# Ridge Regression Method and Bayesian Estimators under Composite LINEX Loss Function to Estimate the Shape Parameter in Lomax Distribution

Authors: Mansour F. Yassen; Fuad S. Al-Duais; Mohammed M. A. Almazah
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

In this paper, the Ridge Regression method is employed to estimate the shape parameter of the Lomax distribution (LD). In addition to that, the approaches of both classical and Bayesian are considered with several loss functions as a squared error (SELF), Linear Exponential (LLF), and Composite Linear Exponential (CLLF). As far as Bayesian estimators are concerned, informative and noninformative priors are used to estimate the shape parameter. To examine the performance of the Ridge Regression method, we compared it with classical estimators which included Maximum Likelihood, Ordinary Least Squares, Uniformly Minimum Variance Unbiased Estimator, and Median Method as well as Bayesian estimators. Monte Carlo simulation compares these estimators with respect to the Mean Square Error criteria (MSE’s). The result of the simulation mentioned that the Ridge Regression method is promising and can be used in a real environment. where it revealed better performance the than Ordinary Least Squares method for estimating shape parameter.


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## Body

## 1. Introduction

Ridge Regression is a popular parameter estimation method for analyzing multiple regression data which has multicollinearity. When Multicollinearity happens, Least squares estimates are unbiased, but their variances are large. As a result, the estimator of the Ordinary Least Squares Method (OLS) becomes far from the true value. Ridge Regression decreases the standard errors when a degree of bias is added to the Regression Estimates. Several authors have addressed Ridge Regression as in [1–10].The Lomax distribution (LD) is a proposed distribution of the Pareto distribution of type II; it was used to obtain a good model for biomedical problems. Also, it is an important model for modeling failure times. The Lomax distribution was used as a stochastic model with a decreasing failure rate for the operating times of the electronic vehicles under study. It was also used in studies related to income and studies related to the size of cities. As well as being a useful model in studying queuing theory and in analyzing data related to biostatistics.Many theoretical and statistician’s studies have given great interest to estimating the parameter and survival analysis of LD.Al-Noor and Alwan [11] compared the Nonbayesian, Bayesian, and Empirical Bayes estimate for the parameters of the LD by considering the symmetric and asymmetric loss functions. Al-Duais and Hmood [12] compared the Bayesian estimators and Classical estimators to estimate the parameter and survival analysis depending on record values and by considering the SELF, LLF, and WLLF. Ellah [13] estimated the parameter, reliability, and hazard function of the LD by applying the Bayesian estimators and Classical estimators based on record values by considering the SELF and LLF. Asl et al. [14] applied the Bayesian estimators and Classical estimators of prediction on unidentified parameters of an LD based on a progressively type-I hybrid censoring scheme. Mohie El-Din et al. [15] studied the Classical estimations and Bayesian estimations of the LD based on progressively type-II censored samples and by considering symmetric (SELF) and asymmetric (LLF and GELF), Okasha [16] estimated the parameters, and survival analysis of the LD by applying the E-Bayesian estimators and Bayesian estimators under type-II censored data and by regarding the balanced squared error loss function. Liu and Zhang [17] studied the Bayesian and E-Bayesian estimations of the LD Based on the Generalized Type-I Hybrid Censoring Scheme. to estimate the unknown parameter of LD and by considering SELF and LLF to estimate the parameter and reliability function. Al-Bossly [18] developed a compound LINEX loss function (CLLF) to estimate the shape parameter of the Lomax distribution utilizing the E-Bayes and Bayes estimation methods for the distributional parameters of the LD.In the current study, Ridge regression was employed to estimate the shape parameter of LD and compare it with the classical estimators which included Maximum Likelihood, Ordinary Least Squares, Uniformly Minimum Variance Unbiased Estimator, and Median Method as well as Bayesian estimators. The uniqueness of this work comes from the fact that, to date, no attempt has been made to estimate the shape parameter of the LD using the method of Ridge regression.The pdf of LD is given as follows [19]:(1)fx;ϑ,δ=ϑδ1+xδ−ϑ+1;x≥0;ϑ,δ>0,0;o.w,where x is a random variable, and δ>0, ϑ>0 are the scale and shape parameters, respectively.The CDF and reliability functionRt of (1) are given by the following equation:(2)Fx;ϑ,δ=1−1+xδ−ϑ;x≥0;ϑ,δ>0,(3)Rt=1+tδ−ϑ;t≥0;ϑ,δ>0.


## 2. Classical Methods of Estimation of Lomax Shape Parameter

The Classical methods selected for the comparative study are (i) Maximum Likelihood Estimator (MLE), (ii) Ordinary Least Squares Method (OLS), (iii) Ridge Regression method, (iv) Uniformly Minimum Variance Unbiased Estimator (UMVUE), and (v) Median Method (M.M).


### 2.1. MLE’s of the Shape Parameterϑ

Suppose thatx¯=x1,x2,…xn is a random sample from the LD as in (1), then the Lx¯|ϑ for the sample observation will be as follows:(4)Lx¯|ϑ=∏i=1nϑδ1+xδ−ϑ+1=ϑδnexp−Tϑ+1,where T=∑i=1nIn1+xi/δ.Log likelihood function(5)lnLϑ,δ=nlnϑ−nlnδ−ϑ+1∑i=1nln1+xiδ.The MLE’s ofϑ denoted by ϑ^MLE is given as follows:(6)ϑ^MLE=n∑i=1nln1+xi/δ.


