E-Bayesian Estimation Based on Burr-X Generalized Type-II Hybrid Censored Data

Abstract

In this article, we are concerned with the E-Bayesian (the expectation of Bayesian estimate) method, the maximum likelihood and the Bayesian estimation methods of the shape parameter, and the reliability function of one-parameter Burr-X distribution. A hybrid generalized Type-II censored sample from one-parameter Burr-X distribution is considered. The Bayesian and E-Bayesian approaches are studied under squared error and LINEX loss functions by using the Markov chain Monte Carlo method. Confidence intervals for maximum likelihood estimates, as well as credible intervals for the E-Bayesian and Bayesian estimates, are constructed. Furthermore, an example of real-life data is presented for the sake of the illustration. Finally, the performance of the E-Bayesian estimation method is studied then compared with the performance of the Bayesian and maximum likelihood methods.

1. Introduction

Hybrid censoring scheme (HCS) is introduced by [1] as a mixture of Type-I and Type-II censoring schemes. In such schemes, experiments are stopped as soon as a pre-specified number r of items from n items fails or when a pre-fixed time T runs out. Thus, there have been two types of censored schemes, namely:

Type-I HCS: in which we terminate the test at $T_{*}=min\{Y_{k:n},T\}$, where $Y_{k:n}$, is the time of k-th item failure, T is the maximum time point for the test and $k\in \{1,2,\cdots ,n\}$ and $T\in (0,\infty )$ are determined in advance, i.e., the experiment is terminated when a number k failures out of n items or a pre-fixed time T is reached. Type-II HCS: in this type, the experiment is terminated at $T^{*}=max\{Y_{k:n},T\}$, i.e., we end the experiment when the latter of the stopping rules is reached. Hence, we guarantee that at least k failures are obtained.

There are advantages and disadvantages in these schemes. HCS Type-I possesses the advantage of having a fixed time. However, there are a small number of failures obtained before the stopping time of the life-testing. These disadvantages may impact the accuracy of estimators. However, a drawback of Type-I HCS is overcome by Type-II HCS, and we guarantee The obtaining of a specified number of failures. It also has a disadvantage in that the experimenter cannot know when the required specified failures are observed. Ref. [2] proposed two generalized HCSs to overcome these drawbacks, described as follows. Generalized Type-I HCS: Let $r,k\in \{1,2,\cdots ,n\}$, $k<r<n$, and time $T\in (0,\infty )$. When k-th failure is obtained before time T, the experiment is terminated at $min\{Y_{r:n},T\}$. If the k-th failure is obtained after time T, the experiment is ended at $Y_{k:n}$.

Generalized Type-II HCS: Fix $r\in \{1,2,\cdots ,n\}$ and $T_1,T_2\in (0,\infty )$ where $T_1<T_2$. The experiment is terminated at $T_1$, if r-th failure occurs before $T_1$. When the r-th failure occurs between $T_1$ and $T_2$, we terminate at $Y_{r:n}$. In the third case, if r-th failure is observed after time $T_2$, the experiment is terminated at $T_2$. This scheme has been studied by many authors, such as [3], who presented details on censoring scheme developments in addition to generalized and unified HCS. Ref. [4] discussed Bayesian analysis and prediction based on generalized Type-II HCS for exponential and Pareto models. Ref. [5] studied maximum likelihood, Bayes and percentile bootstrap methods for unknown parameters, failure rate function, the survival function and the coefficient of variation of the exponential Rayleigh distribution with generalized Type-II HCS.

Here, generalized Type-II HCS is considered. We observe one of these types of the censored data.

Case 1: $\{Y_{1:n}<Y_{2:n}<\cdots <Y_{r:n}<\cdots <Y_{d1:n}<T_1\}$ if $Y_{r:n}<\cdots <Y_{d1:n}<T_1$. In this case, the r-th failure is obtained before $T_1$, so the experiment is stopped at $T_1$ and $d_1$ number of failures is obtained at time $T_1$.

Case 2: $\{Y_{1:n}<Y_{2:n}<\cdots <Y_{d1:n}<\cdots <Y_{r:n}\}$ if $T_1<Y_{r:n}<T_2$. In this case, the r-th failure occurs after $T_1$, so the experiment is terminated at $Y_{r:n}$, and r number of failures is obtained.

Case 3: $\{Y_{1:n}<Y_{2:n}<\cdots <Y_{d2:n}<T_2\}$ if $Y_{r:n}>T_2$. In this case, the r-th failure occurs after $T_2$, so the experiment is ended at $T_2$ and $d_2$ number of failures is obtained at time $T_2$. where $T_1$ and $T_2$ are time points determined by the experimenter according to how the experiment should continue based on the information about the product.

Burr-X as a Lifetime Model

Burr-X model is part of Burr distribution family suggested by [6]. This distribution is important in many fields such as operations research and statistics. It is widely used in health, agriculture, and biology. For more details on the applications of this model, one can refer to [7], as they discussed the cumulative distribution function (CDF) of Burr-X distribution with the cumulative damage process and the shock model. Also, they assumed a mathematical model for the expected lifetime of AIDS patients, then fitted the observed data of infected persons for Burr-X distribution. The probability density function (PDF) of one-parameter Burr-X distribution is given by

$$f(y;\theta )=2\theta y{e}^{-y^2}{(1-e^{-y^2})}^{\theta },y>0,\theta >0,$$

(1)

and the CDF takes the form

$$F(y;\theta )={(1-e^{-y^2})}^{\theta },y>0,\theta >0,$$

(2)

where $\theta$ is the shape parameter. The reliability function $R\left(t\right)$ and the hazard rate function $h\left(t\right)$ for one-parameter Burr-X distribution are, respectively, expressed as follows:

$$R\left(t\right)=1-{(1-e^{-t^2})}^{\theta },t>0,$$

(3)

$$h\left(t\right)=\frac{2\theta t{e}^{-t^2}{(1-e^{-t^2})}^{\theta -1}}{1-{(1-e^{-t^2})}^{\theta }},t>0.$$

(4)

Many publications discuss Burr-X distribution, such as [8], who discussed maximum likelihood estimate (MLE) and Bayesian estimates for parameters of Burr-X model under double Type-II censored sample of dual generalized order statistics. Ref. [9] considered the Bayesian and non-Bayesian estimators of the parameter of the Burr-X model based on grouped data. Ref. [10] studied inferences and predictions for the Burr Type-X model based on records. Ref. [11] considered the statistical analysis of the parameter through different priors and loss functions. Ref. [12] discussed E-Bayesian method and ML and Bayesian methods for the parameter and the reliability function of the Burr-X model under Type-I HCS.

We assume $Y_{1:n},Y_{2:n},\cdots ,Y_{n:n}$ are n lifetimes of failure observations under a generalized Type-II hybrid censored sample following one-parameter Burr-X distribution. Let $D_1$ and $D_2$ denote several items that fail before $T_1$ and $T_2$, respectively; the likelihood function for three cases is given by

$$L\left(\theta \right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\left(2\theta \right)}^{D_1}n!}{(n-D_1)!}{\left\{1-W_{T_1}^{\theta }\right\}}^{n-D_1}\prod _{{}_{i=1}}^{D_1}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle \frac{{\left(2\theta \right)}^{r}n!}{(n-r)!}{\left\{1-W_{y_{r:n}}^{\theta }\right\}}^{n-r}\prod _{{}_{i=1}}^r{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle \frac{{\left(2\theta \right)}^{D_2}n!}{(n-D_2)!}{\left\{1-W_{T_2}^{\theta }\right\}}^{n-D_2}\prod _{{}_{i=1}}^{D_2}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_2=0,1,\cdots ,r-1.\hfill \end{array}\right.$$

(5)

where $W_{T_1}=1-e^{-T_1^2}$, $W_{y_{i:n}}=1-e^{-y_{i:n}^2}$ and $W_{T_2}=1-e^{-T_2^2}$.

