Bayesian Shrinkage Estimator of Burr XII Distribution

In this paper, we derive the generalized Bayesian shrinkage estimator of parameter of Burr XII distribution under three loss functions: squared error, LINEX, and weighted balance loss functions. 'erefore, we obtain three generalized Bayesian shrinkage estimators (GBSEs). In this approach, we find the posterior risk function (PRF) of the generalized Bayesian shrinkage estimator (GBSE) with respect to each loss function.'e constant formula of GBSE is computed by minimizing the PRF. In special cases, we derive two new GBSEs under the weighted loss function. Finally, we give our conclusion.


Introduction
e Burr type XII distribution was first introduced in the literature [1] and has gained special attention in the last two decades or so due to its broad applications in different fields including the area of reliability, failure time modeling, and acceptance sampling plan. e failure of optimum properties of the natural estimator in certain special problems with the risk usually measured by the mean squared error or, in the case of several parameters, by a quadratic function of the estimators, is introduced in [2]. e exact definition of shrinkage estimators is hard to come by, and in [3], shrinkage estimators are characterized as the ones obtained through modification of some standard estimators. e statistical properties and prediction analysis of some estimators are studied in [4][5][6], and they suggested that the use of symmetric loss function may not be appropriate in some estimation and prediction problems. e LINEX loss function is introduced in [7] that it is approximately exponential on one side of zero and approximately linear on the other side. In most of the available literature, the LINEX loss function has been considered as a comparison criterion for comparing competing estimators in the linear regression model. e performance of the least-square estimator of the regression coefficient in a regression model is examined in [8], using the Bayesian approach under asymmetric loss function. e performance properties of some conventional estimators of error variance are studied in [9], under asymmetric loss function. e approach in [7] was modified in [10]. Also, a modified version of the LINEX loss function exists, which is the general entropy loss function proposed in [11]. A minimum mean square error (MMSE) estimator of parameter in the exponential distribution is obtained in [12]. Searls's estimator has been used in [13], and it was inadmissible under the LLF. e optimal shrinkage estimations in [14] are derived for the parameters of exponential distribution based on recorded values. e shrinkage estimation of the parameter of exponential type-II censored data in [15] is presented under LLF. A new methodology for Bayesian analysis of mixture models in [16] has been considered under doubly censored samples. In [17], some Bayesian estimators and Bayesian shrinkage estimators for a family of probability density functions are suggested, and the properties of the suggested estimators in terms of relative efficiencies under two different loss functions are studied. A Bayesian estimate of reliability for the exponential case is developed which utilizes the basic notion of loss in estimation theory [18]. In [19], it is investigated that whether the dominance of the OLS-based estimator of the disturbance estimator over the Stein rule-based estimator still holds when compared under the LINEX loss function using smallsigma asymptotics. A shrinkage estimator is derived for the parameter of exponential distribution contaminated with outliers and in the presence of LINEX loss function in [20]. In this case, an admissible estimator based on the LINEX loss function is also compared with different methods of estimations. In [21], a multivariate normal mean under natural modifications of balanced loss functions is estimated. In [22], explicit expressions for the quantiles, moments, moment-generating function, conditional moments, hazard rates, mean residual lifetime, mean past lifetime, mean deviation about mean and median, the stochastic ordering, various entropies, stress-strength parameters, Bonferroni and Lorenz curves, and order statistics are found. e properties of the SE of the parameter of the simple linear regression model are investigated in [23] under the LINEX loss function.
In this paper, we study the generalized Bayesian shrinkage estimator of the Burr XII distribution under three special loss functions: squared error, LINEX, and weighted balance loss functions. We obtain a new class of Bayesian shrinkage estimators (GBSEs) under SELF, LLF, and WBLF. Under WBLF, we find a special GBSE.
Our work is divided into six sections. In Section 2, we derive the posterior risk function with respect to the squared error loss function (SELF); in this case, we calculate the generalized Bayesian estimator (GBE) and the generalized Bayesian shrinkage estimator (GBSE) under SELF. In Section 3, we find the posterior risk function with respect to the LINEX loss function (LLF); in this case, we compute the GBE and GBSE under LLF. In Section 4, the posterior risk function with respect to the weighted balance loss function (WBLF) is derived, and GBE and GBSE are derived under WBLF. In Section 5, the GBSE under WBLF has two special cases: if ω � 0, then we calculate a new GBSE; if ω � 1, then we obtain another new GBSE. Finally, we give our conclusion in Section 6.

Baysian Shrinkage Estimator under Squared Error Loss Function
In this section, we derive the generalized Bayesian shrinkage estimator (GBSE) under the squared error loss function (SELF). e first step to find GBSE is to compute the generalized Bayesian estimator for the parameter b under the assumption that the parameter c is known. e Burr XII distribution, Burr(b, c), has the following pdf and cdf [24], respectively: (1) e posterior distribution of the parameter b is the gamma distribution G(n, t) which has pdf: where t � n i�0 ln(1 + x c i ) and the improper prior distribution 1/b. e squared error loss function (SELF) is defined as (4) e posterior risk function (PRF) of b can be calculated as where b ∼ gamma(n, t). Taking the derivative of the posterior risk function (PRF) with respect to b s1 , we obtain By minimizing the posterior risk function (PRF) with respect to b s1 , we obtain Equation (7) is the generalized Bayesian estimator of the parameter b under the squared error loss function. Now, we will calculate the GBSE under SELF, and the shrinkage estimator is defined as where b s1 is the generalized Bayesian estimator. e risk function of b sh1 is defined as Equation (9) is the posterior risk function with respect to the shrinkage estimator. To calculate the constant k, we take the derivative w.r.t. k, so we have We minimize ρ 1 (b, b sh1 ) by assuming zρ 1 (b, b sh1 )/ zk � 0; thus, one can obtain 2 International Journal of Mathematics and Mathematical Sciences Substituting equations (7) and (11) in equation (8), we get Equation (12) is the generalized Bayesian shrinkage estimator (GBSE) under the squared error loss function (SELF).

