Parameter and Reliability Inferences of Inverted Exponentiated Half-Logistic Distribution under the Progressive First-Failure Censoring

Abstract

Using progressive first-failure censored samples, we mainly study the inferences of the unknown parameters and the reliability and failure functions of the Inverted Exponentiated Half-Logistic distribution. The progressive first-failure censoring is an extension and improvement of progressive censoring, which is of great significance in the field of lifetime research. Besides maximum likelihood estimation, we use Bayesian estimation under unbalanced and balanced losses: General Entropy loss function, Squared Error loss function and Linex loss function. Approximate explicit expression of Bayesian estimation is given using Lindley approximation method for point estimation and Metropolis-Hastings method for point and interval estimation. Bayesian credible intervals and asymptotic confidence intervals are derived in the form of average length and coverage probability. To show the research effects, a simulation study and practical data analysis are carried out. Finally, we discuss the optimal censoring mode under four different criteria.

1. Introduction

Usually, in lifetime experiment and reliability research, there are limitations like time, money, resources, or due to personnel transfer and accidents, data cannot be recorded completely. In these cases, we often use censored samples. At present, many censoring schemes have been applied to lifetime tests. There exist two most common schemes named Type-I censoring and Type-II censoring. They have been studied by many scholars such as Fujii [1] and Kateri [2]. In terms of content, Type-I censoring means that the test ends at a pre-fixed time, while in Type-II censoring, the test ends when the m-th$(N>m)$ failure occurs. The common disadvantage is that no unit in the test can be removed before the test is completely finished. Thus, the progressive censoring was proposed, which has better efficiency in lifetime experiments. Under this censoring scheme, one can remove the test units at various stages of the experiment. For more details, refer to Balakrishnan [3].

Although the progressive censoring can significantly improve the experimental efficiency, in many cases, the duration of the experiment is still too long. Then the first-failure censoring was proposed. One of the most remarkable features of this censoring is grouping data, which can greatly save time and cost.

To better improve the experimental efficiency, Wu and Ku [4] proposed the progressive first-failure censoring. This mode is of great significance in reliability research. It is a combination of first-failure censoring and progressive censoring, and the two main features of this scheme are the grouped samples and the m-dimension random vector $\Re =(R_1,R_2,\cdots ,R_m)$. The vector ℜ is corresponding to the number of additional groups removed when each failure occurs. In this scheme, we randomly divide N$(N=n\times k)$ units into n groups, each group contains k independent units. Observing these n groups simultaneously and independently. When the i-th failure appears, remove the randomly selected $R_i$ groups together with the group containing the i-th failure. The experiment ends when the m-th failure unit is observed. At this point, all remaining surviving units are removed. m and $\Re =(R_1,\cdots ,R_m)$ are pre-allocated constants and ${\sum }_{i=1}^m{R}_i+m=n$. Thus, considerable time and resources are saved, and a large part of risky units may be removed at various period of the test. In addition, this mode can be easily transformed into other modes. Thus, it is widely used in experimental design due to its flexibility.

For example:

• The first-failure censoring can be obtained when $\Re =(R_1,\cdots ,R_m)=(0,\cdots ,0)$.
• Let $k=1$, this scheme is converted to progressive censoring.
• When $\Re =(R_1,\cdots ,R_{m-1},R_m)=(0,\cdots ,0,n-m)$ and $k=1$, the Type-II censoring is presented.

Assume $x_{1;m:n:k},x_{2;m:n:k},\cdots ,x_{m;m:n:k}$ are the progressive first-failure censoring samples from a continuous function. Then the joint density function is

$$\begin{array}{c}\hfill f_{X_{1:n:k},\cdots ,X_{m:n:k}}\left(x_1,\cdots ,x_m\right)=A\prod _{i=1}^{m}kf\left(x_i\right){[1-F\left(x_i\right)]}^{k(1+R_i)-1}\end{array}$$

(1)

where $F(·)$ means the cumulative distribution function(c.d.f) and $f(·)$ represents the density function (p.d.f). Here we define $R_0=0$ and $A={\prod }_{i=0}^{m-1}\left(n-i-{\sum }_{k=0}^i{R}_k\right)$ is a constant determined by n, m and ℜ.

The Half-Logistic distribution (HLD) is a very important distribution in lifetime test. It is derived from Logistic distribution and has an increasing hazard rate function. Many scholars have studied this distribution such as Balakrishnan [5,6]. It can be used in the failure model of life-testing research. Adatia [7] studied the parameters of HLD using generalized ranked set sampling technique. Asgharzadeh [8] compared several methods of estimating the parameters of HLD. Jung-In and Kang [9] derived the entropy of generalized HLD under Type-II censoring scheme using Bayes estimators and compared them through mean square error and bias.

The Inverted Exponentiated Half-Logistic distribution (IEHLD) is the inverse of exponentiated Half-Logistic distribution (EHLD). At present, there are many studies and applications on EHLD. Rastogi [10] estimated the parameters and reliability characteristics of EHLD using MLE and Bayes estimation under progressive Type-II censoring scheme. Gui [11] considered the joint confidence regions and the MLE, inverse moment and modified inverse moment estimators of EHLD.