By taking the natural logarithm of Equation (5), ignoring the constant term, as it does not depend on $\theta$, we get the log-likelihood function as follows:

$$\ell =lnL\left(\theta \right)=\left\{\begin{array}{c}{\displaystyle D_{1}ln\left(2\theta \right)+(n-D_1)ln(1-W_{T_1}^{\theta })+\sum _{{}_{i=1}}^{D_1}ln{y}_{i:n}-\sum _{{}_{i=1}}^{D_1}{y}_{i:n}^2+(\theta -1)ln{W}_{y_{i:n}},}\hfill \\ D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle rln\left(2\theta \right)+(n-r)ln(1-W_{y_{r:n}}^{\theta })+\sum _{{}_{i=1}}^{r}ln{y}_{i:n}-\sum _{{}_{i=1}}^r{y}_{i:n}^2+(\theta -1)ln{W}_{y_{i:n}},}\hfill \\ D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle D_{2}ln\left(2\theta \right)+(n-D_2)ln(1-W_{T_2}^{\theta })+\sum _{{}_{i=1}}^{D_2}ln{y}_{i:n}-\sum _{{}_{i=1}}^{D_2}{y}_{i:n}^2+(\theta -1)ln{W}_{y_{i:n}},}\hfill \\ D_2=0,1,\cdots ,r-1.\hfill \end{array}\right.$$

(6)

The MLE of the unknown parameter $\theta$ is obtained by differentiating Equation (6) regarding $\theta$ then equating the derivative to zero (see Appendix A) then solving numerically the following equation:

$$\frac{\partial \ell }{\partial \theta }=0,$$

(7)

we get $\widehat{\theta }$, the MLE of the parameter $\theta$. Also, we can obtain $\widehat{R\left(t\right)}$ the MLE of $R\left(t\right)$ by replacing $\theta$ with $\widehat{\theta }$ in Equation (3), as follows:

$$\widehat{R\left(t\right)}=1-{(1-e^{-t^2})}^{\widehat{\theta }}.$$

(8)

The rest of the article is presented as follows. The Bayesian estimation under SEL and LINEX functions is studied in Section 2. Section 3 discusses the E-Bayesian method under SEL and LINEX loss functions. The Markov chain Monte Carlo (MCMC) algorithm is presented in Section 4. Section 5 provides an example of real-life data sets. The results are compared and concluded in Section 6.

2. Bayesian Estimation Method

Here, the Bayesian estimation of the parameter $\theta$ and the reliability function $R\left(t\right)$ of one-parameter Burr-X distribution based on generalized Type-II HCS is considered. The Bayesian approach needs to specify the prior PDF of the parameter $\theta$. The gamma PDFs are widely used in the Bayesian analysis, so we suppose the gamma conjugates prior PDF for $\theta$ (as suggested by [8,9]) with the PDF of the form.

$$\pi \left(\theta \right)=\frac{b^a}{\Gamma \left(a\right)}{\theta }^{a-1}{e}^{-b\theta },a,b>0,$$

(9)

the posterior PDF of $\theta$ can be given from (5) and (9), as follows:

$$\begin{array}{c}\hfill g\left(\theta \right|y)=\left\{\begin{array}{cc}{\displaystyle K^{-1}{\theta }^{D_1+a-1}{e}^{-b\theta }{\left\{1-W_{T_1}^{\theta }\right\}}^{n-D_1}\prod _{{}_{i=1}}^{D_1}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle K^{-1}{\theta }^{r+a-1}{e}^{-b\theta }{\left\{1-W_{y_{r:n}}^{\theta }\right\}}^{n-r}\prod _{{}_{i=1}}^r{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle K^{-1}{\theta }^{D_2+a-1}{e}^{-b\theta }{\left\{1-W_{T_2}^{\theta }\right\}}^{n-D_2}\prod _{{}_{i=1}}^{D_2}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_2=0,1,\cdots ,r-1.\hfill \end{array}\right.\end{array}$$

(10)

where K is the normalizing constant given as

$${\displaystyle K={\int }_0^{\infty }g\left(\theta \right|y)d\theta .}$$

2.1. Estimates under Squared Error Loss Function

The Bayesian estimate of $\theta$ under SEL function is given by

$$\begin{array}{cc}\hfill {\widehat{\theta }}_{BS}& =E\left[\theta \right|y]\hfill \\ & {\displaystyle ={\int }_0^{\infty }\theta g\left(\theta \right|y)d\theta }\hfill \\ & =\left\{\begin{array}{cc}{\displaystyle K^{-1}{\int }_0^{\infty }{\theta }^{D_1+a}{e}^{-b\theta }{\left\{1-W_{T_1}^{\theta }\right\}}^{n-D_1}\prod _{{}_{i=1}}^{D_1}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill & D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle K^{-1}{\int }_0^{\infty }{\theta }^{r+a}{e}^{-b\theta }{\left\{1-W_{y_{r:n}}^{\theta }\right\}}^{n-r}\prod _{{}_{i=1}}^r{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill & D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle K^{-1}{\int }_0^{\infty }{\theta }^{D_2+a}{e}^{-b\theta }{\left\{1-W_{T_2}^{\theta }\right\}}^{n-D_2}\prod _{{}_{i=1}}^{D_2}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill & D_2=0,1,\cdots ,r-1.\hfill \end{array}\right.\hfill \end{array}$$

(11)

The Bayesian estimate for the reliability function $R\left(t\right)$ by using the SEL function is given as follows:

$$\begin{array}{cc}\hfill {\widehat{R\left(t\right)}}_{BS}& =E\left[R\right(t\left)\right|y]\hfill \\ & {\displaystyle ={\int }_0^{\infty }R\left(t\right)g\left(\theta \right|y)d\theta }\hfill \\ & =\left\{\begin{array}{c}{\displaystyle K^{-1}{\int }_0^{\infty }\left(1-{(1-e^{-t^2})}^{\theta }\right){\theta }^{D_1+a-1}{e}^{-b\theta }{\left\{1-W_{T_1}^{\theta }\right\}}^{n-D_1}\prod _{{}_{i=1}}^{D_1}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle K^{-1}{\int }_0^{\infty }\left(1-{(1-e^{-t^2})}^{\theta }\right){\theta }^{r+a-1}{e}^{-b\theta }{\left\{1-W_{y_{r:n}}^{\theta }\right\}}^{n-r}\prod _{{}_{i=1}}^r{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle K^{-1}{\int }_0^{\infty }\left(1-{(1-e^{-t^2})}^{\theta }\right){\theta }^{D_2+a-1}{e}^{-b\theta }{\left\{1-W_{T_2}^{\theta }\right\}}^{n-D_2}\prod _{{}_{i=1}}^{D_2}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_2=0,1,\cdots ,r-1.\hfill \end{array}\right.\hfill \end{array}$$

(12)

