E-Bayesian and Bayesian Estimation for the Lomax Distribution under Weighted Composite LINEX Loss Function

The main contribution of this work is the development of a compound LINEX loss function (CLLF) to estimate the shape parameter of the Lomax distribution (LD). The weights are merged into the CLLF to generate a new loss function called the weighted compound LINEX loss function (WCLLF). Then, the WCLLF is used to estimate the LD shape parameter through Bayesian and expected Bayesian (E-Bayesian) estimation. Subsequently, we discuss six different types of loss functions, including square error loss function (SELF), LINEX loss function (LLF), asymmetric loss function (ASLF), entropy loss function (ENLF), CLLF, and WCLLF. In addition, in order to check the performance of the proposed loss function, the Bayesian estimator of WCLLF and the E-Bayesian estimator of WCLLF are used, by performing Monte Carlo simulations. The Bayesian and expected Bayesian by using the proposed loss function is compared with other methods, including maximum likelihood estimation (MLE) and Bayesian and E-Bayesian estimators under different loss functions. The simulation results show that the Bayes estimator according to WCLLF and the E-Bayesian estimator according to WCLLF proposed in this work have the best performance in estimating the shape parameters based on the least mean averaged squared error.


Introduction
e expected Bayesian estimator is a new criterion for estimating the parameters, reliability and hazard functions, which consist of obtaining the expectation of Bayesian estimates with respect to the distributions of hyperparameters [1]. Monte Carlo simulation is used to compare the E-Bayesian estimator with the associated Bayesian estimator in terms of mean averaged squared error (MASE) [2,3]. e E-Bayesian estimation method is efficient and easy to implement on real data [4]. Monte Carlo simulation is also used to compare new methods with corresponding Bayesian and maximum likelihood techniques [5]. e E-Bayesian method is used to obtain the likelihood function of the LD in the right-censored data type II and the parameter estimators of the LD in the right-censored data type II [6]. A new method is developed, to estimate failure probability which is defined based on formulas of the E-Bayesian estimate of the failure probability by [7]. e estimation under the LLF has a smaller deviation than the loss of the square error [8]. e E-Bayesian estimators are attained and built on the balanced squared error loss function by the gamma distribution as a conjugate solution prior for the indefinite scale parameter also using three diverse distributions for the hyperparameters [9]. E-Bayesian and hierarchical Bayesian estimation methods are used for estimating the scale parameter and reversed hazard rate of inverse Rayleigh distribution. ese estimators are derived under squared error, entropy, and prophylactic loss functions [10]. e main purpose of this study is to develop a CLLF and use Bayesian and E-Bayesian estimators to estimate the shape parameters of the LD. en, it will compare the proposed estimator with other methods including, maximum likelihood estimation (MLE), and Bayesian and E-Bayesian estimators under SELF, AS LF, ENLF, and CLLF.
LD is a widely used statistical model in reliability and life test research, especially in analyzing the data of life-testing experiments in engineering sciences, queuing theory, medicine, and physics. e probability density function (P.D.F) is Hence, the C.D.F. is where σ > 0 is a scale parameter and β > 0 is a shape parameter. Also, the reliability function R(t) for the LD has been specified as follows:

Maximum Likelihood Estimation (MLE)
Suppose that x � (x 1 , x 2 , . . . , x n ), distributed according to the LD, is defined in (1). e likelihood of β can be described as where c � n i�1 ln(1 + (x i /σ)) e logarithm of likelihood (4) is As the parameter σ is assumed to be known, the MLE estimator of β is obtained by solving the equation us, the maximum likelihood estimates (MLEs) β MLE of β is given by

Loss Functions
e Bayes estimation of a parameter β is based in minimization of a Bayesian loss (risk) function; L(β, β) is defined as an average cost-of-error function:

LINEX Loss Function (LLF).
e LLF can be expressed as [12,13] e Bayes estimator of β, based on LLF and denoted by β BL , is given by provided that E β � (e − cβ ) exists and is finite.

