# E-Bayesian Prediction for the Burr XII Model Based on Type-II Censored Data with Two Samples

Authors: Hassan M. Okasha; Chuanmei Wang; Jianhua Wang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Type-II censored data is an important scheme of data in lifetime studies. The purpose of this paper is to obtain E-Bayesian predictive functions which are based on observed order statistics with two samples from two parameter Burr XII model. Predictive functions are developed to derive both point prediction and interval prediction based on type-II censored data, where the median Bayesian estimation is a novel formulation to get Bayesian sample prediction, as the integral for calculating the Bayesian prediction directly does not exist. All kinds of predictions are obtained with symmetric and asymmetric loss functions. Two sample techniques are considered, and gamma conjugate prior density is assumed. Illustrative examples are provided for all the scenarios considered in this article. Both illustrative examples with real data and the Monte Carlo simulation are carried out to show the new method is acceptable. The results show that Bayesian and E-Bayesian predictions with the two kinds of loss functions have little difference for the point prediction, and E-Bayesian confidence interval (CI) with the two kinds of loss functions are almost similar and they are more accurate for the interval prediction.


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## Body

## 1. Introduction

The Burr type XII distribution with two parameters was first introduced by Burr [1]. The probability density function (PDF) and cumulative distribution function (CDF) of this distribution can be, respectively, written as
(1)fx;θ,α=θαxα−11+xα−θ+1,x>0,α>0,θ>0,(2)Fx;θ,α=1−1+xα−θ,x>0.In the following, we shall denote it byBurr(θ,α), where θis the shape parameter and α is the scale parameter. In fact, it is basically a Pareto (type IV) model with the scale parameter α set to 1 in Equation (1). The inference problems with Burr distribution have been extensively investigated in the literature, and it is extremely important in the study of biological, industrial, reliability, and life testing and quality control. In the life or quality tests or experiments, a random sample X1,X2,⋯,Xn of size n is chosen from the distribution using CDF Fx and PDF fx for the test or experiment, but instead of continuing until all n samples have failed, the test is terminated at the time of the rth (1<r<n) failure. The order statistics of the data is X1≤X2≤⋯≤Xr. This kind of data is called type-II censored data. Only the smallest observed values are observed, because it takes a long time to observe all the failure of n individuals in some cases, and such a censoring experiment is both time-saving and cost saving. The number of the censored samples is determined before the test or experiment. Tekindal et al. evaluated left-censored data through substitution, parametric, semiparametric, and nonparametric methods [2]. The Bayesian inference is highly recommended by scholars to study censoring data. Feroze and Aslam [3] studied Bayesian analysis of Gumbel type II distribution under censored data. Tabassum et al. [4] discussed Bayesian inference from the mixture of half-normal distributions under censoring. Singh et al. [5] provided “Bayesian Estimation and Prediction for Flexible Weibull Model under Type-II Censoring Scheme.”Lewis [6] proposed the use of the Burr(θ,α) distribution as a model in accelerated life test data representing times of breakdown of an insulating fluid. Inferences and predictions for the Burr(θ,α) distribution and some of its testing measures based on complete and censored samples were discussed by many authors. Evans and Ragab [7] obtained Bayes estimates of θ and the reliability function based on type-II censored samples. AL-Hussaini and Jaheen [8, 9] obtained Bayesian estimation for the two parameters and reliability and failure rate functions of the Burr XII distribution. Ali Mousa [10] obtained empirical Bayes estimation of the parameter θ and the reliability function based on accelerated type-II censored data. Based on complete samples, Moore and Papadopoulos [11] obtained Bayesian estimates of θ and the reliability function when the parameter α is assumed to be known. A1i Mousa and and Jaheen [12] obtained Bayes approximate estimates for the two parameters and reliability function of the Burr(θ,α) distribution based on progressive type-II censored samples. Jaheen [13] used the generalized order statistics to obtain Bayesian inference for the Burr XII model. Based on progressive samples from the Burr(θ,α) distribution, Soliman [14] obtained the Bayesian estimates using both the symmetric (squared error) loss function and asymmetric (LINEX, general entropy) loss functions.The E-Bayesian method is a special Bayesian method which was developed by Han [15], and it is more and more popular now. The E-Bayesian method can be used to estimate statistical distribution parameters. Gonzalez-Lopez et al. used E-Bayesian to gain flexibility in the reliability-availability system estimation based on exponential distribution under the squared error loss function [16]. Han estimated the system failure probability with the E-Bayesian method, and the relationship of E-Bayesian estimators with three different prior distributions of hyperparameters was revealed [17]. Jaheen and Okasha [18] provided the E-Bayesian parameter and reliability estimation for the Burr type XII model based on type-II censoring. However, those literatures only discussed the E-Bayesian parameter estimation or reliability for some models and lack of prediction research for Burr type XII model with type-II censoring data.Prediction of future events on the basis of the past and present information is a fundamental problem of statistics, arising in many contexts and producing varied solutions. As in estimation, a predictor can be either a point or an interval predictor. Parametric and nonparametric predictions have been considered in the literatures. In many practical data-analytic situations, we are interested in getting the prediction interval of the statistical distribution parameters.Prediction has been applied in medicine, engineering, business, and other areas as well. Many authors discussed prediction problems for many distributions, references to research done and review papers on prediction, in nonparametric and parametric settings, such as Al-Hussaini and Ahmad [19, 20], Al-Hussaini and Jaheen [8, 9], Ashour and El-Wakeel [21], Dunsmore [22], Guilbaud [23], Johnson et al. [24], Nigm et al. [25, 26], Patel [27], Sindhu et al. [28], and Singh et al. [5]. For more details, one can refer to Aitchison and Dunsmore [29] and Geisser [30].In this article, an effort has been made to find Bayesian prediction bounds for future order statistics from the two-parameter Burr XII model based on type-II censored data using the two-sample prediction technique. E-Bayesian and Bayesian predictive function approaches have been used for obtaining the estimates of the unknown parameter, and some other lifetime characteristics such as the reliability and hazard functions. Bayesian estimation has been developed under symmetric and asymmetric loss functions in Section2. E-Bayesian predictive functions are derived based on a conjugate prior for the parameter of interest and symmetric and asymmetric loss functions in Section 3. Properties of E-Bayesian predictive functions are carried out in Section 4. Finally, comparison between the new method and the corresponding Bayes techniques is made using the Monte Carlo simulation in Section 5.