2.2. Estimates under LINEX Loss Function

The LINEX loss function with parameters k, h is given by

$$L(\widehat{\theta },\theta )=k\{e^{h(\widehat{\theta }-\theta )}-h(\widehat{\theta }-\theta )-1\},$$

(13)

where $\widehat{\theta }$ is the estimate of $\theta$ and h is the shape parameter of loss function. For more details, see [13]. The Bayesian estimate ${\widehat{\theta }}_{BL}$ of the parameter $\theta$ using the LINEX loss function is $\widehat{\theta }$, which minimizes Equation (13) given by

$$\begin{array}{cc}\hfill {\widehat{\theta }}_{BL}& =\frac{-1}{h}lnE[exp(-h\theta )|y]\hfill \\ & {\displaystyle =\frac{-1}{h}ln{\int }_0^{\infty }exp(-h\theta )g\left(\theta \right|y)d\theta }\hfill \\ & =\left\{\begin{array}{c}{\displaystyle \frac{-1}{Kh}ln{\int }_0^{\infty }{\theta }^{D_1+a-1}{e}^{-(b+h)\theta }{\left\{1-W_{T_1}^{\theta }\right\}}^{n-D_1}\prod _{{}_{i=1}}^{D_1}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle \frac{-1}{Kh}ln{\int }_0^{\infty }{\theta }^{r+a-1}{e}^{-(b+h)\theta }{\left\{1-W_{y_{r:n}}^{\theta }\right\}}^{n-r}\prod _{{}_{i=1}}^r{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle \frac{-1}{Kh}ln{\int }_0^{\infty }{\theta }^{D_2+a-1}{e}^{-(b+h)\theta }{\left\{1-W_{T_2}^{\theta }\right\}}^{n-D_2}\prod _{{}_{i=1}}^{D_2}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_2=0,1,\cdots ,r-1,h\ne 0,\hfill \end{array}\right.\hfill \end{array}$$

(14)

By using the LINEX loss, the Bayesian estimate of $R\left(t\right)$ is given by

$$\begin{array}{cc}\hfill {\widehat{R\left(t\right)}}_{BL}& =\frac{-1}{h}lnE[exp(-hR\left(t\right))|y]\hfill \\ & {\displaystyle =\frac{-1}{h}ln{\int }_0^{\infty }exp(-hR\left(t\right))g\left(\theta \right|y)d\theta }\hfill \\ & =\left\{\begin{array}{c}{\displaystyle =\frac{-1}{Kh}ln{\int }_0^{\infty }{\theta }^{D_1+a-1}{e}^{h[{(1-exp(-t^2))}^{\theta }-1]-b\theta }{\left\{1-W_{T_1}^{\theta }\right\}}^{n-D_1}\prod _{{}_{i=1}}^{D_1}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle =\frac{-1}{Kh}ln{\int }_0^{\infty }{\theta }^{r+a-1}{e}^{h[{(1-exp(-t^2))}^{\theta }-1]-b\theta }{\left\{1-W_{y_{r:n}}^{\theta }\right\}}^{n-r}\prod _{{}_{i=1}}^r{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle =\frac{-1}{Kh}ln{\int }_0^{\infty }{\theta }^{D_2+a-1}{e}^{h[{(1-exp(-t^2))}^{\theta }-1]-b\theta }{\left\{1-W_{T_2}^{\theta }\right\}}^{n-D_2}\prod _{{}_{i=1}}^{D_2}{y}_{i:n}{e}^{-y_{i:n}^2}{W}_{y_{i:n}}^{\theta -1}d\theta ,}\hfill \\ D_2=0,1,\cdots ,r-1,h\ne 0.\hfill \end{array}\right.\hfill \end{array}$$

(15)

3. E-Bayesian Estimation Method

The possibility of the rate of failure for high-reliability products is less than the possibility of the failure rate of low-reliability products. For this, according to [14], we select the hyper-parameters a and b to prove that $\pi \left(\theta \right)$ is a decreasing function of $\theta$. The first derivative of $\pi \left(\theta \right)$ regarding $\theta$ is as follows:

$$\frac{d\pi \left(\theta \right)}{d\theta }=\frac{b^a}{\Gamma \left(a\right)}{\theta }^{a-2}exp(-b\theta )\left\{(a-1)-b\theta \right\}.$$

Thus, for $0<a<1$ and $b>0$, the prior PDF $\pi \left(\theta \right)$ is a decreasing function of $\theta$. The thinner- tailed prior PDF often decreases the robustness of the Bayesian estimate; therefore, b should be smaller than an upper bound c, where $c>0$ is a constant. Accordingly, hyper-parameters a and b should be selected with the restriction of $0<a<1$ and $0<b<c$. We assume that a and b are independent with bi-variate PDF given by

$$\begin{array}{ccc}\hfill p(a,b)& =& p_1\left(a\right)p_2\left(b\right),\hfill \end{array}$$

the E-Bayesian estimates of $\theta$ and $R\left(t\right)$ are, respectively, obtained as follows:

$${\widehat{\theta }}_{EB}=E\left[{\widehat{\theta }}_B\right|y]=\underset{D}{\int \int }\widehat{{\theta }_B}(a,b)p(a,b)dadb,$$

(16)

$${\widehat{R\left(t\right)}}_{EB}=E\left[{\widehat{R\left(t\right)}}_B\right|y]=\underset{D}{\int \int }{\widehat{R\left(t\right)}}_{B}p(a,b)dadb.$$

(17)

where D stands for the set of possible values of a and b, ${\widehat{\theta }}_B$ and ${\widehat{R\left(t\right)}}_B$ are the Bayesian estimates of $\theta$ and $R\left(t\right)$, respectively, under the SEL and the LINEX loss functions. See, for more details, [15,16,17,18].

3.1. The E-Bayesian Estimate of $\theta$

To obtain the E-Bayesian estimate of $\theta$, we use three prior PDFs of a and b to study the influence of these prior PDFs on the E-Bayesian estimation. We suggest the following prior PDFs

$$\begin{array}{cc}\hfill p_1(a,b)& =\frac{2a}{c},0<a<1,0<b<c,\hfill \\ \hfill p_2(a,b)& =\frac{2b}{c^2},0<a<1,0<b<c,\hfill \\ \hfill p_3(a,b)& =\frac{3b^2}{c^3},0<a<1,0<b<c,\hfill \end{array}$$

(18)

The E-Bayesian estimate of $\theta$, under the SEL are computed from (11), (16) and (18) as follows:

$$\begin{array}{cc}\hfill {\widehat{\theta }}_{EB{S}_j}& =\underset{D}{\int \int }{\widehat{\theta }}_{BS}(a,b)p_j(a,b)dadb,j=1,2,3,\hfill \end{array}$$

(19)

also, under the LINEX loss function, the E-Bayesian estimate of $\theta$, can be derived from (14), (16) and (18) as

$$\begin{array}{cc}\hfill {\widehat{\theta }}_{EB{L}_j}& =\underset{D}{\int \int }{\widehat{\theta }}_{BL}(a,b)p_j(a,b)dadb,j=1,2,3.\hfill \end{array}$$

(20)

3.2. The E-Bayesian Estimate of $R\left(t\right)$

The E-Bayesian estimate can be computed by adopting (18) based on the SEL and LINEX loss functions as follows.