### 2.2. Ordinary Least Squares Method (OLS)

The CDF in equation (2) satisfies(7)ln1−Fx=−ϑln1+xδ=−ϑlnδ+x+ϑlnδ.Now, suppose thatX1,X2,……Xn form a random sample from LD defined by (1), and that X1<X2<…<Xn are the order statistics. With observed ordered observations x1<x2<…<xn (2) gives the following equation:(8)ln1−Fxi=−ϑln1+xiδ=−ϑlnδ+xi+ϑlnδ.(8) represents a simple linear regression function corresponding to Fxi(9)Yi=α+βXi+εi,where Yi=ln1−F^i and F^ii it is a point estimator of Fxi many estimators for F^i are used.For example, the Median Rank estimatorF^i=i−0.3/n+0.4 or F^i=i−3/8/n+0.25, the mean rank estimator F^i=i/n+1. where i denotes the ith smallest value of x1,x2,…xn, i=1,2,…,n. εi is the random error with expected value Eεi=0. Xi=lnδ+xi, β=−ϑ, α=ϑlnδ.The estimatesα^ and β^ of the regression parameters, α and β minimize the function,(10)Qα,β=∑i=1nYi−α−βlnδ+xi2.Therefore, the estimatesβ^OLS of the parameter, β is given by the following equation:(11)β^OLS=n∑i=1nlnδ+xiln1−F^i−∑i=1nlnδ+xi∑i=1nln1−F^in∑i=1nln2δ+xi−∑i=1nlnδ+xi2,α^OLS=1n∑i=1nln1−F^i−β^OLS1n∑i=1nlnδ+xi.The estimateϑ^OLS of the parameter ϑ is given by the following equation:(12)ϑ^OLS=−β^OLS.


### 2.3. Ridge Regression Method

Ridge Regression estimates can be obtained by minimizing the function.(13)Q∗α,β=∑i=1nln1−F^i−α−βlnδ+xi2.According to the following constraint(14)α2+β2=ϕ,where ϕ is a definite positive constant.The Lagrange multiples method requires that we derive the following:(15)L=∑i=1nln1−F^i−α−βlnδ+xi2+λα2+β2−ϕ,∂lnL∂α=−2∑i=1nln1−F^i−α−βlnδ+xilnδ+xi+2αλ=0,∂lnL∂β=−2∑i=1nln1−F^i−α−βlnδ+xilnδ+xi+2βλ.Therefore, the estimatesα^Rid and β^Rid of the parameters, α and β are given by the following equation:(16)β^Rid=n+λ∑i=1nlnδ+xiln1−F^i−∑i=1nlnδ+xi∑i=1nln1−F^in+λ∑i=1nln2δ+xi−∑i=1nlnδ+xi2,α^Rid=∑i=1nln1−F^i−β∑i=1nlnδ+xin+λ,and(17)λ=ρσ2β′β;0<λ<1,where ρ represents the number of parameter of the distribution and β′β represents the covariance matrix.NOT whenλ=0, we get the estimations of the OLS.The Ridge Regression estimate ofϑ denoted by ϑ^Rid is given as follows:(18)ϑ^Rid=−β^Rid.


### 2.4. UMVUE Estimator of the Shape Parameterϑ

The pdf of LD belongs to the exponential family. Therefore,T=∑i=1nln1+xi/δ is a complete sufficient statistic for ϑ. Then, depending on the theorem of Lehmann-Scheffe [20], the uniformly minimum variance unbiased estimator ϑ^UMVUE of ϑ, may be given by the following equation:(19)ϑ^UMVUE=n−1∑i=1nln1+xi/δ.


### 2.5. Median Method (M.M)

This method is dependent on the basis that the median divides the data into two equal parts(20)Fxmed=0.5.By substituting into the cumulative distribution function defined by (2). The equation will become(21)1−1+xmedδ−ϑ=0.5.Therefore, the estimatesϑ^Med of ϑ, can be obtained as follows:(22)ϑ^Med=−log0.5log1+xmed/δ,where xmed is the median of the data.


## 2.1. MLE’s of the Shape Parameterϑ

Suppose thatx¯=x1,x2,…xn is a random sample from the LD as in (1), then the Lx¯|ϑ for the sample observation will be as follows:(4)Lx¯|ϑ=∏i=1nϑδ1+xδ−ϑ+1=ϑδnexp−Tϑ+1,where T=∑i=1nIn1+xi/δ.Log likelihood function(5)lnLϑ,δ=nlnϑ−nlnδ−ϑ+1∑i=1nln1+xiδ.The MLE’s ofϑ denoted by ϑ^MLE is given as follows:(6)ϑ^MLE=n∑i=1nln1+xi/δ.


## 2.2. Ordinary Least Squares Method (OLS)

The CDF in equation (2) satisfies(7)ln1−Fx=−ϑln1+xδ=−ϑlnδ+x+ϑlnδ.Now, suppose thatX1,X2,……Xn form a random sample from LD defined by (1), and that X1<X2<…<Xn are the order statistics. With observed ordered observations x1<x2<…<xn (2) gives the following equation:(8)ln1−Fxi=−ϑln1+xiδ=−ϑlnδ+xi+ϑlnδ.(8) represents a simple linear regression function corresponding to Fxi(9)Yi=α+βXi+εi,where Yi=ln1−F^i and F^ii it is a point estimator of Fxi many estimators for F^i are used.For example, the Median Rank estimatorF^i=i−0.3/n+0.4 or F^i=i−3/8/n+0.25, the mean rank estimator F^i=i/n+1. where i denotes the ith smallest value of x1,x2,…xn, i=1,2,…,n. εi is the random error with expected value Eεi=0. Xi=lnδ+xi, β=−ϑ, α=ϑlnδ.The estimatesα^ and β^ of the regression parameters, α and β minimize the function,(10)Qα,β=∑i=1nYi−α−βlnδ+xi2.Therefore, the estimatesβ^OLS of the parameter, β is given by the following equation:(11)β^OLS=n∑i=1nlnδ+xiln1−F^i−∑i=1nlnδ+xi∑i=1nln1−F^in∑i=1nln2δ+xi−∑i=1nlnδ+xi2,α^OLS=1n∑i=1nln1−F^i−β^OLS1n∑i=1nlnδ+xi.The estimateϑ^OLS of the parameter ϑ is given by the following equation:(12)ϑ^OLS=−β^OLS.