## 2. Bayesian Two-Sample Predictions

Suppose thatX1,X2,⋯,Xn form a random sample from the distribution with CDF Fx and PDF fx. The order statistics is X1≤X2≤⋯≤Xn. Let X1,X2,⋯,Xnhave a joint CDF F~x1,x2,⋯,xn and PDF f~x1,x2,⋯,xn. Then to the k observations xi1,xi2,⋯,xik of Xi1≤Xi2≤⋯≤Xik, we can get the joint PDF:
(3)f~xi1,x2,⋯,xn;θ,α=n!i1−1!i2−i1−1!⋯ik−ik−1−1!n−ik!Fxi1i1−1·Fxi2−Fxi1i2−i1−1⋯Fxik−Fxik−1ik−ik−1−11−Fxikn−ikfxi1fxi2⋯fxik.In particular,(4)f~x1,x2,⋯,xn;θ,α=n!fx1fx2⋯fxn.Suppose thatX1≤X2≤⋯≤Xr is a type-II censored sample of size r obtained from a life test on n items, then the joint PDF is also the likelihood function (LF), which can be written as
(5)f~x1,x2,⋯,xr;θ,α=n!n−r!1−Fxrn−rfx1fx2⋯fxr.To the Burr XII distribution with the PDF (1) and CDF (2), we can get the likelihood function (LF):
(6)Lθ,α∣x¯=f~x1,x2,⋯,xr;θ,α=n!θαrn−r!1+xrα−n−rθ∏i=1rxiα−11+xiα−θ+1=n!n−r!θrxiα−1αr∏i=1r1+xiα1+xrα−n−rθ∏i=1r1+xiα−θ=n!vn−r!θre−Tθ,wherev=∏i=1rxiα−1αr/∏i=1r1+xiα,x¯=x1,x2,⋯,xr,T=∑i=1nln1+xiα+n−rln1+xrα.Whenα is known, the above functions with parametersθ and α can be rephrased only with parameter θ, and we suppose θ is a random variable. According to the Bayesian theory, we use the gamma conjugate prior density for parameter θ, which can be written as
(7)gθ∣c,k=kcΓcθc−1e−kθ,θ>0,where c>0 and k>0. This prior was first used by Papadopoulos [31]. The posterior density of θ given x¯ can be obtained from (6) and (7) as follows:
(8)qθ∣x¯=ηθr+c−1e−k+Tθ,θ>0,where η=k+Tc+r/Γr+c.


### 2.1. Bayesian Prediction Bounds

Assume thatXrr=1,⋯,n is the rth ordered observation in the same sample of size n independent of the informative sample of Xs′, and they have the same distribution. Denote Yss=1,⋯,m as a future independent type-II censored sample from the same population with censoring scheme s, and suppose that Y1≤Y2≤⋯≤Ys is a type-II censored sample of size s obtained from a life test on m items. Our aim is to develop a method to construct a Bayesian prediction about the sth (1≤s≤m) ordered lifetime Ys in a future sample of size m. The PDF of Ysis given as
(9)hysθ=m!s−1!m−s!Fyss−11−Fysm−sfys=m!s−1!m−s!1−1−Fyss−11−Fysm−sfys=smsfys∑j=0s−1s−1j−1j1−Fysm−s+j.Substitution offys andFys, given by (1) and (2), respectively, yields
(10)hysθ=smsθαysα−11+ysα−θ+1∑j=0s−1s−1j−1j1+ysα−m−s+jθ=smsθαysα−11+ysα∑j=0s−1s−1j−1j1+ysα−m−s+j+1θ=θτsαysα−11+ysα∑j=0s−1ρj1+ysα−σjθ,where
(11)τs=sms,ρj=s−1j−1j,σj=m−s+j+1.The Bayes predictive PDF ofYs is defined as
(12)h∗ys∣x¯=∫0∞hysθqθ∣x¯dθ.Thus, combined withhysθ (10) and the posterior PDF qθ∣x¯ (8), one has
(13)h∗ys∣x¯=∫0∞hysθqθ∣x¯dθ=r+cτsk+Tαysα−11+ysα∑j=0s−1ρj1+σjk+Tln1+ysα−r+c+1,ys>0.To obtain the prediction bounds ofys, we first need to find the predictive survival function PYs>v∣x¯. It follows from (13) that
(14)PYs>vx¯=∫v∞h∗ysx¯dys=τs∑j=0s−1ρjσj1+σjk+Tln1+vα−r+c,v>0.A two-sided 100δ% predictive interval for Ys, 1≤s≤m is given by PL<Ys<U=δ and denote Lx¯ and Ux¯ are the confidence lower and upper limits which satisfy
(15)PYs>Lx¯=1+δ2PYs>Ux¯=1−δ2.In this case, it is also not possible to obtain the solutions analytically, and one needs a suitable simulation technique for solving these nonlinear equations. And sample fractiles are used to replace the population fractiles during the simulation process.By applying the following formula due to Lingappaiah [32],
(16)∑j=0n−1jnjj+m−1=n!∏j=0nj+m,one can get the simulation confidence limits from (14).


### 2.2. Special Cases

Case 1.
To predict the first failure timeY1 in the sample of size m, we set s=1 in (14), so that
(17)PY1>v∣x¯=1+mk+Tln1+vα−r+c.The cases=1 is of particular interest; for instance, a lower limit for the first failure in a fleet of m items is called a safe warranty life or an assurance limit for the fleet.Hence, the lower and upper 100δ% Bayesian prediction bounds for Y1are given, respectively,
(18)L1x¯=exp−k+Tm1−1+δ2−1/r+c−11/α,U1x¯=exp−k+Tm1−1−δ2−1/r+c−11/α.Case 2.
The predictive survival function ofYm (the last lifetime in a future sample of size m) can be obtained by setting s=m in (14), yielding
(19)PYm>vx¯=∑j=0m−1−1jmj+11+j+1k+Tln1+vα−r+c.It resulted from (15) and (19) by replacing v by the lower bound Lx¯ and the upper bound Ux¯. With a suitable numerical technique for solving these nonlinear equations, we can get the lower and upper bounds of Ym.


### 2.3. The Bayesian Predictor ofYs

Withh∗ys∣x¯ given by (13), the two sample Bayesian predictive PDF of Ys under squared error loss function can be obtained as
(20)y^sBS=∫0∞ysh∗ysx¯dys.However, this integral of (20) tends to infinity, so the integral does not exist. To solve this problem, we apply the median Bayesian estimation for the two-sample Bayes prediction of Ys. Under the symmetric (Squared error (SE)) loss function, according to the definition of median, we can see the median y^sBS is the solution of the equation:
(21)τs∑j=0s−1ρjσj1+σjk+Tln1+vα1+yrα−r+c=12,because
(22)PYs>vx¯=∫v∞h∗ysx¯dys=τs∑j=0s−1ρjσj1+σjk+Tln1+vα1+yrα−r+c.And from now on, the Bayesian estimationy^sBS of ys is the median estimator; for convenience, we use the same token y^sBS.And the Bayes point predictor ofYs under asymmetric (LINEX (BL)) loss function is given by
(23)y^sBL=−1aln∫0∞e−aysh∗ysx¯dys=−1alnτs∑j=0s−1ρjσj1−a∫0∞e−ays1+σjk+Tln1+ysα−r+cdys.


## 2.1. Bayesian Prediction Bounds

Assume thatXrr=1,⋯,n is the rth ordered observation in the same sample of size n independent of the informative sample of Xs′, and they have the same distribution. Denote Yss=1,⋯,m as a future independent type-II censored sample from the same population with censoring scheme s, and suppose that Y1≤Y2≤⋯≤Ys is a type-II censored sample of size s obtained from a life test on m items. Our aim is to develop a method to construct a Bayesian prediction about the sth (1≤s≤m) ordered lifetime Ys in a future sample of size m. The PDF of Ysis given as
(9)hysθ=m!s−1!m−s!Fyss−11−Fysm−sfys=m!s−1!m−s!1−1−Fyss−11−Fysm−sfys=smsfys∑j=0s−1s−1j−1j1−Fysm−s+j.Substitution offys andFys, given by (1) and (2), respectively, yields
(10)hysθ=smsθαysα−11+ysα−θ+1∑j=0s−1s−1j−1j1+ysα−m−s+jθ=smsθαysα−11+ysα∑j=0s−1s−1j−1j1+ysα−m−s+j+1θ=θτsαysα−11+ysα∑j=0s−1ρj1+ysα−σjθ,where
(11)τs=sms,ρj=s−1j−1j,σj=m−s+j+1.The Bayes predictive PDF ofYs is defined as
(12)h∗ys∣x¯=∫0∞hysθqθ∣x¯dθ.Thus, combined withhysθ (10) and the posterior PDF qθ∣x¯ (8), one has
(13)h∗ys∣x¯=∫0∞hysθqθ∣x¯dθ=r+cτsk+Tαysα−11+ysα∑j=0s−1ρj1+σjk+Tln1+ysα−r+c+1,ys>0.To obtain the prediction bounds ofys, we first need to find the predictive survival function PYs>v∣x¯. It follows from (13) that
(14)PYs>vx¯=∫v∞h∗ysx¯dys=τs∑j=0s−1ρjσj1+σjk+Tln1+vα−r+c,v>0.A two-sided 100δ% predictive interval for Ys, 1≤s≤m is given by PL<Ys<U=δ and denote Lx¯ and Ux¯ are the confidence lower and upper limits which satisfy
(15)PYs>Lx¯=1+δ2PYs>Ux¯=1−δ2.In this case, it is also not possible to obtain the solutions analytically, and one needs a suitable simulation technique for solving these nonlinear equations. And sample fractiles are used to replace the population fractiles during the simulation process.By applying the following formula due to Lingappaiah [32],
(16)∑j=0n−1jnjj+m−1=n!∏j=0nj+m,one can get the simulation confidence limits from (14).