The E-Bayesian estimate of $R\left(t\right)$ under SEL function is written from (12), (17) and (18) as

$${\widehat{R\left(t\right)}}_{EB{S}_j}=\underset{D}{\int \int }{\widehat{R\left(t\right)}}_{BS}{p}_j(a,b)dadb,j=1,2,3,$$

(21)

also, the E-Bayesian estimate of $R\left(t\right)$ by using the LINEX loss function can be computed from (15), (17) and (18) as follows:

$${\widehat{R\left(t\right)}}_{EB{L}_j}=\underset{D}{\int \int }{\widehat{R\left(t\right)}}_{BL}{p}_j(a,b)dadb,j=1,2,3.$$

(22)

It is noted that E-Bayesian and Bayesian estimates cannot be obtained analytically; therefore, we use the MCMC method for deriving the Bayesian and E-Bayesian estimates of $\theta$ and $R\left(t\right)$.

4. The MCMC Algorithm

In this work, we consider the MCMC method for generating a sample of $\theta$ from the posterior PDF; after that, we derive Bayesian and E-Bayesian estimates of $\theta$ and $R\left(t\right)$ under LINEX loss and SEL functions. A Metropolis$\widehat{\mathrm{a}}$€“Hastings sampler is used to obtain the posterior sample of the parameter $\theta$ from the full conditional posterior PDF. From (10), discarding all terms do not depend on $\theta$; we get the full conditional posterior PDF as follows.

$$g^{*}\left(\theta \right|y)=\left\{\begin{array}{cc}{\displaystyle {\theta }^{D_1+a-1}{e}^{-b\theta }{\left\{-W_{T_1}^{\theta }\right\}}^{n-D_1}\prod _{{}_{i=1}}^{D_1}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_1=r,r+1,\cdots ,n,\hfill \\ {\displaystyle {\theta }^{r+a-1}{e}^{-b\theta }{\left\{-W_{y_{r:n}}^{\theta }\right\}}^{n-r}\prod _{{}_{i=1}}^r{W}_{y_{i:n}}^{\theta -1},}\hfill & D_1=0,1,\cdots ,r-1,D_2=r,\hfill \\ {\displaystyle {\theta }^{D_2+a-1}{e}^{-b\theta }{\left\{-W_{T_2}^{\theta }\right\}}^{n-D_2}\prod _{{}_{i=1}}^{D_2}{W}_{y_{i:n}}^{\theta -1},}\hfill & D_2=0,1,\cdots ,r-1.\hfill \end{array}\right.$$

(23)

where $W_{y_{i:n}}=1-e^{-y_{i:n}^2}$, $W_{T_1}=1-e^{-T_1^2}$, $W_{T_2}=1-e^{-T_2^2}$.

For more details, see, for example, [19,20]. The following steps indicate the Metropolis†“Hastings algorithm for simulating the posterior samples, then derive the Bayesian estimates and the credible related intervals (CRIs) as follows.

(1) 

Set the initial value of $\theta$, say ${\theta }^{\left(0\right)}(={\widehat{\theta }}_{MLE})$, to guarantee a rapid convergence of the Markov chain.

(2) 

Set $j=1$.

(3) 

By applying a Metropolis–Hastings sampler, ${\theta }^{\left(j\right)}$ is simulated from $g^{*}\left({\theta }^{(j-1)}\right|y)$

• Generate a proposal ${\theta }^{(*)}$ from a normal distribution as a proposal distribution ignoring negative draws, as they lead to a high rejection rate.
• A sample u is generated from the $Uniform(0,1)$ distribution.
• Compute the acceptance probability

  $${\psi }_{\theta }=min\left[1,\frac{g^{*}\left({\theta }^{(*)}\right|y)}{g^{*}\left({\theta }^{(j-1)}\right|y)}\right].$$

  (24)
• If $u<{\psi }_{\theta }$ accept ${\theta }^{(*)}$ as ${\theta }^{\left(j\right)}$, or else, ${\theta }^{\left(j\right)}$=${\theta }^{(j-1)}$

(4) 

Also, $R\left(t\right)$ is computed from Equation (3) as follows.

$$R^{\left(j\right)}\left(t\right)=1-{(1-exp(-t^2))}^{{\theta }^{\left(j\right)}}.$$

(25)

(5) 

Set $j=j+1$.

(6) 

Steps (3–6) are repeated N times to get a sequence of the parameter $\theta$ with optional burn-in period.

(7) 

The Bayesian estimates of $\theta$ and $R\left(t\right)$ under SEL function are, respectively, given as

$${\displaystyle {\widehat{\theta }}_{BS}=\frac{1}{N-M}\sum _{j=M+1}^N{\theta }^{\left(j\right)}.}$$

(26)

$${\displaystyle {\widehat{R\left(t\right)}}_{BS}=\frac{1}{N-M}\sum _{j=M+1}^N{R}^{\left(j\right)}\left(t\right),}$$

(27)

where M represents a burn-in period of Markov chain discarded to get to the rapid convergence and cancel the effect of choosing the starting point of the parameter.

(8) 

The Bayesian estimates for $\theta$ and $R\left(t\right)$ by the LINEX loss function are, respectively, given by,

$$\begin{array}{c}{\displaystyle {\widehat{\theta }}_{BL}=\frac{-1}{h}ln\left[\frac{1}{N-M}\sum _{j=M+1}^N{e}^{-h{\theta }^{\left(j\right)}}\right],h\ne 0,}\end{array}$$

(28)

$$\begin{array}{c}{\displaystyle {\widehat{R\left(t\right)}}_{BL}=\frac{-1}{h}ln\left[\frac{1}{N-M}\sum _{j=M+1}^N{e}^{-h{R}^{\left(j\right)}\left(t\right)}\right],h\ne 0.}\end{array}$$

(29)

(9) 

A $100(1-\gamma )\%$, confidence intervals (CIs) for MLEs of $\theta$ and $R\left(t\right)$ are obtained as follows:

$$\begin{array}{c}\widehat{\theta }\pm z_{\frac{\gamma }{2}}\left(\sqrt{Var\left(\widehat{\theta }\right)}\right)and\widehat{R\left(t\right)}\pm z_{\frac{\gamma }{2}}\left(\sqrt{Var\left(\widehat{R\left(t\right)}\right)}\right),\hfill \end{array}$$

(30)

where $z_{\frac{\gamma }{2}}$ is the $100(1-\frac{\gamma }{2})\%$ upper percentile of standard normal variate.

(10) 

A $100(1-\gamma )\%$, CRIs of E-Bayesian and Bayesian estimates of $\theta$ and $R\left(t\right)$ are constructed from the ($\frac{\gamma }{2}$) and ($1-\frac{\gamma }{2}$) quantiles sample of the empirical posterior PDF of MCMC draws, given by,

$$\begin{array}{c}\left[{\theta }_{N\left[\frac{\gamma }{2}\right]},{\theta }_{N[1-\frac{\gamma }{2}]}\right]and\left[R_{N\left[\frac{\gamma }{2}\right]},R_{N[1-\frac{\gamma }{2}]}\right],\hfill \end{array}$$

(31)

where N stands for the number of draws.