## 2.3. Ridge Regression Method

Ridge Regression estimates can be obtained by minimizing the function.(13)Q∗α,β=∑i=1nln1−F^i−α−βlnδ+xi2.According to the following constraint(14)α2+β2=ϕ,where ϕ is a definite positive constant.The Lagrange multiples method requires that we derive the following:(15)L=∑i=1nln1−F^i−α−βlnδ+xi2+λα2+β2−ϕ,∂lnL∂α=−2∑i=1nln1−F^i−α−βlnδ+xilnδ+xi+2αλ=0,∂lnL∂β=−2∑i=1nln1−F^i−α−βlnδ+xilnδ+xi+2βλ.Therefore, the estimatesα^Rid and β^Rid of the parameters, α and β are given by the following equation:(16)β^Rid=n+λ∑i=1nlnδ+xiln1−F^i−∑i=1nlnδ+xi∑i=1nln1−F^in+λ∑i=1nln2δ+xi−∑i=1nlnδ+xi2,α^Rid=∑i=1nln1−F^i−β∑i=1nlnδ+xin+λ,and(17)λ=ρσ2β′β;0<λ<1,where ρ represents the number of parameter of the distribution and β′β represents the covariance matrix.NOT whenλ=0, we get the estimations of the OLS.The Ridge Regression estimate ofϑ denoted by ϑ^Rid is given as follows:(18)ϑ^Rid=−β^Rid.


## 2.4. UMVUE Estimator of the Shape Parameterϑ

The pdf of LD belongs to the exponential family. Therefore,T=∑i=1nln1+xi/δ is a complete sufficient statistic for ϑ. Then, depending on the theorem of Lehmann-Scheffe [20], the uniformly minimum variance unbiased estimator ϑ^UMVUE of ϑ, may be given by the following equation:(19)ϑ^UMVUE=n−1∑i=1nln1+xi/δ.


## 2.5. Median Method (M.M)

This method is dependent on the basis that the median divides the data into two equal parts(20)Fxmed=0.5.By substituting into the cumulative distribution function defined by (2). The equation will become(21)1−1+xmedδ−ϑ=0.5.Therefore, the estimatesϑ^Med of ϑ, can be obtained as follows:(22)ϑ^Med=−log0.5log1+xmed/δ,where xmed is the median of the data.


## 3. Prior and Posterior Density Functions

### 3.1. Prior Distribution

The Bays estimators demand an appropriate selection of priors for the parameter. If we do not have sufficient knowledge about the parameter, in this case, the noninformative priors are better chosen. Or else, it is desirable to use informative priors. In this research, we study both types of priors: informative priors and noninformative priors.


#### 3.1.1. Non-Informative Prior

Let us assume thatϑ has noninformative prior density defined as using extended Jeffrey's prior h1ϑ which is given by the following equation:(23)h1ϑ∝Iϑc;c>0,where Iϑ represented Fisher information matrix which defined as follows:(24)Iϑ=−nE∂2logfx;ϑ,δ∂ϑ,h1ϑ=1ϑ2c;c>0.


#### 3.1.2. Informative Priors (The Natural Conjugate Prior)

In this work, three types of prior distributions were used to study the effect of the different prior distributions on a Bayesian estimate ofϑ.(a)
Chi-squared prior(25)h2ϑ=dk/22k/2Γk/2ϑk/2−1exp−dϑ2;ϑ>0;k,d>0.(b)
Inverted levy prior(26)h3ϑ=k2πϑ−1/2exp−dϑ2;ϑ>0;k>0,(c)
Gamma Prior(27)h4ϑ=dkΓkϑk−1exp−dϑ;ϑ>0;k,d>0.


### 3.2. Posterior Density Functions

The posterior distribution for the shape parameterϑ can be expressed as follows:(28)πϑ|x¯=Lϑ,δ|x¯hϑ∫0∞Lϑ,δ|x¯hϑdϑ.Combining theL|x¯|ϑ in (4) and the prior distribution of extended Jeffrey’s prior (16), chi-square prior (17), inverted Levy prior (18), and gamma prior (19). The posterior density of ϑ It can be found on respectively as follows:(29)π1ϑ|x¯=Tn−2c+1Γn−2c+1ϑn−2cexp−Tϑ;ϑ>0;c>0,π2ϑ|x¯=T+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑ;ϑ>0;k,d>0,π3ϑ|x¯=T+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑ;ϑ>0;k,d>0,π4ϑ|x¯=T+dn+kΓn+kϑn+k−1exp−T+dϑ;ϑ>0;k,d>0.


## 3.1. Prior Distribution

The Bays estimators demand an appropriate selection of priors for the parameter. If we do not have sufficient knowledge about the parameter, in this case, the noninformative priors are better chosen. Or else, it is desirable to use informative priors. In this research, we study both types of priors: informative priors and noninformative priors.


