# Bayesian Shrinkage Estimator of Burr XII Distribution

Authors: N. J. Hassan; J. Mahdi Hadad; A. Hawad Nasar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

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. Therefore, 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. The 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.


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## Body

## 1. Introduction

The 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. The 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]. The 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.The statistical properties and prediction analysis of some estimators are studied in [4–6], and they suggested that the use of symmetric loss function may not be appropriate in some estimation and prediction problems. The 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. The 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. The performance properties of some conventional estimators of error variance are studied in [9], under asymmetric loss function. The 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. The optimal shrinkage estimations in [14] are derived for the parameters of exponential distribution based on recorded values. The 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 small-sigma 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. The 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 Section2, 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.


## 2. 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). The first step to find GBSE is to compute the generalized Bayesian estimator for the parameterb under the assumption that the parameter c is known. The Burr XII distribution, Burrb,c, has the following pdf and cdf [24], respectively:(1)fx;b;c=bcxc−11+xcb+1,Fx;b;c=1−1+xc−b.The MLE ofBurrb,c is defined as(2)b^=n∑i=0nln1+xic.The posterior distribution of the parameterb is the gamma distribution Gn,t which has pdf:(3)fbx=tnbn−1exp−tbΓn,where t=∑i=0nln1+xic and the improper prior distribution 1/b. The squared error loss function (SELF) is defined as(4)L1b,b^s=b−b^s2.The posterior risk function (PRF) ofb can be calculated as(5)ρ1b,b^s1=Ebb^s1−b2=b^s12−2b^s1nt+nn+1t2,where b∼gamman,t. Taking the derivative of the posterior risk function (PRF) with respect to b^s1, we obtain(6)∂ρ1b,b^s1∂b^s1=2b^s12−2nt.By minimizing the posterior risk function (PRF) with respect tob^s1, we obtain(7)b^s1=n∑i=1nlog1+xic.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(8)b^sh1=kb^s1−b0+b0,where b^s1 is the generalized Bayesian estimator. The risk function of b^sh1 is defined as(9)ρ1b,b^sh1=Eb^sh1−b2=Ebkb^s1−b0+b0−b2=k2b^s12+2k+2k2b0b^s1+1−k2b02−2kb^s1nt+2k−2ntb0+nn+1t2.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(10)∂ρ1b,b^sh1∂k=2kb^s12+2−4kb0b^s1−21−kb02−2nb^s1+b0t.We minimizeρ1b,b^sh1 by assuming ∂ρ1b,b^sh1/∂k=0; thus, one can obtain(11)k=tb02−b0n/∑i=1nlog1+xic+nn/∑i=1nlog1+xic+b0tn/∑i=1nlog1+xic−b02.Substituting equations (7) and (11) in equation (8), we get(12)b^sh1=tb02−b0n/∑i=1nlog1+xic+nn/∑i=1nlog1+xic+b0tn/∑i=1nlog1+xic−b02n∑i=1nlog1+xic−b0+b0.Equation (12) is the generalized Bayesian shrinkage estimator (GBSE) under the squared error loss function (SELF).