4.1. Simulation Study

• Choose values of $n,r,T_1,T_2,h.$
• By specifying the value of c, values of a and b are generated from (18).
• For known values of a and b, the true value of $\theta$ is generated from gamma (a,b).
• MLEs of the parameter $\theta$ and the reliability function $R\left(t\right)$ are, respectively, obtained from Equations (7) and (8).
• A Metropolis–Hastings sampler is used for generating a Markov chain with 11,000 values of $\theta$, ignoring the first 1000 values as a “burn-in” period of the Markov chain.
• Compute the Bayesian estimates of $\theta$ and $R\left(t\right)$ under SEL function from (26) and (27), respectively.
• Also, the Bayesian estimates of the parameter $\theta$ and the reliability function $R\left(t\right)$ with the LINEX loss function are, respectively, computed from (28) and (Section 4.1).
• Compute the E-Bayesian estimates of the parameter $\theta$ and the reliability function $R\left(t\right)$ under SEL function from (19) and (21), respectively.
• The E-Bayesian estimates of parameter $\theta$ and the reliability function $R\left(t\right)$ under LINEX loss function are, respectively, obtained from (20) and (22).
• The 95% CIs of MLEs of $\theta$ and $R\left(t\right)$ are constructed from (30).
• The 95% CRIs of E-Bayesian and Bayesian estimates of the parameter $\theta$ and the reliability function $R\left(t\right)$ are computed from (31).
• The mean squared error (MSE) of $\theta$ and $R\left(t\right)$ estimates are, respectively, given by,

  $$\begin{array}{c}{\displaystyle MSE\left(\widehat{\theta }\right)=\frac{1}{1000}\sum _{j=1}^{1000}{({\widehat{\theta }}_j-\theta )}^2}\\ {\displaystyle MSE\left(\widehat{R\left(t\right)}\right)=\frac{1}{1000}\sum _{j=1}^{1000}{({\widehat{R\left(t\right)}}_j-R\left(t\right))}^2,}\end{array}$$

  where $\widehat{\theta }$ and $\widehat{R\left(t\right)}$ denote the estimates of $\theta$ and $R\left(t\right)$, respectively. To study the impact of values of $n,r,T_1,$ and $T_2$ on MSEs of estimators, we computed estimates across different values of $n,r,T_1,$ and $T_2$. The numerical results are computed by MATHEMATICA 8 codes, such as (FindRoot, NMaximize, NIntegrate and RandomReal) and listed in

  Table 1. Estimates, MSE, and $95\%$CIs and CRIs of MLEs, Bayesian, and E-Bayesian methods for $\theta$ under LINEX and SEL functions when $\theta =1.22,a=0.8,b=0.7,h=1.5,c=1$.

  $(\mathit{n},\mathit{r})$	$({\mathit{T}}_1,{\mathit{T}}_2)$	Criteria	${\widehat{\mathit{\theta }}}_{\mathit{MLE}}$	SEL Function				LINEX Loss
  				${\widehat{\mathit{\theta }}}_{\mathit{BS}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{1}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{2}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{3}}$	${\widehat{\mathit{\theta }}}_{\mathit{BL}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{1}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{2}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{3}}$
  $(25,15)$	$(0.7,1.5)$	Mean	1.82415	1.81515	1.62638	1.42308	1.49423	1.72759	1.55551	1.36835	1.43408
  		MSE	0.56645	0.39228	0.19571	0.06464	0.10101	0.28806	0.13755	0.04162	0.06726
  		Lower	0.89686	1.1236	1.0148	0.8420	0.3892	0.5593	0.6968	0.2524	0.4161
  		Upper	2.75144	2.5067	2.4404	2.4107	2.457	2.4291	2.4142	2.4843	2.452
  		Length	1.85458	1.3831	1.4256	1.5687	2.0678	1.8698	1.7174	2.2318	2.0359
  	$(1.5,2)$	Mean	1.88381	1.72198	1.54289	1.35003	1.41753	1.64524	1.48081	1.30211	1.36485
  		MSE	0.49302	0.2907	0.13534	0.04071	0.06526	0.21197	0.09359	0.0268	0.04291
  		Lower	1.41099	1.0782	0.9838	0.8095	0.3772	0.5395	0.682	0.2571	0.4137
  		Upper	2.35663	2.3658	2.3067	2.2763	2.3229	2.2956	2.2796	2.3471	2.316
  		Length	0.945648	1.2876	1.3229	1.4668	1.9457	1.7561	1.5975	2.09	1.9022
  $(45,30)$	$(0.7,1.5)$	Mean	1.81167	1.77186	1.58758	1.38913	1.45859	1.72116	1.5467	1.35768	1.42397
  		MSE	0.56434	0.32622	0.15252	0.04193	0.07161	0.27055	0.12248	0.03116	0.0550
  		Lower	0.855328	1.2527	1.1926	0.9552	0.4769	0.6545	0.8653	0.3941	0.5670
  		Upper	2.76801	2.291	2.2498	2.220	2.3014	2.2627	2.2281	2.3213	2.2810
  		Length	1.91268	1.0382	1.0572	1.2648	1.8245	1.6081	1.3628	1.9272	1.714
  	$(1.5,2)$	Mean	1.76545	1.68398	1.50884	1.32024	1.38625	1.63871	1.47236	1.29218	1.35537
  		MSE	0.39961	0.2318	0.0967	0.02021	0.03884	0.18977	0.07545	0.01435	0.02835
  		Lower	1.10533	1.1955	1.1421	0.9117	0.4559	0.6251	0.8315	0.3820	0.5469
  		Upper	2.42557	2.1725	2.1353	2.1060	2.1846	2.1474	2.1132	2.2024	2.1638
  		Length	1.32023	0.9770	0.9931	1.1942	1.7286	1.5222	1.2817	1.8204	1.6168
  $(60,45)$	$(0.7,1.5)$	Mean	1.83636	1.65933	1.48676	1.30091	1.36596	1.62674	1.46051	1.28075	1.34376
  		MSE	0.41976	0.2004	0.07709	0.01109	0.02631	0.17223	0.06335	0.00794	0.01999
  		Lower	1.4239	1.2453	1.2077	0.952	0.4854	0.6573	0.8919	0.4310	0.5994
  		Upper	2.24882	2.0734	2.0457	2.0215	2.1164	2.0746	2.0291	2.1305	2.0881
  		Length	0.824922	0.8281	0.8380	1.0694	1.6310	1.4173	1.1373	1.6996	1.4888
  	$(1.5,2)$	Mean	1.62609	1.59426	1.42846	1.2499	1.3124	1.56302	1.40329	1.23055	1.29109
  		MSE	0.18972	0.15881	0.0585	0.01241	0.02124	0.13462	0.04734	0.01076	0.01674
  		Lower	1.28096	1.190	1.1541	0.9097	0.4630	0.6277	0.8523	0.4109	0.5723
  		Upper	1.97122	1.9985	1.9719	1.9473	2.0368	1.9971	1.9543	2.0502	2.0099
  		Length	0.690261	0.8085	0.8178	1.0376	1.5738	1.3694	1.1019	1.6392	1.4375
  $(80,65)$	$(0.7,1.5)$	Mean	1.62173	1.64368	1.47274	1.28865	1.35308	1.61869	1.45263	1.27321	1.33607
  		MSE	0.2717	0.18689	0.06981	0.00925	0.02272	0.16581	0.05966	0.00711	0.01818
  		Lower	0.893912	1.2825	1.2542	0.9801	0.5046	0.6786	0.9324	0.4622	0.6332
  		Upper	2.34955	2.0048	1.9832	1.9654	2.0727	2.0275	1.9729	2.0842	2.0389
  		Length	1.45564	0.7223	0.7290	0.9853	1.5681	1.3489	1.0405	1.6220	1.4057
  	$(1.5,2)$	Mean	1.53225	1.63107	1.46144	1.27876	1.34269	1.60709	1.44215	1.26396	1.3264
  		MSE	0.1890	0.17092	0.05986	0.00465	0.01637	0.15162	0.05079	0.00305	0.01255
  		Lower	0.869392	1.2779	1.2508	0.9763	0.5031	0.6762	0.9304	0.4624	0.6326
  		Upper	2.19511	1.9843	1.9634	1.9465	2.0544	2.0092	1.9539	2.0655	2.0202
  		Length	1.32572	0.7064	0.7126	0.9702	1.5513	1.3331	1.0235	1.6031	1.3876