## 2.2. Special Cases

Case 1.
To predict the first failure timeY1 in the sample of size m, we set s=1 in (14), so that
(17)PY1>v∣x¯=1+mk+Tln1+vα−r+c.The cases=1 is of particular interest; for instance, a lower limit for the first failure in a fleet of m items is called a safe warranty life or an assurance limit for the fleet.Hence, the lower and upper 100δ% Bayesian prediction bounds for Y1are given, respectively,
(18)L1x¯=exp−k+Tm1−1+δ2−1/r+c−11/α,U1x¯=exp−k+Tm1−1−δ2−1/r+c−11/α.Case 2.
The predictive survival function ofYm (the last lifetime in a future sample of size m) can be obtained by setting s=m in (14), yielding
(19)PYm>vx¯=∑j=0m−1−1jmj+11+j+1k+Tln1+vα−r+c.It resulted from (15) and (19) by replacing v by the lower bound Lx¯ and the upper bound Ux¯. With a suitable numerical technique for solving these nonlinear equations, we can get the lower and upper bounds of Ym.


## 2.3. The Bayesian Predictor ofYs

Withh∗ys∣x¯ given by (13), the two sample Bayesian predictive PDF of Ys under squared error loss function can be obtained as
(20)y^sBS=∫0∞ysh∗ysx¯dys.However, this integral of (20) tends to infinity, so the integral does not exist. To solve this problem, we apply the median Bayesian estimation for the two-sample Bayes prediction of Ys. Under the symmetric (Squared error (SE)) loss function, according to the definition of median, we can see the median y^sBS is the solution of the equation:
(21)τs∑j=0s−1ρjσj1+σjk+Tln1+vα1+yrα−r+c=12,because
(22)PYs>vx¯=∫v∞h∗ysx¯dys=τs∑j=0s−1ρjσj1+σjk+Tln1+vα1+yrα−r+c.And from now on, the Bayesian estimationy^sBS of ys is the median estimator; for convenience, we use the same token y^sBS.And the Bayes point predictor ofYs under asymmetric (LINEX (BL)) loss function is given by
(23)y^sBL=−1aln∫0∞e−aysh∗ysx¯dys=−1alnτs∑j=0s−1ρjσj1−a∫0∞e−ays1+σjk+Tln1+ysα−r+cdys.


## 3. E-Bayesian Estimation ofθ

According to Han [33], the prior parameters c and k should be selected to guarantee that the prior gθ∣c,k in (7) is a decreasing function of θ. The derivative of gθ∣c,k with respect to θ is
(24)dgθ∣c,kdθ=kcΓcθc−2e−kθc−1−kθ.Thus, for0<c<1, k>0, the prior gθ∣c,k is a decreasing function of θ.Assuming that the hyperparametersc and k in (7) are independent and πc,k=π1cπ2k, the E-Bayesian estimate of parameter θ (expectation of the Bayesian estimate of θ) is
(25)θ^EB=Eθx¯=∬Dθ^Bc,kπc,kdcdk,where D is the domain of c and k for which the prior density is decreasing in θ, and θ^B is the Bayes estimate ofθ . For more details, see Han [34] and Jaheen and Okasha [18].


### 3.1. The E-Bayesian Predictor ofYs with Squared Error Loss Function

E-Bayesian estimate ofθ is obtained based on three different distributions of the hyperparameters c and k. These distributions are used to investigate the influence of the different prior distributions on the E-Bayesian estimation of θ.The following distributions ofc and k may be used
(26)π1c,k=1bBu,vcu−11−cv−1,0<c<1,0<k<bπ2c,k=2b−kb2Bu,vcu−11−cv−1,0<c<1,0<k<bπ3c,k=2kb2Bu,vcu−11−cv−1,0<c<1,0<k<b,where Bu,v is the beta function. Forπic,k,i=1,2,3, the E-Bayesian estimate of Ys with squared error loss function is obtained from (21) and (26) as
(27)y^sEBSi=∬Dy^sBSπic,kdkdc.


### 3.2. E-Bayesian Point Predictor ofYs with LINEX Loss Function

Based on the LINEX loss function, the E-Bayesian estimation ofθ can be computed for the three different distributions of the hyperparameters c and k given by (26). For πic,k, i = 1, 2, 3, the E-Bayesian estimator of Ys with LINEX loss function is obtained from (23) and (26) as below:
(28)y^sEBLi=∬Dy^sBLπic,kdkdc.Analytical and numerical computations for the integrals in (27) and (28) are very complicated. With Monte Carlo simulation, we can get all of the estimators, and the samples are shown in part 4.


### 3.3. Properties of E-Bayesian Point Predictor ofYs

As the integral ofy^sBS (21) does not exist, we cannot get the expression of real y^sBS, and we can only get the median estimation of y^sBS with (22). So, we cannot prove the properties of E-Bayesian point predictor of Ys, but with our experience of E-Bayesian estimation [18], we can only guess the relations among y^sEBSiandy^sEBLi,i=1,2,3as follows:
(i)
y^sEBS2<y^sEBS1<y^sEBS3
(ii)
limT→∞y^sEBS1=limT→∞y^sEBS2=limT→∞y^sEBS3
(iii)
y^sEBL2<y^sEBL1<y^sEBL3
(iv)
limT→∞y^sEBL1=limT→∞y^sEBL2=limT→∞y^sEBL3The relationship ofy^sEBSi, y^sEBLii=1,2,3can be observed only, and we cannot give the proof. However, the examples in part 4 can confirm this relationship.In order to verify our parameter estimation, sample prediction, and the above relationship, the following examples are given to illustrate them.


## 3.1. The E-Bayesian Predictor ofYs with Squared Error Loss Function

E-Bayesian estimate ofθ is obtained based on three different distributions of the hyperparameters c and k. These distributions are used to investigate the influence of the different prior distributions on the E-Bayesian estimation of θ.The following distributions ofc and k may be used
(26)π1c,k=1bBu,vcu−11−cv−1,0<c<1,0<k<bπ2c,k=2b−kb2Bu,vcu−11−cv−1,0<c<1,0<k<bπ3c,k=2kb2Bu,vcu−11−cv−1,0<c<1,0<k<b,where Bu,v is the beta function. Forπic,k,i=1,2,3, the E-Bayesian estimate of Ys with squared error loss function is obtained from (21) and (26) as
(27)y^sEBSi=∬Dy^sBSπic,kdkdc.


