73828κ^2.287882.301452.481442.172752.453472.293862.29980AVBρ^0.661040.667070.772550.556860.786400.708410.66816δ^0.031980.032970.038220.029170.039440.035470.03311κ^0.441980.463070.649020.387140.630410.475450.46638MSEρ^0.607850.627470.846100.527060.862830.702270.63059δ^0.001580.001700.002280.001420.002380.001920.00172κ^0.453890.551231.320590.371791.103660.528120.63115MREρ^0.264420.266830.309020.222740.314560.283360.26727δ^0.042640.043970.050950.038900.052580.047290.04414κ^0.220990.231540.324510.193570.315210.237720.23319200AVEρ^2.383192.366812.326892.380312.269722.269972.35139δ^0.742840.742500.740390.744900.737780.738050.74156κ^2.147382.160182.235832.079872.263822.180582.17066AVBρ^0.542580.566240.642090.393550.665610.589060.57429δ^0.025480.026930.031060.020740.031930.028320.02726κ^0.288460.315290.405220.244170.424830.328900.32415MSEρ^0.398840.435660.571620.312070.614960.474330.45050δ^0.000970.001080.001460.000800.001520.001190.00111κ^0.166550.210340.388180.132950.423110.211350.22558MREρ^0.217030.226500.256830.157420.266240.235620.22972δ^0.033970.035910.041410.027650.042570.037760.03635κ^0.144230.157650.202610.122080.212420.164450.16208400AVEρ^2.465652.442242.380622.436422.376392.383232.40852δ^0.747310.746430.743250.746810.742870.744080.74470κ^2.065352.079952.144202.040992.141492.093052.10390AVBρ^0.427140.460760.563250.247060.542400.470870.48153δ^0.019360.021670.026170.013210.025340.022150.02216κ^0.201330.229960.301800.151090.291890.233580.24279MSEρ^0.244660.284920.416300.173690.409170.303570.31548δ^0.000540.000670.000970.000410.000960.000710.00072κ^0.074720.094830.182020.057410.177590.096950.11244MREρ^0.170860.184300.225300.098830.216960.188350.19261δ^0.025820.028890.034900.017620.033790.029530.02955κ^0.100660.114980.150900.075540.145950.116790.12139Table 8
Simulation results of the AVE, AVB, MSE, and MRE of BXLE distribution forρ=3,δ=1.5,κ=0.5.
nEst.Est. Par.MLEADECVMEMPSELSEPCEWLSE20AVEρ^2.259222.271612.303862.565172.312452.220162.11469δ^1.288651.339281.337801.466751.375581.331831.31017κ^1.444431.067991.479470.896271.250291.043111.34574AVBρ^1.241021.151631.100230.652761.043421.143321.28963δ^0.380190.378410.362270.284770.362590.383760.42134κ^0.992570.642301.043870.518670.848030.632930.92532MSEρ^2.493212.189572.286681.294952.145462.245512.64590δ^4.249121.787425.285951.981673.576781.733293.57683κ^0.220300.209380.202490.153870.203290.217020.24799MREρ^0.413670.383880.366740.217590.347810.381110.42988δ^0.253460.252270.241510.189850.241720.255840.28089κ^1.985141.284592.087741.037341.696061.265871.8506330AVEρ^2.377952.352312.305912.587772.268342.311862.22954δ^1.327961.344651.337081.436561.340681.348331.31035κ^0.991710.913471.148390.752911.154540.861020.99014AVBρ^1.124861.103411.123910.596101.117831.102761.21249δ^0.342910.345900.356950.243230.368000.362610.37609κ^0.541420.476750.709680.346570.730