  Where ${\widehat{\theta }}_{EBS1}$, ${\widehat{\theta }}_{EBS2}$ and ${\widehat{\theta }}_{EBS3}$ are the E-Bayesian estimates of $\theta$ relative to SEL based on $p_1(a,b)$, $p_2(a,b)$ and $p_3(a,b)$, respectively, ${\widehat{\theta }}_{EBL1}$, ${\widehat{\theta }}_{EBL2}$ and ${\widehat{\theta }}_{EBL3}$ are the E-Bayesian estimates of $\theta$ relative to LINEX based on $p_1(a,b)$, $p_2(a,b)$ and $p_3(a,b)$, respectively. In addition, ${\widehat{\theta }}_{BS}$ and ${\widehat{\theta }}_{BL}$ denote the Bayesian estimates of $\theta$ under SEL and LINEX loss, respectively.

  and

  Table 2. Estimates, MSE, and $95\%$CIs and CRIs of MLEs, Bayesian, and E-Bayesian methods for $R(t=1.3)$ under LINEX and SEL functions when $\theta =1.22,a=0.8,b=0.7,h=1.5,c=1$.

  $(\mathit{n},\mathit{r})$	$({\mathit{T}}_1,{\mathit{T}}_2)$	Criteria	${\widehat{\mathit{R}}}_{\mathit{MLE}}$	SEL Function				LINEX Loss
  				${\widehat{\mathit{R}}}_{\mathit{BS}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{1}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{2}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{3}}$	${\widehat{\mathit{R}}}_{\mathit{BL}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{1}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{2}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{3}}$
  $(25,15)$	$(0.7,1.5)$	Mean	0.307776	0.30712	0.27518	0.24078	0.25282	0.30532	0.27373	0.23967	0.2516
  		MSE	0.011812	0.00825	0.00359	0.00086	0.00154	0.00793	0.00342	0.00081	0.00146
  		Lower	0.174502	0.2112	0.2094	0.1607	0.0792	0.1096	0.1577	0.0764	0.1066
  		Upper	0.44105	0.4030	0.4013	0.3897	0.4023	0.3961	0.3898	0.4030	0.3966
  		Length	0.266548	0.1917	0.1919	0.2290	0.3231	0.2865	0.2321	0.3266	0.2901
  	$(1.5,2)$	Mean	0.318297	0.29404	0.26346	0.23052	0.24205	0.29241	0.26215	0.22952	0.24095
  		MSE	0.010628	0.00623	0.0025	0.00059	0.00101	0.00598	0.00238	0.00056	0.00095
  		Lower	0.252122	0.2031	0.2015	0.1546	0.0764	0.1055	0.1518	0.0738	0.1028
  		Upper	0.384472	0.3849	0.3834	0.3724	0.3847	0.3786	0.3725	0.3853	0.3791
  		Length	0.132349	0.1818	0.1819	0.2178	0.3083	0.2731	0.2206	0.3115	0.2764
  $(45,30)$	$(0.7,1.5)$	Mean	0.302633	0.30199	0.27058	0.23676	0.2486	0.30094	0.26974	0.23612	0.24789
  		MSE	0.009276	0.0071	0.00287	0.00053	0.00109	0.00692	0.00278	0.00051	0.00105
  		Lower	0.193104	0.2288	0.2278	0.1750	0.0895	0.1209	0.1731	0.0877	0.1191
  		Upper	0.412162	0.3751	0.3741	0.3662	0.3841	0.3763	0.3664	0.3845	0.3767
  		Length	0.219058	0.1463	0.1463	0.1912	0.2946	0.2554	0.1933	0.2968	0.2576
  	$(1.5,2)$	Mean	0.300882	0.28955	0.25944	0.22701	0.23836	0.28859	0.25867	0.22642	0.23771
  		MSE	0.008729	0.00514	0.0018	0.00025	0.00056	0.0050	0.00174	0.00024	0.23771
  		Lower	0.203186	0.2194	0.2184	0.1678	0.0858	0.1159	0.1660	0.0842	0.1142
  		Upper	0.398578	0.3597	0.3587	0.3511	0.3682	0.3608	0.3513	0.3687	0.3612
  		Length	0.195393	0.1402	0.1403	0.1833	0.2825	0.2448	0.1853	0.2845	0.2469
  $(60,45)$	$(0.7,1.5)$	Mean	0.298925	0.28809	0.25813	0.22586	0.23715	0.28737	0.25755	0.22542	0.23667
  		MSE	0.008271	0.00491	0.00168	0.00022	0.0005	0.00481	0.00164	0.00022	0.000479
  		Lower	0.204471	0.2275	0.2268	0.1738	0.0897	0.1204	0.1724	0.0885	0.1191
  		Upper	0.393379	0.3486	0.3479	0.3425	0.3620	0.3539	0.3427	0.3624	0.3543
  		Length	0.188907	0.1211	0.1212	0.1687	0.2724	0.2335	0.1703	0.2739	0.2352
  	$(1.5,2)$	Mean	0.29833	0.28406	0.25451	0.2227	0.23383	0.28336	0.25396	0.22227	0.23337
  		MSE	0.007858	0.00436	0.00141	0.00019	0.00038	0.00427	0.00137	0.00019	0.00037
  		Lower	0.211413	0.2245	0.2238	0.1715	0.0885	0.1188	0.1702	0.0873	0.1175
  		Upper	0.385247	0.3436	0.3429	0.3376	0.3569	0.3489	0.3378	0.3572	0.3492
  		Length	0.173834	0.1190	0.1191	0.1661	0.2684	0.2301	0.1676	0.2699	0.2316
  $(80,65)$	$(0.7,1.5)$	Mean	0.298976	0.28424	0.25468	0.22285	0.23399	0.28371	0.25425	0.22252	0.23363
  		MSE	0.007187	0.00424	0.00131	0.0001	0.00029	0.00417	0.00128	0.0001	0.00028
  		Lower	0.233718	0.2318	0.2313	0.1765	0.0916	0.1224	0.1755	0.0907	0.1214
  		Upper	0.364234	0.3367	0.3362	0.3328	0.3541	0.3456	0.333	0.3544	0.3458
  		Length	0.130516	0.1049	0.1049	0.1563	0.2625	0.2232	0.1576	0.2637	0.2244
  	$(1.5,2)$	Mean	0.300122	0.2825	0.25312	0.22148	0.23256	0.28198	0.25271	0.22116	0.23221
  		MSE	0.006783	0.00391	0.00111	0.00003	0.00018	0.00384	0.00108	0.00003	0.00017
  		Lower	0.25815	0.2310	0.2305	0.1759	0.0913	0.1219	0.1748	0.0904	0.1210
  		Upper	0.342094	0.3340	0.3335	0.3304	0.3517	0.3432	0.3306	0.3520	0.3434
  		Length	0.0839436	0.1030	0.1030	0.1545	0.2604	0.2212	0.1557	0.2616	0.2225

  Where ${\widehat{R}}_{EBS1}$, ${\widehat{R}}_{EBS3}$ and ${\widehat{R}}_{EBS2}$ are the E-Bayesian estimates of $R\left(t\right)$ relative to SEL based on $p_1(a,b)$, $p_2(a,b)$ and $p_3(a,b)$, respectively, ${\widehat{R}}_{EBL1}$, ${\widehat{R}}_{EBL2}$ and ${\widehat{R}}_{EBL3}$ are the E-Bayesian estimates of $R\left(t\right)$ relative to LINEX based on $p_1(a,b)$, $p_2(a,b)$ and $p_3(a,b)$, respectively. In addition, ${\widehat{R}}_{BS}$ and ${\widehat{R}}_{BL}$ denote the Bayesian estimates of $R\left(t\right)$ under SEL and LINEX loss, respectively.