## 3.2. E-Bayesian Point Predictor ofYs with LINEX Loss Function

Based on the LINEX loss function, the E-Bayesian estimation ofθ can be computed for the three different distributions of the hyperparameters c and k given by (26). For πic,k, i = 1, 2, 3, the E-Bayesian estimator of Ys with LINEX loss function is obtained from (23) and (26) as below:
(28)y^sEBLi=∬Dy^sBLπic,kdkdc.Analytical and numerical computations for the integrals in (27) and (28) are very complicated. With Monte Carlo simulation, we can get all of the estimators, and the samples are shown in part 4.


## 3.3. Properties of E-Bayesian Point Predictor ofYs

As the integral ofy^sBS (21) does not exist, we cannot get the expression of real y^sBS, and we can only get the median estimation of y^sBS with (22). So, we cannot prove the properties of E-Bayesian point predictor of Ys, but with our experience of E-Bayesian estimation [18], we can only guess the relations among y^sEBSiandy^sEBLi,i=1,2,3as follows:
(i)
y^sEBS2<y^sEBS1<y^sEBS3
(ii)
limT→∞y^sEBS1=limT→∞y^sEBS2=limT→∞y^sEBS3
(iii)
y^sEBL2<y^sEBL1<y^sEBL3
(iv)
limT→∞y^sEBL1=limT→∞y^sEBL2=limT→∞y^sEBL3The relationship ofy^sEBSi, y^sEBLii=1,2,3can be observed only, and we cannot give the proof. However, the examples in part 4 can confirm this relationship.In order to verify our parameter estimation, sample prediction, and the above relationship, the following examples are given to illustrate them.


