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

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.


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 In the following, we shall denote it by Burr(θ, α), where θ is 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 X 1 , X 2 , ⋯, X n of size n is chosen from the distribution using CDF FðxÞ and PDF f ðxÞ for the test or experiment, but instead of continuing until all n samples have failed, the test is terminated at the time of the r th (1 < r < n) failure. The order statistics of the data is X ð1Þ ≤ X ð2Þ ≤⋯≤X ðrÞ : 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.
In this article, an effort has been made to find Bayesian prediction bounds for future order statistics from the twoparameter 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 Section 2. 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.

Bayesian Two-Sample Predictions
Suppose that X 1 , X 2 , ⋯, X n form a random sample from the distribution with CDF FðxÞ and PDF f ðxÞ. The order statistics is X ð1Þ ≤ X ð2Þ ≤ ⋯ ≤ X ðnÞ : Let X ð1Þ , X ð2Þ , ⋯, X ðnÞ have a joint CDFFðx ð1Þ , x ð2Þ , ⋯, x ðnÞ Þ and PDFf ðx ð1Þ , x ð2Þ , ⋯, x ðnÞ Þ. Then to the k observations we can get the joint PDF: In particular, Suppose that X ð1Þ ≤ X ð2Þ ≤ ⋯ ≤ X ðrÞ 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

Advances in Mathematical Physics
To the Burr XII distribution with the PDF (1) and CDF (2), we can get the likelihood function (LF): When α is known, the above functions with parameters θ 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 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: where η = ððk + TÞ c+r Þ/ðΓðr + cÞÞ: 2.1. Bayesian Prediction Bounds. Assume that X ðrÞ ðr = 1, ⋯, nÞ is the r th ordered observation in the same sample of size n independent of the informative sample of X s ′, and they have the same distribution. Denote Y s ð s = 1, ⋯, m Þ as a future independent type-II censored sample from the same population with censoring scheme s, and suppose that Y 1 ≤ Y 2 ≤ ⋯ ≤ Y s 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 s th (1 ≤ s ≤ m) ordered lifetime Y s in a future sample of size m. The PDF of Y s is given as ð9Þ Substitution of f ðy s Þ and Fðy s Þ, given by (1) and (2), respectively, yields where The Bayes predictive PDF of Y s is defined as Thus, combined with hðy s jθÞ (10) and the posterior PDF qðθ | xÞ (8), one has To obtain the prediction bounds of y s , we first need to find the predictive survival function P½Y s > v | x. It follows from (13) that A two-sided 100δ% predictive interval for Y s , 1 ≤ s ≤ m is given by P½L < Y s < U = δ and denote LðxÞ and UðxÞ are the confidence lower and upper limits which satisfy

Advances in Mathematical Physics
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], one can get the simulation confidence limits from (14).

Special Cases
Case 1. To predict the first failure time Y 1 in the sample of size m, we set s = 1 in (14), so that The case s = 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 Y 1 are given, respectively,  (14), yielding It resulted from (15) and (19) 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 Y s . Under the symmetric (Squared error (SE)) loss function, according to the definition of median, we can see the medianŷ BS s is the solution of the equation: because And from now on, the Bayesian estimationŷ BS s of y s is the median estimator; for convenience, we use the same tokenŷ BS s . And the Bayes point predictor of Y s under asymmetric (LINEX (BL)) loss function is given bŷ 3. E-Bayesian Estimation of θ According to Han [33], the prior parameters c and k should be selected to guarantee that the prior gðθ | c, kÞ in (7) is a decreasing function of θ. The derivative of gðθ | c, kÞ with respect to θ is Thus, for 0 < c < 1, k > 0, the prior gðθ | c, kÞ is a decreasing function of θ.
Assuming that the hyperparameters c and k in (7) are independent and πðc, kÞ = π 1 ðcÞπ 2 ðkÞ, the E-Bayesian estimate of parameter θ (expectation of the Bayesian estimate where D is the domain of c and k for which the prior density is decreasing in θ, and b θ B is the Bayes estimate of θ . For more details, see Han [34] and Jaheen and Okasha [18]. The following distributions of c and k may be used where Bðu, vÞ is the beta function. For π i ðc, kÞ, i = 1, 2, 3, the E-Bayesian estimate of Y s with squared error loss function is obtained from (21) and (26) aŝ

E-Bayesian Point Predictor of Y s with LINEX Loss
Function. Based on the LINEX loss function, the E-Bayesian estimation of θ can be computed for the three different distributions of the hyperparameters c and k given by (26). For π i ðc, kÞ, i = 1, 2, 3, the E-Bayesian estimator of Y s with LINEX loss function is obtained from (23) and (26) as below: Analytical and numerical computations for the integrals in (27) and (28)  In order to verify our parameter estimation, sample prediction, and the above relationship, the following examples are given to illustrate them.

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 b α = 5:115 and b θ = 7:0651.

80.
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  Table 1. The procedure for estimating them is as follows: (i) For given values of the prior parameters u = v = 3, b = 1, a = 1, δ = 0:9, we generate samples from the beta distribution c~Bðu, vÞ and uniform priors kŨ ð0, bÞ, respectively (ii) Repeat the above step (i) 10,000 times. Using the real data above, we obtain the E-Bayes estimates of Y s and its bounds based on the BS function and BL function by simulation (c) Figure 1: The prediction curves.  Figure 1 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)

81.
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 Tables 1 and 2 is computed and put together as  Tables 3 and 4.

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 with α = 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:

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 size n 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.