  , where Table 1, gives estimates, MSE, and $95\%$ CIs and CRIs for MLE, Bayesian, and E-Bayesian estimates of the parameter $\theta$ with $n=25,45,60,80$; $r=15,30,45,65$; $T_1=0.7,1.5$ and $T_2=1.5,2$. Table 2 shows average estimates, MSE and $95\%$ CIs and CRIs of MLE, Bayesian, and E-Bayesian estimates for the reliability function $R(t=1.3)$ with $n=25,45,60,80$; $r=15,30,45,65$; $T_1=0.7,1.5$, and $T_2=1.5,2$.

5. Illustrative Example (Real Data Set)

An illustrative example is provided to investigate the performance of proposed methods and clarify how they behave in practice. These data were reported by [21], representing minority electron mobility for $G{a}_{1-x}A{l}_{x}As$. Two data sets at the mole fractions of $0.25$ and $0.30$ were used by [22], who stated that Burr-X distribution presents a good fit for Sets 1 and 2 of data. Each set contains 21 observations, as given below:

Data Set 1 ( mole fraction 0.25): $0.7948,0.7007,0.6292,3.051,2.779,2.604,2.371,1.525,1.296,1.154,2.214,2.045,1.715,1.016,0.6175,0.6449,0.8881,1.115,1.397,$ $1.506$ and $1.528.$

Data Set 2 ( mole fraction 0.30): $2.092,1.959,1.814,2.658,2.434,2.288,1.530,1.366,1.165,1.041,1.002,1.250,$ $1.347,0.9198,0.7241,0.6403,0.576,0.5647,0.5873,0.8013,$ and $1.368.$

By using generalized Type-II HCS, taking $n=21$, $r=15$, $\theta =2$ and $R(t=1.2)$, we observe the following cases, for data Set 1:

• When $T_1=0.7$ and $T_2=1.5$, we observe that $Y_{r:n}=1.715>T_2$, hence the experiment is terminated at random time $T_2=1.5$, i.e., only 11 items fail at a random time $T_2=1.5$.
• When $T_1=0.9$ and $T_2=2$, we observe that $T_1<Y_{r:n}=1.715<T_2$, hence the experiment is finished at random time $Y_{r:n}=1.715$, i.e., 15 items fail at random time $Y_{r:n}=1.715$.

For data Set 2:

• When $T_1=0.7$ and $T_2=1.5$, we observe that $Y_{r:n}=1.53>T_2$, hence the experiment is terminated at random time $T_2=1.5$. Therefore, only 14 items fail out of 21 items at random time $T_2=1.5$.
• When $T_1=0.9$ and $T_2=1.7$, we observe that $T_1<Y_{r:n}=1.153<T_2$, hence the experiment is terminated at random time $Y_{r:n}=1.53$, i.e., 15 items are obtained out of 21 items at random time $Y_{r:n}=1.53$.

With respect to real data Sets 1 and 2, all estimates are obtained according to the same procedure as before and results are displayed in

Table 3. For real data Set 1: Estimates, MSE and $95\%$CIs and CRIs of MLEs, Bayesian and E-Bayesian methods for $\theta$ under LINEX and SEL functions when $\theta =2,a=0.8,b=0.7,h=1.5,c=1$.

$(\mathit{n},\mathit{r})$	$({\mathit{T}}_1,{\mathit{T}}_2)$	Criteria	${\widehat{\mathit{\theta }}}_{\mathit{MLE}}$	SEL Function				LINEX Loss
				${\widehat{\mathit{\theta }}}_{\mathit{BS}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{1}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{2}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{3}}$	${\widehat{\mathit{\theta }}}_{\mathit{BL}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{1}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{2}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{3}}$
$(21,15)$	$(0.7,1.5)$	Mean	2.71786	2.81461	2.52189	2.20665	2.31698	2.31698	2.32603	2.0548	2.1503
		MSE	0.7076	0.6636	0.27238	0.04271	0.10049	0.32886	0.10631	0.00301	0.0226
		Lower	1.66523	1.6376	1.3051	1.2125	0.5317	0.7883	0.8086	0.1566	0.3951
		Upper	3.77049	3.9916	3.8418	3.8313	3.8816	3.8456	3.8434	3.953	3.9055
		Length	2.10526	2.3541	2.5368	2.6188	3.3498	3.0573	3.0348	3.7964	3.5104
	$(0.9,1.7)$	Mean	2.83501	2.7219	2.43882	2.13397	2.24067	2.49547	2.25497	1.99146	2.08423
		MSE	0.70233	0.52121	0.19262	0.01799	0.05797	0.24558	0.06508	0.00013	0.00715
		Lower	2.68764	1.5827	1.2729	1.1717	0.5135	0.7616	0.7937	0.1618	0.3931
		Upper	2.98238	3.8611	3.7181	3.706	3.7544	3.7197	3.7162	3.8211	3.7753
		Length	0.294742	2.2784	2.4452	2.5343	3.2408	2.958	2.9226	3.6592	3.3822

,

Table 4. For real data Set 1: Estimates, MSE and $95\%$CIs and CRIs of MLEs, Bayesian and E-Bayesian methods for $R(t=1.2)$ under LINEX and SEL functions, $\theta =2,a=0.8,b=0.7,h=1.5,c=1$.

$(\mathit{n},\mathit{r})$	$({\mathit{T}}_1,{\mathit{T}}_2)$	Criteria	${\widehat{\mathit{R}}}_{\mathit{MLE}}$	SEL Function				LINEX Loss
				${\widehat{\mathit{R}}}_{\mathit{BS}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{1}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{2}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{3}}$	${\widehat{\mathit{R}}}_{\mathit{BL}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{1}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{2}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{3}}$
$(21,15)$	$(0.7,1.5)$	Mean	0.517062	0.52677	0.47198	0.41299	0.43363	0.52255	0.4686	0.41039	0.43078
		MSE	0.013175	0.01189	0.00294	0.00002	0.00025	0.01099	0.00259	0.00005	0.00017
		Lower	0.379033	0.3799	0.3755	0.2901	0.1460	0.1994	0.2827	0.1391	0.1921
		Upper	0.655091	0.6736	0.6696	0.6539	0.6800	0.6679	0.6545	0.6817	0.6694
		Length	0.276057	0.2936	0.2941	0.3638	0.5340	0.4685	0.3718	0.5425	0.4773
	$(0.9,1.7)$	Mean	0.535322	0.51515	0.46157	0.40387	0.42407	0.5110	0.45824	0.40133	0.42126
		MSE	0.013907	0.00949	0.00192	0.00019	0.00004	0.0087	0.00164	0.00027	0.000013
		Lower	0.517135	0.3695	0.3652	0.282	0.1416	0.1937	0.2748	0.1349	0.1866
		Upper	0.55350	0.6608	0.6568	0.6411	0.6661	0.6544	0.6417	0.6677	0.6559
		Length	0.0363737	0.2912	0.2917	0.3591	0.5245	0.4608	0.3668	0.5328	0.4693

,

Table 5. For real data Set 2: Estimates, MSE and $95\%$CIs and CRIs of MLEs, Bayesian, and E-Bayesian methods for $\theta$ under LINEX and SEL functions when $\theta =2,a=0.8,b=0.7,h=1.5,c=1$.