600.449390.55049MSEρ^1.996941.964542.205191.166562.248972.022862.26525δ^0.177570.176680.191660.115190.202740.191300.20194κ^1.436811.000282.289480.785012.482220.764711.18992MREρ^0.374950.367800.374640.198700.372610.367590.40416δ^0.228610.230600.237960.162160.245330.241740.25073κ^1.082830.953511.419360.693151.461210.898781.1009850AVEρ^2.415632.447592.339702.659012.348532.468122.33817δ^1.338711.360301.330531.434641.344341.379021.33240κ^0.817020.726350.924250.632960.888700.720360.80824AVBρ^1.078051.024761.088370.504261.081371.014651.13162δ^0.318700.315300.327070.198140.334650.322990.34249κ^0.364290.280730.478010.205020.454840.298650.36362MSEρ^1.775791.593341.991080.928711.967951.643111.92550δ^0.152940.142110.164980.085970.167870.151830.16718κ^0.543100.235120.975450.224400.912360.304090.44341MREρ^0.359350.341590.362790.168090.360460.338220.37721δ^0.212470.210200.218040.132090.223100.215330.22833κ^0.728590.561470.956010.410050.909690.597300.72724100AVEρ^2.504972.544962.509032.829092.356252.591212.51501δ^1.362221.389651.374291.471821.336731.408151.37591κ^0.664220.640910.712430.545290.741510.624690.66829AVBρ^0.958290.931290.989600.283081.068640.912620.97376δ^0.279930.282310.293900.126940.315290.284490.28950κ^0.208410.194720.263680.098630.294720.194490.21911MSEρ^1.331831.268981.532540.456691.760011.238691.40130δ^0.114030.112080.129610.042570.144760.113400.12035κ^0.106380.089900.204650.048850.234980.093890.12633MREρ^0.319430.310430.329870.094360.356210.304210.32459δ^0.416820.389450.527350.197250.589440.388990.43823κ^0.186620.188210.195930.084630.210190.189660.19300200AVEρ^2.686222.639292.544492.913412.526552.779922.57827δ^1.412911.403481.383001.487871.382361.454661.39272κ^0.587010.596700.645830.515050.643040.560150.61597AVBρ^0.792670.820700.932970.134180.937810.763200.86433δ^0.229120.240480.274040.075190.274610.234400.25492κ^0.132580.142850.195320.049780.194040.127680.16375MSEρ^0.877720.952461.284150.196871.277740.831111.07875δ^0.074030.081900.107020.018940.106020.075140.09153κ^0.037210.044210.092420.013870.091750.034140.05970MREρ^0.264220.273570.310990.044730.312600.254400.28811δ^0.152750.160320.182700.050130.183070.156270.16995κ^0.265160.285700.390630.099550.388080.255360.32749400AVEρ^2.814002.786512.676282.978412.684542.844822.72588δ^1.451461.447241.418571.499841.420691.466421.42676κ^0.549130.556690.591350.499160.586880.539570.56863AVBρ^0.657470.700650.803080.037020.798830.666050.73811δ^0.191070.205800.232580.039200.233580.198980.21271κ^0.096960.107190.139370.023150.137000.098320.11442MSEρ^0.593680.685210.926460.042800.915850.606490.75292δ^0.050230.058020.075570.005090.075740.053080.06228κ^0.018490.022620.044600.003080.040000.018280.02615MREρ^0.219160.233550.267690.012340.266280.222020.24604δ^0.127380.137200.155050.026130.155720.132650.14180κ^0.193920.214390.278740.046300.274000.196630.22884