Bayesian Shrinkage Estimator under LINEX Loss Function
In this section, we calculate the generalized Bayesian shrinkage estimator (GBSE) under the LINEX loss function (LLF); in this case, we find the posterior risk function (PRF) of GBSE. To prove the above approach, first we must prove the generalized Bayesian estimator GBE under LLF; so, the LINEX loss function is defined as where, in general, Δ � θ/θ, but in our theory, Δ � b s2 /b. e posterior risk function of the parameter b under LLF can be found as Since b ∼ gamma(n, t)and then 1/b ∼ inversegamma (n, t), so the pdf of 1/b is (14) as

We calculate the quantities E[exp ab s2 /b] and E[1/b] in equation
en, equation (14) becomes where t > ab s2 . Taking the derivative of the posterior risk function ρ 2 (b, b s2 ) w.r.t b s2 , we have Equation (19) is the generalized Bayesian estimator under the LINEX loss function (LLF). Now, to find the generalized Bayesian shrinkage estimator b sh2 under LLF, we must define the posterior risk function ρ 2 (b, b sh2 ) w.r.t b sh2 as e generalized Bayesian shrinkage b sh2 can be defined as ]. We substitute equation (21) in equation (20), and then we get Taking the derivative of ρ 2 (b, b sh2 ) with respect to k, we obtain By minimizing ρ 2 (b, b s2 ), which means taking zρ 2 (b, b s2 )/zk � 0, we have International Journal of Mathematics and Mathematical Sciences Substituting equations (19) and (24) in equation (21), one can obtain where t � n i�1 log(1 + x c i ). Equation (25) is the generalized Bayesian shrinkage estimator (GBSE) under the LINEX loss function (LLF).

Bayesian Shrinkage Estimator under Weighted Balance Loss Function
In this section, we derive the generalized Bayesian shrinkage estimator (GBSE) under the weighted balance loss function (WBLF). To prove the above theory, first we must derive the generalized Bayesian estimator (GBE) under WBLF; so, we consider the posterior loss function with respect to the weighted balance loss function which is defined as where 0 ≤ ω ≤ 1. e posterior risk function with respect to WBLF is We take the derivative of ρ 3 (b, b s3 ) w.r.t. b s3 and assume zρ 3 /zb s3 , thus one can obtain where t � n i�1 log(1 + x c i ). Equation (29) is the generalized Bayesian estimator under the weighted balance loss function. Now, we have two special cases with respect to b s3 : first case: if ω � 0, then equation (29) becomes Equation (30) is the special generalized Bayesian estimator b s31 under WBLF, when ω � 0. Second case: if ω � 1, then equation (29) becomes b s32 � X. (31) Equation (31) is the special generalized Bayesian estimator b s32 under WBLF, when ω � 1.
e posterior risk function of the shrinkage estimator b sh3 is calculated as Taking the derivative of ρ 3 (b, b sh3 ) with respect to k and assuming zρ 3 /zk � 0, one can obtain e shrinkage estimator is defined as Substituting equations (29) and (33) in equation (34), we get Equation (35) is the generalized Bayesian shrinkage estimator b sh3 under the weighted balance loss function.

Special Cases of Bayesian Shrinkage Estimator under Weighted Balance Loss Function
In this section, we derive special generalized Bayesian and shrinkage estimators under the weighted balance loss function, by depending on the value of ω. e first case: if ω � 0, we have the generalized Bayesian estimator b s31 as in equation (30); to calculate the special generalized Bayesian shrinkage estimator b sh31 , we substitute equation (30) in equation (35), then we get Equation (36)  (37) Equation (37) is the second special generalized Bayesian shrinkage estimator b sh32 under WBLF when ω � 0.

Conclusion
In this paper, we discussed the generalized Bayesian and Bayesian shrinkage estimators of parametric Burr XII distribution under three special loss functions: squared error, LINEX, and weighted balance loss functions. Because we have that Jeffrey's prior is improper, we obtained the generalized Bayesian estimator and the generalized Bayesian shrinkage estimator. In this approach, we derived PRF with respect to SELF, LLF, and WBLF. e main results are derived from the generalized Bayesian estimators b s1 , b s2 , and b s3 with respect to SELF, LLF, and WBLF, respectively. e generalized Bayesian shrinkage estimators b sh1 , b sh2 , and b sh3 are calculated by using the above generalized Bayesian estimators, and the constant k in the formula expression of shrinkage estimators was found by minimizing theposterior risk functions of the GBSEs with respect to k. e PRFs are considered with respect to the squared error loss function L 1 , LINEX loss function L 2 , and weighted balance loss function L 3 , respectively. e generalized Bayesian shrinkage estimator b sh3 under WBLF has two special cases: first case: we derived a new GBE b s31 when ω � 0 and another new GBE b s32 when ω � 1. By substituting b s31 and b s32 in the shrinkage estimator formula and computing the constant k in this formula, we obtain two new special generalized Bayesian shrinkage estimators b sh31 and b sh32 .

Data Availability
e data used to support the findings of this study are included within the article.

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