Asymmetric Loss Function (ASLF).
Asymmetric loss function is defined as [14] L e Bayes estimator of β, based on ASLF and denoted by β BAS , is given by

Entropy Loss Function (ANLF).
e ENLF for β can be expressed as [15] e Bayes estimator of β, denoted by β BEN is the value β which minimizes equation (15) and is given as provided that E β (β − 1 ) exists and is finite.

Composite LINEX Loss Function (CLLF).
CLLF was introduced by Zhang [16] as follows: e Bayes estimator of β, denoted by β BCL , is given by 2 Computational Intelligence and Neuroscience

Weighted Composite LINEX Loss Function.
e researcher proposes this loss function depending on weighting CLLF as follows: where w(β) represents the proposed weighted function, which is given by (20) According to the abovementioned loss function, we drive the corresponding Bayes estimators for β using Risk function R(β − β), which minimizes the posterior risk: e Bayes estimator for the parameter β under the WCLLF, denoted by β WBCL , is given by Note: composite CLLF is a special case of WCLLF when ω � 0 in equation (8). It means the WCLLF is a generalizing of CLLF.

Bayesian Estimation
is section spotlights to derive Bayesian estimates of the shape parameter β of the LD. We use six different loss functions, including SELF, ASLF, ENLF, LLF, CLLF, and WCLLF. We use the gamma (z, k) as a conjugate prior of β and its density function as follows: Based on equations (4) and (23), the posterior density function of β given as x is Computational Intelligence and Neuroscience where c � n i�1 ln(1 + (x i /σ)).

Bayesian Estimation Based on SELF.
e Bayesian estimator β BSE , of β with SELF, is defined as

Bayesian Estimation Based on LLF.
Based on LLF, we can give the Bayesian estimation, β BL , of β as

Bayesian Estimation Based on ASLF.
Under ASLF, the Bayesian estimation, β BS , of β can be expressed as where

Bayesian Estimation-Based ENLF.
Based on ENLF, the Bayesian estimation, β BE , of β can be shown to be

Bayesian Estimation Based on WCLLF.
Under the WCLLF, the Bayesian estimation, β WBCL , of β, can be shown as where Computational Intelligence and Neuroscience So, (36)

Simulation and Results
In order to examine the performance of the estimators obtained in Sections 4 and 5, we used a Monte Carlo simulation study, according to the following steps: (1) Select sample size n � 25, 50, 75, and 100 with the parameter (β � 1, 1.5, and 2).
where β is the estimate at the i th run. (8) e computational results are displayed in Tables 1-4.

Computational Intelligence and Neuroscience
From Tables 1-4, we have the following observations: (1) e estimated values of β is very close to the real values when the sample size increases for all cases; also, the differences between average estimates and the true value of the different estimates decrease as n increases (2) e E-Bayesian estimation of β with the proposed loss function WCLLF has the best estimate due to the

Conclusion
In this work, CLLF is developed to estimate the shape parameter of LD. e development occurred through merging the weights into the CLLF to generate a new loss function called the weighted compound LINEX loss function (WCLLF). We used WCLLF to estimate the LD shape parameter, through Bayesian and expected Bayesian estimation. Subsequently, six different types of loss functions are discussed, including SELF, LLF, ASLF, ENLF, and CLLF and the proposed loss function WCLLF. en, Bayesian and expected Bayesian estimations are compared based on proposed loss function with the other methods, including MLE, Bayesian, and E-Bayesian estimators under different loss functions. e simulation results show that the Bayes estimator according to WCLLF and the E-Bayesian estimator according to WCLLF proposed in this work have the best performance in estimating the shape parameters based on the least mean averaged squared error. E-Bayesian estimators perform better than the Bayesian estimator in terms of MASE, for all sample sizes n and all cases. e results of the simulation showed that the E-Bayesian estimation method is both efficient and easy to perform.

Data Availability
e data used to support the findings of the study were generated by simulation done by using mathematical software.

Conflicts of Interest
e authors declare that they have no conflicts of interest.