### 3.1.1. Non-Informative Prior

Let us assume thatϑ has noninformative prior density defined as using extended Jeffrey's prior h1ϑ which is given by the following equation:(23)h1ϑ∝Iϑc;c>0,where Iϑ represented Fisher information matrix which defined as follows:(24)Iϑ=−nE∂2logfx;ϑ,δ∂ϑ,h1ϑ=1ϑ2c;c>0.


### 3.1.2. Informative Priors (The Natural Conjugate Prior)

In this work, three types of prior distributions were used to study the effect of the different prior distributions on a Bayesian estimate ofϑ.(a)
Chi-squared prior(25)h2ϑ=dk/22k/2Γk/2ϑk/2−1exp−dϑ2;ϑ>0;k,d>0.(b)
Inverted levy prior(26)h3ϑ=k2πϑ−1/2exp−dϑ2;ϑ>0;k>0,(c)
Gamma Prior(27)h4ϑ=dkΓkϑk−1exp−dϑ;ϑ>0;k,d>0.


## 3.1.1. Non-Informative Prior

Let us assume thatϑ has noninformative prior density defined as using extended Jeffrey's prior h1ϑ which is given by the following equation:(23)h1ϑ∝Iϑc;c>0,where Iϑ represented Fisher information matrix which defined as follows:(24)Iϑ=−nE∂2logfx;ϑ,δ∂ϑ,h1ϑ=1ϑ2c;c>0.


## 3.1.2. Informative Priors (The Natural Conjugate Prior)

In this work, three types of prior distributions were used to study the effect of the different prior distributions on a Bayesian estimate ofϑ.(a)
Chi-squared prior(25)h2ϑ=dk/22k/2Γk/2ϑk/2−1exp−dϑ2;ϑ>0;k,d>0.(b)
Inverted levy prior(26)h3ϑ=k2πϑ−1/2exp−dϑ2;ϑ>0;k>0,(c)
Gamma Prior(27)h4ϑ=dkΓkϑk−1exp−dϑ;ϑ>0;k,d>0.


## 3.2. Posterior Density Functions

The posterior distribution for the shape parameterϑ can be expressed as follows:(28)πϑ|x¯=Lϑ,δ|x¯hϑ∫0∞Lϑ,δ|x¯hϑdϑ.Combining theL|x¯|ϑ in (4) and the prior distribution of extended Jeffrey’s prior (16), chi-square prior (17), inverted Levy prior (18), and gamma prior (19). The posterior density of ϑ It can be found on respectively as follows:(29)π1ϑ|x¯=Tn−2c+1Γn−2c+1ϑn−2cexp−Tϑ;ϑ>0;c>0,π2ϑ|x¯=T+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑ;ϑ>0;k,d>0,π3ϑ|x¯=T+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑ;ϑ>0;k,d>0,π4ϑ|x¯=T+dn+kΓn+kϑn+k−1exp−T+dϑ;ϑ>0;k,d>0.


## 4. Loss Functions

In Bayes estimation, we will consider three types of loss functions including SELL, LLF, and CLLF.


### 4.1. Squared Error Loss Function

The SELF is defined as follows [21]:(30)Lϑ^,ϑ=ϑ^−ϑ2.The Bayes estimator ofϑ relative to SELF, signified by ϑ^BSE is(31)ϑ^BSE=Ehϑ|x¯.


### 4.2. LINEX Loss Function

The LINEX loss function forϑ can be written as follows [22, 23]:(32)Lϑ^,ϑ∝expaϑ^−ϑ−aϑ^−ϑ−1;a≠0.The Bayes estimator ofϑ relative to LLF, denoted by ϑ^BL is(33)ϑ^BL=−1aLnEϑexp−aϑ;a≠0.Provided thatEϑ=exp−aϑ exists and is finite, where Eϑ denotes the expected value.


### 4.3. Composite LINEX Loss Function

CLLF is given by the following formula [24].(34)Lϑ^,ϑ=Laϑ^,ϑ+L−aϑ^,ϑ=exp−aϑ^,ϑ+expaϑ^,ϑ−2a>0.The Bayes estimator ofϑ relative to CLLF, denoted by β^BCL, is(35)β^BCL=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯.Provided thatEϑ=expaϑ|x¯ and Eϑ|x¯exp−aϑ exist and are finite.


## 4.1. Squared Error Loss Function

The SELF is defined as follows [21]:(30)Lϑ^,ϑ=ϑ^−ϑ2.The Bayes estimator ofϑ relative to SELF, signified by ϑ^BSE is(31)ϑ^BSE=Ehϑ|x¯.


## 4.2. LINEX Loss Function

The LINEX loss function forϑ can be written as follows [22, 23]:(32)Lϑ^,ϑ∝expaϑ^−ϑ−aϑ^−ϑ−1;a≠0.The Bayes estimator ofϑ relative to LLF, denoted by ϑ^BL is(33)ϑ^BL=−1aLnEϑexp−aϑ;a≠0.Provided thatEϑ=exp−aϑ exists and is finite, where Eϑ denotes the expected value.


## 4.3. Composite LINEX Loss Function

CLLF is given by the following formula [24].(34)Lϑ^,ϑ=Laϑ^,ϑ+L−aϑ^,ϑ=exp−aϑ^,ϑ+expaϑ^,ϑ−2a>0.The Bayes estimator ofϑ relative to CLLF, denoted by β^BCL, is(35)β^BCL=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯.Provided thatEϑ=expaϑ|x¯ and Eϑ|x¯exp−aϑ exist and are finite.