## 3. 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(13)L2Δ=expaΔ−aΔ−1,where, in general, Δ=θ^/θ, but in our theory, Δ=b^s2/b. The posterior risk function of the parameter b under LLF can be found as(14)ρ2b,b^s2=Ebexpab^s2b−1−ab^s2b−1−1=expaEbexpab^s2b−ab^s2Eb1b+a−1.Sinceb∼gamman,tand then 1/b∼inversegamman,t, so the pdf of 1/b is(15)f1b=tnΓnb−n−1exp−tb.We calculate the quantitiesEexpab^s2/b and E1/b in equation (14) as(16)Eexpab^s2b=tnΓn∫0∞exp−t−ab^s2bb−n−1db=tt−ab^s2n,E1b=tn−1.Then, equation (14) becomes(17)ρ2b,b^s2=expatt−ab^s2n−ab^s2tn−1+a−1,where t>ab^s2. Taking the derivative of the posterior risk function ρ2b,b^s2 w.r.t b^s2, we have(18)∂ρ2b,b^s2∂b^s2=antnexp−at−ab^s2−n−1−atn−1.By minimizingρ2b,b^s2 with respect to b^s2, which means ∂ρ2b,b^s2/∂b^s2=0, one can obtain(19)b^s2=1at−ntn−1exp−a1/n+1.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 ρ2b,b^sh2 w.r.t b^sh2 as(20)ρ2b,b^sh2=Ebexpab^sh2b−1−ab^sh2Eb1b+a−1=exp−att−ab^sh2n−ab^sh2tn−1+a−1.The generalized Bayesian shrinkageb^sh2 can be defined as(21)b^sh2=kb^s2−b0+b0,where b^s2=1/at−ntn−1exp−a1/n+1. We substitute equation (21) in equation (20), and then we get(22)ρ2b,b^sh2=exp−att−ab^sh2n−ab^sh2tn−1+a−1=exp−atnt−akb^s2−b0−ab0−n−atn−1kb^s2−b0−ab0tn−1.Taking the derivative ofρ2b,b^sh2 with respect to k, we obtain(23)∂ρ2b,b^s2∂k=anb^s2−b0exp−atnt−akb^s2−b0−ab0−n−1−atb^s2−b0n−1.By minimizingρ2b,b^s2, which means taking ∂ρ2b,b^s2/∂k=0, we have(24)k=t−ab0−nn−1tn−1exp−a1/n+1a1/at−ntn−1exp−a1/n+1−b0.Substituting equations (19) and (24) in equation (21), one can obtain(25)b^sh2=t−ab0−nn−1tn−1exp−a1/n+1a1/at−ntn−1exp−a1/n+1−b01at−ntn−1exp−a1/n+1−b0+b0,where t=∑i=1nlog1+xic. Equation (25) is the generalized Bayesian shrinkage estimator (GBSE) under the LINEX loss function (LLF).


## 4. 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(26)L3b,b^s3=ω∑i=1nxi−b^s32b2+1−ωb^s3b−12,where 0≤ω≤1. The posterior risk function with respect to WBLF is(27)ρ3b,b^s3=ωn∑i=1nxi−b^s32Eb1b2+1−ωEbb^s3b−12.We take the derivative ofρ3b,b^s3 w.r.t. b^s3 and assume ∂ρ3/∂b^s3, thus one can obtain(28)b^s3=ωX¯+1−ωEb−1Eb−2,where X¯=1/n∑i=1nxi and b−1∼inversegamman,t, soE1/b2=t2/n−1n−2 and E1/b=t/n−1; then, equation (28) becomes(29)b^s3=ωX¯+1−ωn−2t,where t=∑i=1nlog1+xic. 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(30)b^s31=n−2t.Equation (30) is the special generalized Bayesian estimator b^s31 under WBLF, when ω=0. Second case: if ω=1, then equation (29) becomes(31)b^s32=X¯.Equation (31) is the special generalized Bayesian estimator b^s32 under WBLF, when ω=1. The posterior risk function of the shrinkage estimator b^sh3 is calculated as(32)ρ3b,b^sh3=ωn∑i=1nxi−kb^s3−b0−b02Ebb−2+1−ωEbkb^s3−b0+b0b−12.Taking the derivative ofρ3b,b^sh3 with respect to k and assuming ∂ρ3/∂k=0, one can obtain(33)k=ωX¯−b0ωX¯+1−ωn−2/t−b0+1−ωn−2tωX¯+1−ωn−2/t−b0.The shrinkage estimator is defined as(34)b^sh3=kb^s3−b0+b0.Substituting equations (29) and (33) in equation (34), we get(35)b^sh3=ωX¯−b0ωX¯+1−ωn−2/t−b0+1−ωn−2tωX¯+1−ωn−2/t−b0×ωX¯+1−ωn−2t−b0+b0.Equation (35) is the generalized Bayesian shrinkage estimator b^sh3 under the weighted balance loss function.


## 5. 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 ofω. The 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(36)b^sh31=−b0n−2/t−b0+n−2tn−2/t−b0n−2t−b0+b0.Equation (36) is first special generalized Bayesian shrinkage estimator b^sh31 under WBLF when ω=0. The second case: if ω=1, then we have the generalized Bayesian estimator bs32^ as in equation (31); the special generalized Bayesian shrinkage estimator b^sh32 can be found by substituting equation (31) in equation (35), so we have(37)b^sh32=X¯+b0.Equation (37) is the second special generalized Bayesian shrinkage estimator b^sh32 under WBLF when ω=0.


## 6. 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. The main results are derived from the generalized Bayesian estimatorsb^s1, b^s2, and b^s3 with respect to SELF, LLF, and WBLF, respectively. The 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. The PRFs are considered with respect to the squared error loss function L1, LINEX loss function L2, and weighted balance loss function L3, respectively. The 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.



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