## 4. Monte Carlo Simulation and Comparisons

### 4.1. Illustrative Example with Real Data

To verify the estimation and prediction method of this paper, we give two illustrative examples. A complete sample from a clinical trial describes a relief time (in hours) for 50 arthritic patients given by Wingo [35] and used recently by Ahmed et al. [36, 37] and Wu et al. [38]. Wingo [35], and Ahmed et al. [36, 37] showed that the Burr type XII model was acceptable for these data. Ahmed et al. [37] obtained the estimation of the parameters as α^=5.115 and θ^=7.0651.Example 1.
The following data sample was generated from the two parameter Burr type XII distribution withα=5.115.0, θ=7.0651; sample size n=15; 10 observations r=10; 5 censoring data m=5; to predict the 5 censoring data with s=1,2,3,4,5, we set u=v=3, b=1, a=1, δ=0.9. The data is the sample coming from the real data and is arranged as the order statistics, and the last 5 bold ones are censored which we predict in Table 1: 0.29, 0.35, 0.36, 0.44, 0.46, 0.49, 0.50, 0.52, 0.55, 0.55, 0.57, 0.59, 0.61, 0.70, 0.80.Table 1
Prediction of Example1 for the real data with sample size n=15.
s
BL1
BS1
BS1 CI
EBS1
EBS1 CI
EBL1
EBL1 CI
1
0.4263764
0.4316624
(0.2576238, 0.5902167)
0.4257137
(0.3786459, 0.4628089)
0.4174119
(0.3695664, 0.4584008)
2
0.5237016
0.5259481
(0.3823931, 0.6691212)
0.5186851
(0.4608387, 0.5644443)
0.5142769
(0.4546417, 0.5652961)
3
0.5941900
0.5944176
(0.4607947, 0.7375225)
0.5861901
(0.5200581, 0.6387599)
0.5837638
(0.5153327, 0.6425961)
4
0.6616387
0.6603902
(0.5269148, 0.8136975)
0.6512217
(0.5765007, 0.7110461)
0.6503505
(0.5731552, 0.7170255)
5
0.7496692
0.7448745
(0.5982316, 0.9329483)
0.7344847
(0.6474641, 0.8051043)
0.7360397
(0.6461408, 0.8144799)
s
BL2
BS2
BS2 CI
EBS2
EBS2 CI
EBL2
EBL2 CI
1
0.4065425
0.4110962
(0.2454158, 0.5614648)
0.4103752
(0.3748135, 0.4511887)
0.4025163
(0.3647253, 0.4450713)
2
0.4985693
0.5006385
(0.3642232, 0.6357492)
0.4997995
(0.4561466, 0.5500556)
0.4956201
(0.4489861, 0.5483002)
3
0.5652026
0.5654322
(0.4387903, 0.6996264)
0.5645469
(0.514723, 0.6221456)
0.5622232
(0.5088181, 0.6227278)
4
0.6286810
0.6275569
(0.501555, 0.7699742)
0.6266803
(0.5705216, 0.6919983)
0.6257744
(0.5659482, 0.6945529)
5
0.7098164
0.7064558
(0.5690331, 0.8778819)
0.7057107
(0.6406105, 0.7823044)
0.7069225
(0.6379363, 0.7888539)
s
BL3
BS3
BS3 CI
EBS3
EBS3 CI
EBL3
EBL3 CI
1
0.4392170
0.4465164
(0.2664276, 0.6111139)
0.4409658
(0.401057, 0.4644336)
0.4325223
(0.3911746, 0.4624098)
2
0.5399871
0.5442798
(0.3955065, 0.6935378)
0.5374641
(0.4883087, 0.5664592)
0.5332061
(0.4809603, 0.5704103)
3
0.6130198
0.6154904
(0.4766979, 0.765482)
0.6077103
(0.5513518, 0.6410906)
0.6056173
(0.5458406, 0.6482854)
4
0.6832925
0.6843914
(0.5452837, 0.8463613)
0.6756226
(0.6116752, 0.7137244)
0.6752716
(0.6078378, 0.7231206)
5
0.7764433
0.7732495
(0.6194648, 0.9751638)
0.7630916
(0.6880245, 0.8083315)
0.7655518
(0.6860387, 0.8221346)Using this data, we can get the point prediction and bound prediction of the last 5 censored data according to the method given by this paper. With the results of Equations (21), (23), and (27)–(28), different Bayesian and E-Bayesian estimators for Ys and its bounds are computed and are presented in Table 1. The procedure for estimating them is as follows:
(i)
For given values of the prior parametersu=v=3, b=1, a=1, δ=0.9, we generate samples from the beta distribution c~Bu,v and uniform priorsk~U0,b, respectively
(ii)
Repeat the above step (i) 10,000 times. Using the real data above, we obtain the E-Bayes estimates ofYs and its bounds based on the BS function and BL function by simulation
(iii)
The computational results are summarized in Table1, where BLi is the Bayesian prediction, and EBLi is the E-Bayesian prediction, for Ysunder LINEX error loss function with three different distributions of the hyperparameters (i=1,2,3). BSi is the Bayesian estimators (EBSi is the E-Bayesian estimators) for Ysunder squared error loss function with three different distributions of the hyperparameters (i=1,2,3). BS CI corresponds to the 90% Bayesian confidence interval, and EBSi CI corresponds to the 90% E-Bayesian confidence interval under squared error loss function. EBLi CI corresponds to the 90% E-Bayesian confidence interval under LINEX error loss functionFigure1 shows us the prediction curves. Here, BSiL (i=1,2,3) is the lower bound of the corresponding confidence interval, and BSiU (i=1,2,3) is the upper bound of the corresponding confidence interval. EBSiL (i=1,2,3) and EBSiU (i=1,2,3) have the same meaning. These three graphs have similar results: BSi CI (i=1,2,3) totally covers the real data, but they do not have much of degree of confidence. Especially the lower bound of BSi (i=1,2,3) is far from the real data curve; EBSi CI and EBLi CI (i=1,2,3) almost coincide, and their upper bounds are closest to the real data; BLi, BSi, EBLi, and EBSi (i=1,2,3) almost coincide. So we can see that the point prediction BLi, BSi, EBLi, and EBSi (i=1,2,3) are not very different, the interval prediction EBSi CI and EBLi CI (i=1,2,3) are almost similar, and the E-Bayesian interval predictions are more accurate. The next two examples have the similar results, so we will not show the figures again.Figure 1