## 5. Bayes Estimator

In this part, we estimateϑ, using three various loss functions, including SELF, LLF, and CLLF. We assume four different prior distributions for ϑ including; extended Jeffrey's prior, chi-square prior, inverted Levy prior, and gamma prior [25–28].


### 5.1. Bayesian Estimator ofϑ under SELF

The Bayes estimates ofϑ relative to SELF depended on π1ϑ|x¯ which is signified as ϑ^BSE1 and can be acquired by using equations (21) and (26) to be(36)ϑ^BSE1=Eϑ|x¯=∫0∞ϑπ1ϑ|x¯dϑ,ϑ^BSE1=∫0∞Tn−2c+1Γn−2c+1ϑn−2c+1exp−Tϑdϑ=n−2c+1T.Likewise, we can obtain the Bayesian estimates ofϑ relative to SELF depending on π2ϑ|x¯, π3ϑ|x¯, and π4ϑ|x¯, which are signified as ϑ^BSE2, ϑ^BSE3, and ϑ^BSE4 by using equations ((22) and (26)), ((23) and (26)) and ((24) and (26)), respectively, to be(37)ϑ^BSE2=Eϑ|x¯=∫0∞ϑπ2ϑ|x¯dϑ,ϑ^BSE2=∫0∞T+d/2n+k/2Γn+k/2ϑn+k/2exp−T+d2ϑdϑ=n+k/2T+d/2,ϑ^BSE3=Eϑ|x¯=∫0∞ϑπ3ϑ|x¯dϑ,ϑ^BSE3=∫0∞T+d/2n+1/2Γn+1/2ϑn+1/2exp−T+d2ϑdϑ=Γn+3/2Γn+1/2T+d/2,and(38)ϑ^BSE4=Eϑ|x¯=∫0∞ϑπ3ϑ|x¯dϑ,ϑ^BSE4=Eϑ|x¯=∫0∞ϑπ3ϑ|x¯dϑ,ϑ^BSE4=∫0∞T+dn+kΓn+kϑn+kexp−T+dϑdϑ=n+kT+d.


### 5.2. Bayesian Estimator ofϑ under LLF

We can obtain the Bayes estimator ofϑ under the LLF depending on π1ϑ|x¯ signified as ϑ^BL1 by using equations (21) and (28) as follows:(39)ϑ^BL1=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ1ϑ|x¯dϑ,ϑ^BL1=−1aln∫0∞exp−aϑTn−2c+1Γn−2c+1ϑn−2cexp−Tϑdϑ=n−2c+1aln1+aT.In the same way, the Bayes estimates ofϑ relative to LLF depended on π2ϑ|x¯, π3ϑ|x¯, and π4ϑ|x¯, which are signified as ϑ^BL2, ϑ^BL3, and ϑ^BL4 by using equations ((22) and (28)) ((23) and (28)), and ((24) and (28)), respectively, to be(40)ϑ^BL2=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ2ϑ|x¯dϑ,ϑ^BL2=−1aln∫0∞exp−aϑT+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑdϑ=n+0.5kaln1+aT+d/2,ϑ^BL3=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ3ϑ|x¯dϑ,ϑ^BL3=−1aln∫0∞exp−aϑT+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑdϑ=n+0.5aln1+aT+d/2,and(41)ϑ^BL4=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ4ϑ|x¯dϑ,ϑ^BL4=−1aln∫0∞exp−aϑT+dn+kΓn+kϑn+k−1exp−T+dϑdϑ=n+kaln1+aT+d.


### 5.3. Bayesian Estimation ofϑ under CLLF

The Bayes estimate ofϑ under CLLF depended on π1ϑ|x¯, which is signified as ϑ^BCL1 by using equations (21) and (30) to be(42)ϑ^BCL1=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I1I2,where:(43)I1=Eϑexpaϑ|x¯=∫0∞expaϑπ1ϑ|x¯dϑ=∫0∞expaϑTn−2c+1Γn−2c+1ϑn−2cexp−Tϑdϑ=TT−an−2c+1,and(44)I2=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ1ϑ|x¯dϑ=∫0∞exp−aϑTn−2c+1Γn−2c+1ϑn−2cexp−Tϑdϑ=TT+an−2c+1.Therefore, the Bayes estimation of parameterϑ is(45)ϑ^BCL1=12alnT/T−an−2c+1T/T+an−2c+1=n−2c+12alnT+aT−a.Similarly, the Bayesian estimates ofϑ under CLLF depended on π2ϑ|x¯, π3ϑ|x¯, and π4ϑ|x¯, which are signified as ϑ^BCL2, ϑ^BCL3, and ϑ^BCL4 by using equations ((22) and (30)), ((23) and (30)), and ((24) and (30)), respectively, to be(46)ϑ^BCL2=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I3I4,where(47)I3=Eϑexpaϑ|x¯=∫0∞expaϑπ2ϑ|x¯dϑ=∫0∞expaϑT+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑdϑ=T+d/2T+d/2−an+k/2,and(48)I4=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ2ϑ|x¯dϑ=∫0∞exp−aϑT+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑdϑ=T+d/2T+d/2+an+k/2,So(49)ϑ^BCL2=12alnT+d/2/T+d/2−an+k/2T+d/2/T+d/2+an+k/2=n+k/22alnT+d/2+aT+d/2−a,and(50)ϑ^BCL3=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I5I6,where(51)I5=Eϑexpaϑ|x¯=∫0∞expaϑπ3ϑ|x¯dϑ=∫0∞expaϑT+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑdϑ=T+d/2T+d/2−an+1/2,and(52)I6=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ3ϑ|x¯dϑ=∫0∞exp−aϑT+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑdϑ=T+d/2T+d/2+an+1/2,So(53)ϑ^BCL3=12alnT+d/2/T+d/2−an+1/2T+d/2/T+d/2+an+1/2=n+1/22alnT+d/2+aT+d/2−a,ϑ^BCL4=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I7I8,where(54)I7=Eϑexpaϑ|x¯=∫0∞expaϑπ4ϑ|x¯dϑ=∫0∞expaϑT+dn+kΓn+kϑn+k−1exp−T+dϑdϑ=T+dT+d−an+k,and(55)I8=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ4ϑ|x¯dϑ=∫0∞exp−aϑT+dn+kΓn+kϑn+k−1exp−T+dϑdϑ=T+dT+d+an+k.Therefore, the Bayes estimation of parameterϑ is(56)ϑ^BCL4=12alnT+d/T+d/2−an+kT+d/T+d+an+k=n+k2alnT+d+aT+d−a.