## 5. Application

In this part, we will look at two real-world datasets. The first dataset has 74 observations and represents gauge lengths of 20 mm [17]. The second set is made up of 100 observations and reflects the breaking stress of carbon fiber [18].We compare the BXLE model with some other well-known competitive distributions such as the betaE (BE) [19], transmuted generalized-E (TGE) [5], exponentiated E (ExE), alpha power exponentiated E (APExE) transmuted E (TE) [23], exponetial (E), Nadarajah-Haghighi (NH) [4], Fréchet Weibull mixture E (FWME) [21], gamma exponentiated E (GExE) [22], and linear E (LE) [23] distributions.Some discriminatory practice measures, such as Akaike information (AKI), Hannan–Quinn information (HAQUI), Bayesian information (BAI), and consistent Akaike information (CAKI), can be used to compare competing models. Other discrimination measures include the Anderson–Darling (ANDA), Cramér–von Mises (CRVMI), and thep value of the Kolmogorov–Smirnov (KOSM).The estimated parameters using the maximum likelihood method and their standard errors for the BXLE model and other compared models are reported in Tables9 and 10 for the two datasets, respectively. The values of discrimination measures are listed in Tables 11 and 12. The values in Tables 11 and 12 indicate that the proposed BXLE distribution provides better fit for the two analyzed datasets than other competing models.Table 9
Estimated parameters with their standard errors of the BXLE model and other fitted models for first dataset.
ModelEstimated parameters (standard errors)BXLEρ^=16.65476215.44δ^=0.017086.37703κ^=39.52977.64793FWMEα^=2.771361.2456043λ^=2.880910.0012536k^=3.114531.256135a^=3.844240.012989BEλ^=0.073540.00071a^=24.28158.98448b^=121.9987.64793APExEα^=46.323965.8173a^=2.521180.217317c^=91.045752.8461GExEλ^=0.252490.382442α^=35.221814.7076δ^=27.680349.284TGEα^=90.152538.8898λ^=−0.6975290.20628θ^=2.215360.187642ExEb^=89.437433.3448a^=2.019180.002538TEβ^=0.5852650.0672482λ^=−1.000000.459119LEβ^=1.74693×10−220.375895θ^=0.3138990.162698Ea^=0.403670.0469257NHα^=38.526826.7975λ^=0.00824340.0057971Table 10
Estimated parameters with their standard errors of the BXLE model and other fitted models for second dataset.
ModelEstimated parameters (standard errors)BXLEρ^=0.6261490.970701δ^=0.2933850.432376κ^=6.243909.78511FWMEα^=1.905680.024546λ^=2.849170.567416k^=2.160870.005413a^=1.717840.025468BEλ^=0.0568440.000868a^=5.958152.13741b^=37.57630.821257APExEα^=32.217958.4502a^=1.238470.103739c^=4.600252.54393GExEλ^=0.2697240.257315α^=8.075492.2026δ^=6.159796.82518TGEα^=6.187321.92831λ^=−0.6837050.279297θ^=1.101990.094658ExEb^=7.7882433.3448a^=1.013170.002538TEβ^=32.21790.557692λ^=−1.000000.398622LEβ^=4.07405×10−100.0859971θ^=0.2535130.0475148Ea^=0.3814760.0381476NHα^=38.303824.8412λ^=0.0072530.004764Table 11
Discrimination measures of the BXLE model and other competing models for first dataset.
ModelAKICAKIBAIHAQUIANDACRVMIKOSMp valueBXLE110.123110.466117.036112.8810.3803560.0590980.0565290.972044FWME116.084116.664125.301119.7610.560270.07756390.06930950.927044BE112.344112.687119.257115.1020.5651120.0864840.0681300.882162APExE115.308115.651122.22118.0660.7159410.09651860.06970250.864808GExE112.777113.12119.689115.5340.6001830.09201410.07006970.860604TGE119.542119.885126.454122.31.088650.154130.0843450.668357ExE121.607121.776126.215123.4451.502440.2235490.09531390.512098TE237.473237.642242.081239.31115.84573.12660.35916100000LE192.302192.471196.91194.1413.31262.654350.33929800000E284.259284.315286.563285.17822.14074.672180.4494700000NH239.564239.733244.172241.40223.96195.223390.46858900000Table 12
Discrimination measures of the BXLE model and other competing models for second dataset.
ModelAKICAKIBAIHAQUIANDACRVMIKOSMp valueBXLE288.775289.025296.591291.9380.412950.0674190.0608110.853353FWME300.559300.98310.98304.7761.103670.1675550.09028040.388871BE292.486292.736300.301295.6490.7604580.150470.093490.346368APExE291.673291.923299.489294.8370.6647690.1257780.08705560.43464GExE293.482293.732301.297296.6450.8346210.1645510.09639210.310696TGE294.83295.08302.646297.9930.9273320.1748290.09663580.307819ExE296.365296.488301.575298.4731.224630.2291580.1077250.196182TE342.209342.333347.419344.3188.423041.49150.210330.000287LE303.002303.126308.212305.1113.54640.6340980.1383420.0435195E394.742394.783397.347395.79617.30213.434020.32059300000NH347.224347.348352.434349.33313.84962.830650.28898300000The fitted functions are displayed graphically, including the PDF, CDF, SF, and PP plots, in Figures3 and 4. These plots supports the numerical values in Tables 11 and 12 that the proposed BXLE model provides the best fit for the two datasets.Figure 3
The fitted BXLE PDF, CDF, SF, and P-P plots for first dataset.Figure 4
The fitted BXLE PDF, CDF, SF, and P-P plots for second dataset.


## 6. Concluding Remarks

This study proposed a unique three-parameter BurrX logistic-exponential (BXLE) distribution for modeling engineering data and other applications. The BXLE model generalizes and extends the logistic-exponential distribution. The hazard rate of the BXLE distribution might be declining, increasing, bathtub, unimodal, J-shape, or reversed-J shape. In certain circumstances, its mathematical properties were derived. Its density was determined as a mixture of exponential densities. The maximum likelihood estimators, Cramér–von Mises estimators, Anderson–Darling estimators, maximum product of spacing estimators, least-squares estimators, percentile estimators, and weighted least-squares estimators were used to estimate the unknown parameters of the BXLE model. All estimators perform brilliantly in predicting the BXLE parameters, as proved by simulation data. According to our findings, the maximum likelihood approach delivers the best accurate estimations of the parameters of the BXLE distribution. The BXLE distribution’s practical importance was proved using two authentic engineering datasets, proving its acceptable fits and benefits over other competing contemporary models.



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*Source: 4688871-2022-04-28.xml*