The prediction curves.
(a)
(b)
(c)Example 2.
Another example is with the data sample generated from the same distribution as the above Example1, but here, we enlarge the sample size to n=50; the meaning of other indexes is the same as the above example. Here, r=45; m=5; s=1,2,3,4,5; u=v=3; b=1; a=1; δ=0.9. We get the prediction of the last 5 bold ones as Table 2. The data are as follows:
0.29, 0.29, 0.34, 0.34, 0.35, 0.36, 0.36, 0.36, 0.44, 0.44, 0.46, 0.46, 0.49, 0.49, 0.50, 0.50, 0.52, 0.54, 0.55, 0.55, 0.55, 0.56, 0.57, 0.58, 0.59, 0.59, 0.60, 0.60, 0.61, 0.61, 0.62, 0.64, 0.68, 0.70, 0.70, 0.71, 0.71, 0.71, 0.72, 0.73,0.75, 0.75, 0.80, 0.80, 0.81.Table 2
Prediction of Example2 for the real data with sample size n=50.
s
BL1
BS1
BS1 CI
EBS1
EBS1 CI
EBL1
EBL1 CI
1
0.4427966
0.4543341
(0.2722825, 0.6114758)
0.4538649
(0.4466274, 0.460834)
0.4441464
(0.42695, 0.4613846)
2
0.5446005
0.5539914
(0.4060942, 0.690365)
0.5534132
(0.544460, 0.5620414)
0.5477999
(0.527261, 0.5684298)
3
0.6182980
0.6266535
(0.4911443, 0.7592645)
0.6259903
(0.615667, 0.6359494)
0.6224449
(0.5998507, 0.645974)
4
0.6890106
0.6970331
(0.5633366, 0.837345)
0.6962807
(0.6844733, 0.70769)
0.6943775
(0.6694205, 0.719781)
5
0.7831541
0.7878778
(0.641267, 0.9639144)
0.7869955
(0.7729321, 0.800626)
0.7879466
(0.7599305, 0.816337)
s
BL2
BS2
BS2 CI
EBS2
EBS2 CI
EBL2
EBL2 CI
1
0.4393090
0.4508057
(0.270185, 0.6065865)
0.4512354
(0.4461563, 0.458294)
0.4416102
(0.425065, 0.4584106)
2
0.5401777
0.5496258
(0.402952, 0.6846776)
0.5501589
(0.5438776, 0.558895)
0.5446061
(0.5248414, 0.564781)
3
0.6131909
0.6216188
(0.4873143, 0.752772)
0.6222362
(0.6149967, 0.632314)
0.6187336
(0.5970456, 0.641236)
4
0.6831240
0.6912727
(0.558888, 0.8297724)
0.6919834
(0.6837077, 0.703519)
0.6901039
(0.666393, 0.7147314)
5
0.7757520
0.7810124
(0.6360892, 0.954093)
0.7818692
(0.772022, 0.7956296)
0.7827897
(0.756385, 0.8103091)
s
BL3
BS3
BS3 CI
EBS3
EBS3 CI
EBL3
EBL3 CI
1
0.4453643
0.4572579
(0.274020, 0.6155324)
0.4564765
(0.449534, 0.4612051)
0.4467165
(0.429975, 0.4636027)
2
0.5478573
0.5576113
(0.4086974, 0.695090)
0.5566453
(0.548053, 0.5625018)
0.5510368
(0.530939, 0.5708726)
3
0.6220617
0.6308316
(0.494319, 0.7646674)
0.6297189
(0.619806, 0.6364815)
0.6262050
(0.6040166, 0.648505)
4
0.6933625
0.7018194
(0.5670253, 0.843662)
0.7005487
(0.689201, 0.7083007)
0.6987046
(0.674852, 0.7230455)
5
0.7886493
0.7935954
(0.645565, 0.9721487)
0.7920867
(0.7785474, 0.801358)
0.7931679
(0.7661625, 0.819795)To compare the estimators, the mean square error (MSE) is used to measure the estimation accuracy as follows. Here, to compare the bounds of the estimators, the 90% confidence limits of real data is used to calculate the MSE of our estimated bounds. For the convenience of the comparison, the MSE of Tables1 and 2 is computed and put together as Tables 3 and 4.Table 3
MSE of Example1 with the estimators in Table 1.
Estimators
BL1
BL2
BL3
BS1
BS2
BS3
EBS1
EBS2
EBS3
EBL1
MSE
0.005856
0.010061
0.00409
0.005618
0.009844
0.003666
0.006629
0.00999
0.004274
0.007252
Estimators
EBL2
EBL3
LBS1
UBS1
LBS2
UBS2
LBS3
UBS3
LEBS1
UEBS1
MSE
0.010683
0.004789
0.024107
0.002179
0.03017
0.001059
0.020334
0.00521
0.006387
0.008847
Estimators
LEBS2
UEBS2
LEBS3
UEBS3
LEBL1
UEBL1
LEBL2
UEBL2
LEBL3
UEBL3
MSE
0.007143
0.011743
0.0031435
0.0084877
0.0072448
0.0086676
0.0082327
0.0119137
0.0038
0.0078657Table 4
MSE of example2 with the estimators in Table 2.
Estimators
BL1
BL2
BL3
BS1
BS2
BS3
EBS1
EBS2
EBS3
EBL1
MSE
0.003743
0.0040891
0.0035257
0.0030217
0.0032818
0.0028476
0.0030529
0.003247
0.0028904
0.0035903
Estimators
EBL2
EBL3
LBS1
UBS1
LBS2
UBS2
LBS3
UBS3
LEBS1
UEBS1
MSE
0.003803
0.0034056
0.0175073
0.0042639
0.0182517
0.0034877
0.0169131
0.0049894
0.00296
0.0093169
Estimators
LEBS2
UEBS2
LEBS3
UEBS3
LEBL1
UEBL1
LEBL2
UEBL2
LEBL3
UEBL3
MSE
0.0029161
0.0099231
0.003253
0.0092306
0.0026306
0.0082243
0.0025454
0.0088625
0.0028125
0.0078276


### 4.2. Illustrative Example with Simulation

To illustrate the operability of the methods put forward by this paper, we also give an example with simulation. The following data sample was generated from the two parameter Burr type XII distribution withα=1.0, θ=2.0, sample size n=20, 15 observations r=15, 5 censoring data m=5; to predict the 5 censored data with s=1,2,3,4,5, we set u=v=3, b=1, a=1, δ=0.9. The data are listed as follows:0.05280128, 0.05560962, 0.06225709, 0.10117384, 0.10626390, 0.11375182, 0.19062614, 0.24794003, 0.24840441, 0.42198706,0.46646103, 0.48875702, 0.55176242, 0.65954615, 0.66710844.