## 5.1. Bayesian Estimator ofϑ under SELF

The Bayes estimates ofϑ relative to SELF depended on π1ϑ|x¯ which is signified as ϑ^BSE1 and can be acquired by using equations (21) and (26) to be(36)ϑ^BSE1=Eϑ|x¯=∫0∞ϑπ1ϑ|x¯dϑ,ϑ^BSE1=∫0∞Tn−2c+1Γn−2c+1ϑn−2c+1exp−Tϑdϑ=n−2c+1T.Likewise, we can obtain the Bayesian estimates ofϑ relative to SELF depending on π2ϑ|x¯, π3ϑ|x¯, and π4ϑ|x¯, which are signified as ϑ^BSE2, ϑ^BSE3, and ϑ^BSE4 by using equations ((22) and (26)), ((23) and (26)) and ((24) and (26)), respectively, to be(37)ϑ^BSE2=Eϑ|x¯=∫0∞ϑπ2ϑ|x¯dϑ,ϑ^BSE2=∫0∞T+d/2n+k/2Γn+k/2ϑn+k/2exp−T+d2ϑdϑ=n+k/2T+d/2,ϑ^BSE3=Eϑ|x¯=∫0∞ϑπ3ϑ|x¯dϑ,ϑ^BSE3=∫0∞T+d/2n+1/2Γn+1/2ϑn+1/2exp−T+d2ϑdϑ=Γn+3/2Γn+1/2T+d/2,and(38)ϑ^BSE4=Eϑ|x¯=∫0∞ϑπ3ϑ|x¯dϑ,ϑ^BSE4=Eϑ|x¯=∫0∞ϑπ3ϑ|x¯dϑ,ϑ^BSE4=∫0∞T+dn+kΓn+kϑn+kexp−T+dϑdϑ=n+kT+d.


## 5.2. Bayesian Estimator ofϑ under LLF

We can obtain the Bayes estimator ofϑ under the LLF depending on π1ϑ|x¯ signified as ϑ^BL1 by using equations (21) and (28) as follows:(39)ϑ^BL1=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ1ϑ|x¯dϑ,ϑ^BL1=−1aln∫0∞exp−aϑTn−2c+1Γn−2c+1ϑn−2cexp−Tϑdϑ=n−2c+1aln1+aT.In the same way, the Bayes estimates ofϑ relative to LLF depended on π2ϑ|x¯, π3ϑ|x¯, and π4ϑ|x¯, which are signified as ϑ^BL2, ϑ^BL3, and ϑ^BL4 by using equations ((22) and (28)) ((23) and (28)), and ((24) and (28)), respectively, to be(40)ϑ^BL2=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ2ϑ|x¯dϑ,ϑ^BL2=−1aln∫0∞exp−aϑT+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑdϑ=n+0.5kaln1+aT+d/2,ϑ^BL3=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ3ϑ|x¯dϑ,ϑ^BL3=−1aln∫0∞exp−aϑT+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑdϑ=n+0.5aln1+aT+d/2,and(41)ϑ^BL4=−1aLnEϑexp−aϑ=∫0∞exp−aϑπ4ϑ|x¯dϑ,ϑ^BL4=−1aln∫0∞exp−aϑT+dn+kΓn+kϑn+k−1exp−T+dϑdϑ=n+kaln1+aT+d.