$(\mathit{n},\mathit{r})$	$({\mathit{T}}_1,{\mathit{T}}_2)$	Criteria	${\widehat{\mathit{\theta }}}_{\mathit{MLE}}$	SEL Function				LINEX Loss
				${\widehat{\mathit{\theta }}}_{\mathit{BS}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{1}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{2}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBS}\mathbf{3}}$	${\widehat{\mathit{\theta }}}_{\mathit{BL}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{1}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{2}}$	${\widehat{\mathit{\theta }}}_{\mathit{EBL}\mathbf{3}}$
$(21,15)$	$(0.7,1.5)$	Mean	2.30514	2.43127	2.17842	1.90612	2.00142	2.24881	2.03042	1.79153	1.87559
		MSE	0.31672	0.18603	0.03186	0.00884	0.00003	0.06194	0.00095	0.04348	0.0155
		Lower	1.32817	1.4141	1.1706	1.0469	0.4589	0.6806	0.7451	0.1769	0.3853
		Upper	3.28211	3.4485	3.3271	3.310	3.3533	3.3223	3.3157	3.4062	3.3659
		Length	1.95393	2.0344	2.1565	2.2631	2.8943	2.6417	2.5706	3.2292	2.9806
	$(0.9,1.7)$	Mean	2.25925	2.39822	2.1488	1.8802	1.97421	2.22801	2.00479	1.77471	1.85176
		MSE	0.18272	0.15868	0.02223	0.01442	0.00074	0.05203	0.0001	0.05078	0.02204
		Lower	1.55708	1.3943	1.1582	1.0322	0.4524	0.6714	0.7389	0.1748	0.3838
		Upper	2.96142	3.4021	3.2832	3.2654	3.308	3.2923	3.2707	3.3746	3.3197
		Length	1.40434	2.0078	2.1250	2.2332	2.8557	2.6209	2.5318	3.1998	2.936

and

Table 6. For real data Set 2: Estimates, MSE, and $95\%$CIs and CRIs of MLEs, Bayesian, and E-Bayesian methods for $R(t=1.2)$ under LINEX and SEL functions, $\theta =1.22,a=0.8,b=0.7,h=1.5,c=1$.

$(\mathit{n},\mathit{r})$	$({\mathit{T}}_1,{\mathit{T}}_2)$	Criteria	${\widehat{\mathit{R}}}_{\mathit{MLE}}$	SEL Function				LINEX Loss
				${\widehat{\mathit{R}}}_{\mathit{BS}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{1}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{2}}$	${\widehat{\mathit{R}}}_{\mathit{EBS}\mathbf{3}}$	${\widehat{\mathit{R}}}_{\mathit{BL}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{1}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{2}}$	${\widehat{\mathit{R}}}_{\mathit{EBL}\mathbf{3}}$
$(21,15)$	$(0.7,1.5)$	Mean	0.50136	0.47678	0.4272	0.3738	0.39249	0.47292	0.4241	0.37143	0.38987
		MSE	0.012054	0.00349	0.00009	0.00193	0.00064	0.00305	0.00004	0.00214	0.000776
		Lower	0.354413	0.3362	0.3322	0.2563	0.1278	0.1756	0.2497	0.1216	0.1690
		Upper	0.648307	0.6173	0.6137	0.5981	0.6198	0.6094	0.5985	0.6212	0.6107
		Length	0.293894	0.2811	0.2815	0.3418	0.4919	0.4338	0.3488	0.4996	0.4417
	$(0.9,1.7)$	Mean	0.455001	0.47222	0.42311	0.37022	0.38873	0.4684	0.42004	0.36787	0.38614
		MSE	0.003422	0.00297	0.00003	0.00226	0.00084	0.00257	0.00001	0.00249	0.00099
		Lower	0.36187	0.3324	0.3284	0.2533	0.1262	0.1735	0.2468	0.1201	0.1670
		Upper	0.548132	0.6121	0.6084	0.5929	0.6142	0.6040	0.5933	0.6156	0.6053
		Length	0.18626	0.2797	0.2801	0.3395	0.4880	0.4305	0.3465	0.4955	0.4383

. Table 3 and Table 5, give average estimates, MSE, and $95\%$ CIs and CRIs of MLEs, Bayesian, and E-Bayesian estimates for $\theta$ under LINEX and SEL functions belong to real data Sets 1 and 2, respectively. Table 4 and Table 6, display average estimates, MSE and $95\%$ CIs and CRIs of MLEs, Bayesian, and E-Bayesian estimates of $R(t=1.2)$ based on LINEX and SEL functions belong to real data Sets 1 and 2, respectively.

6. Concluding Remarks

In this article, we used the Bayesian and E-Bayesian approaches and the ML method for estimating the parameter $\theta$ and the reliability function $R\left(t\right)$ of one-parameter Burr-X distribution under generalized Type-II HCS. It is noted that Bayesian estimators cannot be derived analytically, so we applied the MCMC method to derive estimates of the parameter $\theta$ and the reliability function $R\left(t\right)$. Based on SEL and LINEX loss functions, Bayesian and E-Bayesian estimates are derived. Also, CIs of MLEs and CRIs of Bayesian and E-Bayesian estimates are constructed. Furthermore, an example of real data testing is provided for illustration.

From Table 1 and Table 2, one can observe that the MSEs decrease when the sample sizes n, $T_1$, and $T_2$ increase. Also, the length of CIs and CRIs decreases when time points $T_1$ and $T_2$ increase for constant sample size n. The MSE of E-Bayesian estimator for the parameter $\theta$ and the reliability function $R\left(t\right)$ is the smallest compared to the MSE of MLEs and Bayesian estimates. Furthermore, the estimator by using LINEX loss function is more efficient than the estimator based on SEL function in terms of MSE. Generally, The LINEX loss function is better than SEL function in most cases. Ref. [23] compared the difference between the LINEX and SEL functions and stated that the LINEX loss is more appropriate than the SEL function. For the shape parameter h of LINEX close to zero, it is approximately SEL and, therefore, almost symmetric. Based on the above results, we can state that a large sample size n gives a better estimate with a smaller MSE. Finally, we can conclude that the E-Bayesian method is more convenient and efficient when compared with ML and Bayesian methods. The prior PDFs affect the E-Bayesian estimation because of selecting the restriction of $0<a<1$ and $0<b<c$.

From Table 3, Table 4, Table 5 and Table 6, we observe that the E-Bayesian method is more efficient than ML, and Bayesian methods in the sense of having smaller MSE. On the other hand, MSEs of all estimators and the length of CIs and CRIs decrease when time points $T_1$ and $T_2$ increase, i.e., the proposed methods perform efficiently in the application as well.