## 4.1. Illustrative Example with Real Data

To verify the estimation and prediction method of this paper, we give two illustrative examples. A complete sample from a clinical trial describes a relief time (in hours) for 50 arthritic patients given by Wingo [35] and used recently by Ahmed et al. [36, 37] and Wu et al. [38]. Wingo [35], and Ahmed et al. [36, 37] showed that the Burr type XII model was acceptable for these data. Ahmed et al. [37] obtained the estimation of the parameters as α^=5.115 and θ^=7.0651.Example 1.
The following data sample was generated from the two parameter Burr type XII distribution withα=5.115.0, θ=7.0651; sample size n=15; 10 observations r=10; 5 censoring data m=5; to predict the 5 censoring data with s=1,2,3,4,5, we set u=v=3, b=1, a=1, δ=0.9. The data is the sample coming from the real data and is arranged as the order statistics, and the last 5 bold ones are censored which we predict in Table 1: 0.29, 0.35, 0.36, 0.44, 0.46, 0.49, 0.50, 0.52, 0.55, 0.55, 0.57, 0.59, 0.61, 0.70, 0.80.Table 1
Prediction of Example1 for the real data with sample size n=15.
s
BL1
BS1
BS1 CI
EBS1
EBS1 CI
EBL1
EBL1 CI
1
0.4263764
0.4316624
(0.2576238, 0.5902167)
0.4257137
(0.3786459, 0.4628089)
0.4174119
(0.3695664, 0.4584008)
2
0.5237016
0.5259481
(0.3823931, 0.6691212)
0.5186851
(0.4608387, 0.5644443)
0.5142769
(0.4546417, 0.5652961)
3
0.5941900
0.5944176
(0.4607947, 0.7375225)
0.5861901
(0.5200581, 0.6387599)
0.5837638
(0.5153327, 0.6425961)
4
0.6616387
0.6603902
(0.5269148, 0.8136975)
0.6512217
(0.5765007, 0.7110461)
0.6503505
(0.5731552, 0.7170255)
5
0.7496692
0.7448745
(0.5982316, 0.9329483)
0.7344847
(0.6474641, 0.8051043)
0.7360397
(0.6461408, 0.8144799)
s
BL2
BS2
BS2 CI
EBS2
EBS2 CI
EBL2
EBL2 CI
1
0.4065425
0.4110962
(0.2454158, 0.5614648)
0.4103752
(0.3748135, 0.4511887)
0.4025163
(0.3647253, 0.4450713)
2
0.4985693
0.5006385
(0.3642232, 0.6357492)
0.4997995
(0.4561466, 0.5500556)
0.4956201
(0.4489861, 0.5483002)
3
0.5652026
0.5654322
(0.4387903, 0.6996264)
0.5645469
(0.514723, 0.6221456)
0.5622232
(0.5088181, 0.6227278)
4
0.6286810
0.6275569
(0.501555, 0.7699742)
0.6266803
(0.5705216, 0.6919983)
0.6257744
(0.5659482, 0.6945529)
5
0.7098164
0.7064558
(0.5690331, 0.8778819)
0.7057107
(0.6406105, 0.7823044)
0.7069225
(0.6379363, 0.7888539)
s
BL3
BS3
BS3 CI
EBS3
EBS3 CI
EBL3
EBL3 CI
1
0.4392170
0.4465164
(0.2664276, 0.6111139)
0.4409658
(0.401057, 0.4644336)
0.4325223
(0.3911746, 0.4624098)
2
0.5399871
0.5442798
(0.3955065, 0.6935378)
0.5374641
(0.4883087, 0.5664592)
0.5332061
(0.4809603, 0.5704103)
3
0.6130198
0.6154904
(0.4766979, 0.765482)
0.6077103
(0.5513518, 0.6410906)
0.6056173
(0.5458406, 0.6482854)
4
0.6832925
0.6843914
(0.5452837, 0.8463613)
0.6756226
(0.6116752, 0.7137244)
0.6752716
(0.6078378, 0.7231206)
5
0.7764433
0.7732495
(0.6194648, 0.9751638)
0.7630916
(0.6880245, 0.8083315)
0.7655518
(0.6860387, 0.8221346)Using this data, we can get the point prediction and bound prediction of the last 5 censored data according to the method given by this paper. With the results of Equations (21), (23), and (27)–(28), different Bayesian and E-Bayesian estimators for Ys and its bounds are computed and are presented in Table 1. The procedure for estimating them is as follows:
(i)
For given values of the prior parametersu=v=3, b=1, a=1, δ=0.9, we generate samples from the beta distribution c~Bu,v and uniform priorsk~U0,b, respectively
(ii)
Repeat the above step (i) 10,000 times. Using the real data above, we obtain the E-Bayes estimates ofYs and its bounds based on the BS function and BL function by simulation
(iii)
The computational results are summarized in Table1, where BLi is the Bayesian prediction, and EBLi is the E-Bayesian prediction, for Ysunder LINEX error loss function with three different distributions of the hyperparameters (i=1,2,3). BSi is the Bayesian estimators (EBSi is the E-Bayesian estimators) for Ysunder squared error loss function with three different distributions of the hyperparameters (i=1,2,3). BS CI corresponds to the 90% Bayesian confidence interval, and EBSi CI corresponds to the 90% E-Bayesian confidence interval under squared error loss function. EBLi CI corresponds to the 90% E-Bayesian confidence interval under LINEX error loss functionFigure1 shows us the prediction curves. Here, BSiL (i=1,2,3) is the lower bound of the corresponding confidence interval, and BSiU (i=1,2,3) is the upper bound of the corresponding confidence interval. EBSiL (i=1,2,3) and EBSiU (i=1,2,3) have the same meaning. These three graphs have similar results: BSi CI (i=1,2,3) totally covers the real data, but they do not have much of degree of confidence. Especially the lower bound of BSi (i=1,2,3) is far from the real data curve; EBSi CI and EBLi CI (i=1,2,3) almost coincide, and their upper bounds are closest to the real data; BLi, BSi, EBLi, and EBSi (i=1,2,3) almost coincide. So we can see that the point prediction BLi, BSi, EBLi, and EBSi (i=1,2,3) are not very different, the interval prediction EBSi CI and EBLi CI (i=1,2,3) are almost similar, and the E-Bayesian interval predictions are more accurate. The next two examples have the similar results, so we will not show the figures again.Figure 1
The prediction curves.
(a)
(b)
(c)Example 2.
Another example is with the data sample generated from the same distribution as the above Example1, but here, we enlarge the sample size to n=50; the meaning of other indexes is the same as the above example. Here, r=45; m=5; s=1,2,3,4,5; u=v=3; b=1; a=1; δ=0.9. We get the prediction of the last 5 bold ones as Table 2. The data are as follows:
0.29, 0.29, 0.34, 0.34, 0.35, 0.36, 0.36, 0.36, 0.44, 0.44, 0.46, 0.46, 0.49, 0.49, 0.50, 0.50, 0.52, 0.54, 0.55, 0.55, 0.55, 0.56, 0.57, 0.58, 0.59, 0.59, 0.60, 0.60, 0.61, 0.61, 0.62, 0.64, 0.68, 0.70, 0.70, 0.71, 0.71, 0.71, 0.72, 0.73,0.75, 0.75, 0.80, 0.80, 0.81.Table 2
Prediction of Example2 for the real data with sample size n=50.
s
BL1
BS1
BS1 CI
EBS1
EBS1 CI
EBL1
EBL1 CI
1
0.4427966
0.4543341
(0.2722825, 0.6114758)
0.4538649
(0.4466274, 0.460834)
0.4441464
(0.42695, 0.4613846)
2
0.5446005
0.5539914
(0.4060942, 0.690365)
0.5534132
(0.544460, 0.5620414)
0.5477999
(0.527261, 0.5684298)
3
0.6182980
0.6266535
(0.4911443, 0.7592645)
0.6259903
(0.615667, 0.6359494)
0.6224449
(0.5998507, 0.645974)
4
0.6890106
0.6970331
(0.5633366, 0.837345)
0.6962807
(0.6844733, 0.70769)
0.6943775
(0.6694205, 0.719781)
5
0.7831541
0.7878778
(0.641267, 0.9639144)
0.7869955
(0.7729321, 0.800626)
0.7879466
(0.7599305, 0.816337)
s
BL2
BS2
BS2 CI
EBS2
EBS2 CI
EBL2
EBL2 CI
1
0.4393090
0.4508057
(0.270185, 0.6065865)
0.4512354
(0.4461563, 0.458294)
0.4416102
(0.425065, 0.4584106)
2
0.5401777
0.5496258
(0.402952, 0.6846776)
0.5501589
(0.5438776, 0.558895)
0.5446061
(0.5248414, 0.564781)
3
0.6131909
0.6216188
(0.4873143, 0.752772)
0.6222362
(0.6149967, 0.632314)
0.6187336
(0.5970456, 0.641236)
4
0.6831240
0.6912727
(0.558888, 0.8297724)
0.6919834
(0.6837077, 0.703519)
0.6901039
(0.666393, 0.7147314)
5
0.7757520
0.7810124
(0.6360892, 0.954093)
0.7818692
(0.772022, 0.7956296)
0.7827897
(0.756385, 0.8103091)
s
BL3
BS3
BS3 CI
EBS3
EBS3 CI
EBL3
EBL3 CI
1
0.4453643
0.4572579
(0.274020, 0.6155324)
0.4564765
(0.449534, 0.4612051)
0.4467165
(0.429975, 0.4636027)
2
0.5478573
0.5576113
(0.4086974, 0.695090)
0.5566453
(0.548053, 0.5625018)
0.5510368
(0.530939, 0.5708726)
3
0.6220617
0.6308316
(0.494319, 0.7646674)
0.6297189
(0.619806, 0.6364815)
0.6262050
(0.6040166, 0.648505)
4
0.6933625
0.7018194
(0.5670253, 0.843662)
0.7005487
(0.689201, 0.7083007)
0.6987046
(0.674852, 0.7230455)
5
0.7886493
0.7935954
(0.645565, 0.9721487)
0.7920867
(0.7785474, 0.801358)
0.7931679
(0.7661625, 0.819795)To compare the estimators, the mean square error (MSE) is used to measure the estimation accuracy as follows. Here, to compare the bounds of the estimators, the 90% confidence limits of real data is used to calculate the MSE of our estimated bounds. For the convenience of the comparison, the MSE of Tables1 and 2 is computed and put together as Tables 3 and 4.Table 3
MSE of Example1 with the estimators in Table 1.
Estimators
BL1
BL2
BL3
BS1
BS2
BS3
EBS1
EBS2
EBS3
EBL1
MSE
0.005856
0.010061
0.00409
0.005618
0.009844
0.003666
0.006629
0.00999
0.004274
0.007252
Estimators
EBL2
EBL3
LBS1
UBS1
LBS2
UBS2
LBS3
UBS3
LEBS1
UEBS1
MSE
0.010683
0.004789
0.024107
0.002179
0.03017
0.001059
0.020334
0.00521
0.006387
0.008847
Estimators
LEBS2
UEBS2
LEBS3
UEBS3
LEBL1
UEBL1
LEBL2
UEBL2
LEBL3
UEBL3
MSE
0.007143
0.011743
0.0031435
0.0084877
0.0072448
0.0086676
0.0082327
0.0119137
0.0038
0.0078657Table 4
MSE of example2 with the estimators in Table 2.
Estimators
BL1
BL2
BL3
BS1
BS2
BS3
EBS1
EBS2
EBS3
EBL1
MSE
0.003743
0.0040891
0.0035257
0.0030217
0.0032818
0.0028476
0.0030529
0.003247
0.0028904
0.0035903
Estimators
EBL2
EBL3
LBS1
UBS1
LBS2
UBS2
LBS3
UBS3
LEBS1
UEBS1
MSE
0.003803
0.0034056
0.0175073
0.0042639
0.0182517
0.0034877
0.0169131
0.0049894
0.00296
0.0093169
Estimators
LEBS2
UEBS2
LEBS3
UEBS3
LEBL1
UEBL1
LEBL2
UEBL2
LEBL3
UEBL3
MSE
0.0029161
0.0099231
0.003253
0.0092306
0.0026306
0.0082243
0.0025454
0.0088625
0.0028125
0.0078276