## 5.3. Bayesian Estimation ofϑ under CLLF

The Bayes estimate ofϑ under CLLF depended on π1ϑ|x¯, which is signified as ϑ^BCL1 by using equations (21) and (30) to be(42)ϑ^BCL1=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I1I2,where:(43)I1=Eϑexpaϑ|x¯=∫0∞expaϑπ1ϑ|x¯dϑ=∫0∞expaϑTn−2c+1Γn−2c+1ϑn−2cexp−Tϑdϑ=TT−an−2c+1,and(44)I2=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ1ϑ|x¯dϑ=∫0∞exp−aϑTn−2c+1Γn−2c+1ϑn−2cexp−Tϑdϑ=TT+an−2c+1.Therefore, the Bayes estimation of parameterϑ is(45)ϑ^BCL1=12alnT/T−an−2c+1T/T+an−2c+1=n−2c+12alnT+aT−a.Similarly, the Bayesian estimates ofϑ under CLLF depended on π2ϑ|x¯, π3ϑ|x¯, and π4ϑ|x¯, which are signified as ϑ^BCL2, ϑ^BCL3, and ϑ^BCL4 by using equations ((22) and (30)), ((23) and (30)), and ((24) and (30)), respectively, to be(46)ϑ^BCL2=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I3I4,where(47)I3=Eϑexpaϑ|x¯=∫0∞expaϑπ2ϑ|x¯dϑ=∫0∞expaϑT+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑdϑ=T+d/2T+d/2−an+k/2,and(48)I4=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ2ϑ|x¯dϑ=∫0∞exp−aϑT+d/2n+k/2Γn+k/2ϑn+k/2−1exp−T+d2ϑdϑ=T+d/2T+d/2+an+k/2,So(49)ϑ^BCL2=12alnT+d/2/T+d/2−an+k/2T+d/2/T+d/2+an+k/2=n+k/22alnT+d/2+aT+d/2−a,and(50)ϑ^BCL3=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I5I6,where(51)I5=Eϑexpaϑ|x¯=∫0∞expaϑπ3ϑ|x¯dϑ=∫0∞expaϑT+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑdϑ=T+d/2T+d/2−an+1/2,and(52)I6=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ3ϑ|x¯dϑ=∫0∞exp−aϑT+d/2n+1/2Γn+1/2ϑn−1/2exp−T+d2ϑdϑ=T+d/2T+d/2+an+1/2,So(53)ϑ^BCL3=12alnT+d/2/T+d/2−an+1/2T+d/2/T+d/2+an+1/2=n+1/22alnT+d/2+aT+d/2−a,ϑ^BCL4=12alnEϑexpaϑ|x¯Eϑexp−aϑ|x¯=I7I8,where(54)I7=Eϑexpaϑ|x¯=∫0∞expaϑπ4ϑ|x¯dϑ=∫0∞expaϑT+dn+kΓn+kϑn+k−1exp−T+dϑdϑ=T+dT+d−an+k,and(55)I8=Eϑexp−aϑ|x¯=∫0∞exp−aϑπ4ϑ|x¯dϑ=∫0∞exp−aϑT+dn+kΓn+kϑn+k−1exp−T+dϑdϑ=T+dT+d+an+k.Therefore, the Bayes estimation of parameterϑ is(56)ϑ^BCL4=12alnT+d/T+d/2−an+kT+d/T+d+an+k=n+k2alnT+d+aT+d−a.