## 4.2. Illustrative Example with Simulation

To illustrate the operability of the methods put forward by this paper, we also give an example with simulation. The following data sample was generated from the two parameter Burr type XII distribution withα=1.0, θ=2.0, sample size n=20, 15 observations r=15, 5 censoring data m=5; to predict the 5 censored data with s=1,2,3,4,5, we set u=v=3, b=1, a=1, δ=0.9. The data are listed as follows:0.05280128, 0.05560962, 0.06225709, 0.10117384, 0.10626390, 0.11375182, 0.19062614, 0.24794003, 0.24840441, 0.42198706,0.46646103, 0.48875702, 0.55176242, 0.65954615, 0.66710844.


## 5. Conclusion

In this paper, the E-Bayes point prediction and prediction bounds for ordered lifetime in a future sample are discussed under symmetric and asymmetric loss functions. Two examples with real data and different choices of sample sizen were illustrated to examine the performance of the different predictions. Comparing Tables 3 and 4, we can find with the increasing of sample size, the MSE of the Bayes predictions and E-Bayes prediction decreases, and the predictions vary with different loss functions.A simulation study was conducted to show the feasibility of the E-Bayes prediction given in this paper. All the predictions computed with different loss functions, different sample size, and different choice of model parameters of the censored scheme are shown in Tables1, 2, and 5, and the predictions with different loss functions are consistent with the relationship we put forward in 3.3. From the results, we can draw the following conclusion:
(i)
From Tables1, 2, and 5, the estimators of the E-Bayes estimates of y^sEBSiandy^sEBLii=1,2,3 satisfy the relationship y^sEBS2<y^sEBS1<y^sEBS3 and y^sEBL2<y^sEBL1<y^sEBL3, and the E-Bayes estimates of different priors are more close to each other. And they are consistent with the assumption in 3.3
(ii)
If one uses the E-Bayesian approach for prediction, one would expect these estimators to be better (in the sense ofMSE’s) than the Bayesian approach. With our examples, this cannot be seen in the results for different choices of loss functions. However, the results are similar. Because the E-Bayesian method uses more information than the ordinary Bayesian approach, it is much more reliable
(iii)
When the sample sizen increases, the MSE of E-Bayesian estimators decrease. So increasing the sample size can get exacter results
(iv)
The results establish that for optimum decision-making, importance should be given on the choice of loss function and not just the choice of prior distribution onlyTable 5
Prediction example with simulation.
s
BL1
BS1
BS1 CI
EBS1
EBS1 CI
EBL1
EBL1 CI
1
0.0989268
0.0643332
(0.0045293, 0.337259)
0.0634986
(0.058912, 0.0681691)
0.0962945
(0.0876987, 0.1050439)
2
0.2310097
0.1844959
(0.034875, 0.7016814)
0.1819962
(0.168183, 0.1961246)
0.2266603
(0.2078706, 0.2459008)
3
0.4209535
0.3657668
(0.0931238, 1.312666)
0.360525
(0.3313396, 0.390554)
0.4154838
(0.3817439, 0.4504281)
4
0.7277203
0.6848347
(0.1923819, 2.69180)
0.6742092
(0.614363, 0.736322)
0.7197569
(0.6624925, 0.7785412)
5
1.3405130
1.515928
(0.393869, 9.209026)
1.488853
(1.332795, 1.653516)
1.3251390
(1.2276826, 1.4256792)
s
BL2
BS2
BS2 CI
EBS2
EBS2 CI
EBL2
EBL2 CI
1
0.0955290
0.062083
(0.004375, 0.3241314)
0.0618931
(0.058546, 0.0664266)
0.0938743
(0.0866777, 0.1019417)
2
0.2228465
0.1777076
(0.0336715, 0.671249)
0.1771502
(0.167088, 0.1908336)
0.2207915
(0.2057193, 0.2390398)
3
0.4056461
0.3513917
(0.089825, 1.247777)
0.3502552
(0.329043, 0.3792505)
0.4044791
(0.3782247, 0.4372029)
4
0.7009810
0.6552626
(0.1852814, 2.531697)
0.6530585
(0.6097058, 0.712764)
0.7006364
(0.6570353, 0.7567613)
5
1.2939372
1.438354
(0.3782452, 8.434845)
1.433241
(1.320916, 1.590156)
1.2919972
(1.2178218, 1.3893391)
s
BL3
BS3
BS3 CI
EBS3
EBS3 CI
EBL3
EBL3 CI
1
0.1014997
0.0660396
(0.004646, 0.347282)
0.0651254
(0.0606048, 0.068627)
0.0987469
(0.0906665, 0.1063018)
2
0.2371997
0.1896604
(0.035787, 0.725086)
0.1869064
(0.173261, 0.1975176)
0.2326070
(0.2146817, 0.2486330)
3
0.4325663
0.3767499
(0.095628, 1.363028)
0.3709303
(0.3420138, 0.393537)
0.4266335
(0.3940571, 0.4542965)
4
0.7479744
0.7075708
(0.197785, 2.817802)
0.6956375
(0.6360885, 0.742565)
0.7391301
(0.6833638, 0.7852044)
5
1.3756176
1.576287
(0.4058139, 9.83703)
1.545187
(1.388652, 1.670446)
1.3587232
(1.2634259, 1.4365982)Burr distribution has extreme importance in the study of biological, industrial, reliability, and life testing and quality control. With the sample prediction functions, we can understand the trends and control them in a time.Algorithm 1
The Code for the Examples
two_R.txt
rm(list = ls())
#set.seed(99)
n = 200
r = 195
af = 1
th = 2
z1 = runif(n,0,1)
z2 = sort(((1-z1)^(-1/th)-1)^(1/af))
x = z2[1:r]
xaf = log(1 + x^(af))
t = sum(xaf) + (n-r)∗xaf[r]
b = 1
u = 3
v = 3
a = 1
yebs = 0
yebl = 0
p = 0
yebs11 = 0
yebl11 = 0
while (p < 5000) {
k = runif(1,0,b) #EBS1
# k1 = runif(1,0,1) #EBS23
# k = b∗(1-sqrt(1-k1)) #EBS2
# k = b∗sqrt(k1) #EBS3
c = rbeta(1,u,v)
m = n-r
s = 1
ybs11 = 0
ybl11 = 0
q = 5000
y = rexp(q,a)
while (s < m + 1) {
ta = 1/beta(s,m-s + 1)
rosg = ((-1)^c(0:(s-1)))/(beta(c(1:s),s + 1-c(1:s))∗s∗(m-s + c(1:s)))
sg = m-s + c(1:s)
ybs = t(((1 + log(1 + y^af)%o%sg/(k + t))^(-r-c))∗exp(a∗y))∗((ta/a)∗rosg)
ybs1 = apply(ybs,2,sum)
ybs11 = rbind(ybs11,ybs1)
ybl = t((1 + log(1 + y^af)%o%sg/(k + t))^(-r-c))∗(ta∗rosg)
ybl1 = apply(ybl,2,sum)
ybl11 = rbind(ybl11,ybl1)
s = s + 1
}
ybs111 = apply(ybs11,1,sum)/q
ybl111 = apply(ybl11,1,sum)/q
ybl111 = -log(1-ybl111)/a
yebs = yebs+ybs111
yebs11 = cbind(yebs11,ybs111)
yebl = yebl+ybl111
yebl11 = cbind(yebl11,ybl111)
two_R.txt
p = p + 1
}
yebs1 = yebs/p
yebl1 = yebl/p
yebs11 = yebs11[,1:p + 1]
yebl11 = yebl11[,1:p + 1]
deta = 0.90
low = ceiling((1-deta)∗p/2)
up = ceiling((1 + deta)∗p/2)
yebs1low = apply(yebs11,1,sort)[low,]
yebs1up = apply(yebs11,1,sort)[up,]
yebl1low = apply(yebl11,1,sort)[low,]
yebl1up = apply(yebl11,1,sort)[up,]
s = 1
ta = 1/beta(s,m-s + 1)
rosg = ((-1)^c(0:(s-1)))/(beta(c(1:s),s + 1-c(1:s))∗s∗(m-s + c(1:s)))
sg = m-s + c(1:s)
z3 = ybs111 [1]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1 + deta)/2)/f2
}
ybslow1 = z3
z3 = ybs111 [2]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1-deta)/2)/f2
}
ybsup1 = z3
s = 2
ta = 1/beta(s,m-s + 1)
rosg = ((-1)^c(0:(s-1)))/(beta(c(1:s),s + 1-c(1:s))∗s∗(m-s + c(1:s)))
sg = m-s + c(1:s)
z3 = ybs111 [2]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1 + deta)/2)/f2
}
ybslow2 = z3
z3 = ybs111 [3]
two_R.txt
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1-deta)/2)/f2
}
ybsup2 = z3
s = 3
ta = 1/beta(s,m-s + 1)
rosg = ((-1)^c(0:(s-1)))/(beta(c(1:s),s + 1-c(1:s))∗s∗(m-s + c(1:s)))
sg = m-s + c(1:s)
z3 = ybs111 [3]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1 + deta)/2)/f2
}
ybslow3 = z3
z3 = ybs111 [4]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1-deta)/2)/f2
}
ybsup3 = z3
s = 4
ta = 1/beta(s,m-s + 1)
rosg = ((-1)^c(0:(s-1)))/(beta(c(1:s),s + 1-c(1:s))∗s∗(m-s + c(1:s)))
sg = m-s + c(1:s)
z3 = ybs111 [4]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1 + deta)/2)/f2
}
ybslow4 = z3
z3 = ybs111 [5]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1-deta)/2)/f2
}
ybsup4 = z3
s = 5
ta = 1/beta(s,m-s + 1)
rosg = ((-1)^c(0:(s-1)))/(beta(c(1:s),s + 1-c(1:s))∗s∗(m-s + c(1:s)))
sg = m-s + c(1:s)
z3 = ybs111 [3]
two_R.txt
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1 + deta)/2)/f2
}
ybslow5 = z3
z3 = ybs111 [6]
z4 = 1
while ((1-z4/z3)^2 > 10^(-16)){
f1 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c))∗ta∗rosg)
f2 = sum(((1 + log(1 + z3^af)∗sg/(k + t))^(-r-c-1)∗(z3^(af-1)/(1 + z3^af)))∗ta∗rosg∗sg∗(r + c)∗af/(k + t)
)
z4 = z3
z3 = z3 + (f1-(1-deta)/2)/f2
}
ybsup5 = z3
x
z2
ybs111[1:m + 1]
ybslow1
ybsup1
ybslow2
ybsup2
ybslow3
ybsup3
ybslow4
ybsup4
ybslow5
ybsup5
ybl111[1:m + 1]
yebs1[1:m + 1]
yebs1low[1:m + 1]
yebs1up[1:m + 1]
yebl1[1:m + 1]
yebl1low[1:m + 1]
yebl1up[1:m + 1]



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*Source: 3510673-2020-02-01.xml*