## 6. Simulation Study and Results

In this part, a simulation study has been conducted to assess and examine the behavior of Classical methods and Bayes estimators for the shape parameter of LD under different cases. The following steps of the simulation are as follows:(1)
Set the true values for the parameters of LD which are varied into three cases to observe their effect on the estimates whenδ>ϑ, δ=ϑ, and δ<ϑ “case I δ=2,ϑ=1.5, case II δ=2,ϑ=2 and case III δ=2,ϑ=2.5”(2)
Determine the sample sizen=20,40,60,80 and 100(3)
Determine the valueλ = 0.75, c=0.5, K,d=0.6,0.2 and a=0.5,1 and 1.5(4)
For a given sample sizen, generate x1,x2,…xn by using the following formula: xi=δ1−Ui−1/ϑ−1, i=1,…,n, where Ui is uniform 0,1(5)
Classical methods estimation,ϑ^MLE, ϑ^OLS , ϑ^Rid, ϑ^UMVUE, and ϑ^Med of ϑ are computed from equations (6), (10), (13)–(15), respectively(6)
Under SELL and based onh1ϑ,h2ϑh3ϑ and h4ϑ priors, Bayesian estimation, ϑ^BSE1, ϑ^BSE2, ϑ^BSE3 and ϑ^BSE.$, of ϑ are computed from equations (31), (32), (33), and (34), respectively(7)
Under LLF and based onh1ϑ,h2ϑ,h3ϑ and h4ϑ priors, Bayesian estimation, ϑ^BL1, ϑ^BL2, ϑ^BL3 and ϑ^BL4, of ϑ are computed from equations (35), (36), (37), and (38), respectively(8)
Under CLLF and based onh1ϑ,h2ϑh3ϑ and h4ϑ priors, ϑ^BCL1, ϑ^BCL2, ϑ^BCL3 and ϑ^BCL4, of ϑ are calculated from equations (40), (42), (44), and (46), respectively(9)
Steps 4 to 8 are replicated 10,000 times. The (MSE’s) for all Estimates of the parameterϑ are obtained, where(57)MSEϑ^=110000∑i=110000ϑ^i−ϑ2.The results are displayed in the following Tables1–5.Table 1
MSE values for non-bayes estimators ofϑ.
Casesnϑ^MLEϑ^OLSϑ^Ridϑ^UMVUEϑ^MedI200.14570.20210.13500.12490.3042400.06450.10050.08700.05990.1440600.04080.06890.06310.03880.0873800.03020.05310.05010.02910.06651000.02370.04170.03990.02290.0507II200.25510.35850.24340.21960.5406400.11260.18140.15210.10470.2412600.07020.12240.11050.06670.1528800.05260.09420.08820.05070.11481000.04120.07700.07320.03990.0878III200.39950.57430.43570.34620.7967400.17430.28180.24550.16210.3770600.11160.18640.17350.10620.2475800.08070.14780.14160.07780.17751000.06680.12150.11750.06490.1440Table 2
MSE values for bayes estimators ofϑ with extended Jeffrey’s prior.
Casesnϑ^BSE1ϑ^BL1ϑ^BCL1a=0.5a=1a=1.5a=0.5a=1a=1.5I200.14570.12980.11770.10810.14640.14760.1511400.06450.06110.05600.05320.06450.06210.0614600.04080.03930.03710.03710.04080.03970.0408800.03020.02940.02920.02750.03030.03070.02961000.02370.02320.02300.02230.02370.02390.0237II200.25510.22000.19610.18030.25740.26280.2776400.11260.10500.09670.09520.11280.11040.1151600.07020.06700.06550.06320.07030.07160.0708800.05260.05080.05040.04800.05260.05360.05261000.04120.04000.03970.03880.04120.04190.0415III200.39950.33470.29220.27270.40510.41890.4535400.17430.16010.15170.14410.17480.17800.1794600.11160.10540.10120.09850.11170.11260.1121800.08070.07740.07290.07230.08080.07860.08051000.06680.06460.06000.05960.06680.06360.0646Table 3
MSE values for bayes estimators ofϑ with chi-square prior.
Casesnϑ^BSE1ϑ^BL1ϑ^BCL1a=0.5a=1a=1.5a=0.5a=1a=1.5I200.14680.13050.11770.10770.14760.14860.1522400.06480.06130.05610.05310.06490.06250.0617600.04100.03950.03710.03700.04100.03990.0410800.03040.02950.02920.02750.03040.03090.02971000.02380.02320.02310.02230.02380.02400.0238II200.25290.21770.19330.17710.25520.26020.2746400.11240.10450.09610.09440.11260.11010.1149600.07020.06680.06530.06290.07020.07150.0708800.05260.05070.05020.04790.05260.05360.05261000.04120.04000.03970.03860.04120.04190.0415III200.38950.32650.28490.26580.39480.40770.4405400.17260.15840.14990.14230.17310.17620.1776600.11090.10470.10050.09770.11100.11190.1114800.08040.07700.07250.07190.08040.07830.08011000.06660.06430.05980.05930.06660.06340.0643Table 4
MSE values for bayes estimators ofϑ with inverted Levy prior.
Casesnϑ^BSE1ϑ^BL1ϑ^BCL1a=0.5a=1a=1.5a=0.5a=1a=1.5I200.15280.13520.12100.11000.15360.15440.1583400.06620.06240.05690.05360.06630.06380.0630600.04160.04000.03750.03730.04160.04050.0416800.03070.02980.02940.02770.03070.03120.03001000.02400.02340.02320.02240.02400.02420.0240II200.26300.22480.19740.17930.26540.27030.2858400.11460.10620.09720.09510.11490.11240.1174600.07120.06760.06580.06310.07120.07260.0717800.05310.05120.05050.04810.05320.05410.05321000.04160.04030.03990.03870.04160.04220.0418III200.40430.33580.28920.26620.40990.42340.4581400.17600.16070.15110.14250.17650.17980.1813600.11240.10570.10110.09770.11260.11350.1128800.08120.07760.07280.07200.08130.07910.08111000.06710.06470.05990.05940.06710.06390.0649Table 5
MSE values for bayes estimators ofϑ with gamma prior.
Casesnϑ^BSE1ϑ^BL1ϑ^BCL1a=0.5a=1a=1.5a=0.5a=1a=1.5I200.14820.13140.11790.10750.14890.14980.1534400.06520.06150.05620.05300.06530.06290.0621600.04120.03960.03720.03700.04120.04010.0412800.03050.02960.02930.02750.03050.03100.02981000.02390.02330.02310.02230.02390.02400.0239II200.25100.21560.19070.17420.25310.25780.2720400.11210.10410.09560.09370.11230.10990.1147600.07010.06670.06510.06260.07020.07150.0707800.05260.05070.05010.04770.05260.05350.05261000.04120.04000.03960.03850.04120.04190.0415III200.38010.31870.27790.25910.38510.39710.4284400.17090.15670.14820.14050.17140.17450.1759600.11020.10400.09970.09690.11040.11130.1107800.08000.07670.07210.07140.08010.07800.07981000.06630.06410.05950.05910.06640.06320.0641


## 7. Conclusions and Recommendations

In this paper, the Ridge Regression method was employed to estimate the shape parameter of LD. Besides, researchers made a Monte Carlo simulation to test the performance of the Ridge Regression method. Then, compared the Ridge Regression estimator with the other estimators, including MLE, OLS, UMVUE, M.M, and Bayesian estimators based on SELF, LLF, and CLLF. However, the major observations are identified in the following points:(1)
Among classical estimators, in Table1, the performance of the UMVUE was shown as better than other estimators: “MLE, OLS, Ridge, and M.M estimators” in all different cases and all samples sizes. Whereas the performance of Ridge Regression was better than MLE. Estimator especially for a small sample size n=10. In the meanwhile, the results showed that the performance of the Ridge estimator was better than OLS estimator in all different cases and sample sizes.(2)
With Bayes estimators, gamma prior records full appearance as best prior based on LLF and CLLF for all different cases and all sample sizes. As well as that is true under SELF withδ=ϑ and δ<ϑ , while extended Jeffrey’s prior record as best prior based on SELF when δ>ϑ.(3)
The MSE values associated with each of the classical and Bayes estimate “corresponding to each prior and every loss function” reduces with the increase in the sample size. Also, the results show a convergence between most of the estimators to increase the sample sizes and this conforms to the statistical theory.(4)
For all cases and all sample sizes, LLFa=1.5 records full appearance as the best loss function associated with Bayes estimates corresponding to gamma prior.(5)
According to the results, MSE values of all classical and Bayes estimators of shape parameters are decreasing as the shape parameter value increase.



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*Source: 1200611-2022-08-29.xml*