However, at the same time, few people study IEHLD. It is an extension to generalized HLD with nonmonotone hazard rate. Shirke [12] studied the maximum likelihood estimations and the asymptotic confidence intervals for parameters of generalized HLD, and considered its inverse as a member of the generalized inverted scale family to give estimates. A scale family of distributions plays a significant role in lifetime research, and the inverses of these distributions have been studied by many scholars. Dey [13] analyzed lifetime data using inverted exponential distribution. Panahi [14] studied the maximum likelihood estimations and Bayesian estimations of parameters of inverted exponentiated Rayleigh distribution under adaptive Type-II progressive hybrid censoring scheme. Moreover, Lee and Cho [15] applied the maximum likelihood estimations, Importance sampling method and Lindley method to estimate the statistical inference of IEHLD using progressive Type II censored samples.

In addition, when the value of the shape parameter is set to be 1, the IEHLD can be transformed into the usual HLD, thus it has greater inclusiveness and flexibility in the practice of reliability research and lifetime test correspondingly.

Furthermore, through the graph of hazard rate function of IEHLD in Figure 1, we can see that it has one maximum. In this respect, IEHLD can be performed as an alternative lifetime model for several well-known distributions such as Generalized Inverted Rayleigh distribution (GIRD) and Inverse Weibull distribution (IWD), which have the same properties in terms of hazard rate.

The p.d.f, c.d.f, hazard function and reliability function of IEHLD are given as below:

$$\begin{array}{cc}\hfill f\left(x;\gamma ,\beta \right)={\displaystyle \frac{2\gamma }{\beta x^2}}exp\left(-{\displaystyle \frac{1}{\beta x}}\right){\displaystyle \frac{{\left[1-exp(-{\displaystyle \frac{1}{\beta x}})\right]}^{\gamma -1}}{{\left[1+exp(-{\displaystyle \frac{1}{\beta x}})\right]}^{\gamma +1}}},& x>0,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill F\left(x;\gamma ,\beta \right)=1-{\left[{\displaystyle \frac{1-exp(-{\displaystyle \frac{1}{\beta x}})}{1+exp(-{\displaystyle \frac{1}{\beta x}})}}\right]}^{\gamma },& x>0,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill R(t)={\left[{\displaystyle \frac{1-exp(-{\displaystyle \frac{1}{\beta x}})}{1+exp(-{\displaystyle \frac{1}{\beta x}})}}\right]}^{\gamma },& t>0,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill H(t;\gamma ,\beta )={\displaystyle \frac{2\gamma }{\beta t^2}}exp\left(-{\displaystyle \frac{1}{\beta t}}\right){\left[1-exp\left(-{\displaystyle \frac{2}{\beta t}}\right)\right]}^{-1},& t>0,\hfill \end{array}$$

(5)

where $\beta >0$ and $\gamma >0$ are the shape and scale parameters. Please note that when $\gamma =1$, the distribution reduce to the HLD. The graph of hazard function is presented in Figure 1.

This paper mainly studies the parameters and the reliability and failure functions of IEHLD using progressive first-failure censored samples. The two main objectives are the maximum likelihood estimation (MLE) and the Bayesian estimation. The asymptotic interval of the parameters are given using MLE, and the Bayesian estimation is prestented under General Entropy loss function, Linex loss function and Squared Error loss function. We use Lindley approximation method and Metropolis-Hastings (M-H) method to approximate the Bayesian estimation to give an explicit solution.

The rest sections in this paper can be summarized as follows: First, we focus on the MLEs of the parameters and the reliability and failure functions in Section 2. The asymptotic intervals of parameters are constructed at the same time. In Section 3, the Bayesian estimations using Lindley approximation method and M-H method are derived, then the Bayesian credible intervals of parameters are given. Next, simulation studies are conducted in Section 4 while practical data analysis is presented in Section 5. In Section 6, an optimal censoring plan under four criteria is derived. And in Section 7, we carry out a brief conclusion.

2. Maximum Likelihood Estimator

Suppose $x_{1;m:n:k},x_{2;m:n:k},\cdots ,x_{m;m:n:k}$ be the progressive first-failure censored samples from IEHLD with $\Re =(R_1,R_2,\cdots ,R_m)$, group n and group size k. For convenience, we replace $(x_{1;m:n:k},x_{2;m:n:k},\cdots ,x_{m;m:n:k})$ with $\overrightarrow{X}$. The likelihood function can be obtained as:

$$\begin{array}{c}\hfill L\left(\gamma ,\beta |\overrightarrow{X}\right)=A{k}^m\prod _{i=1}^m{\displaystyle \frac{2\gamma }{\beta x_i^2}}exp\left(-{\displaystyle \frac{1}{\beta x_i}}\right){\displaystyle \frac{{\left[1-exp(-{\displaystyle \frac{1}{\beta x_i}})\right]}^{\gamma k(1+R_i)-1}}{{\left[1+exp(-{\displaystyle \frac{1}{\beta x_i}})\right]}^{\gamma k(1+R_i)+1}}}\end{array}$$

(6)

Then we can obtain the log-likelihood function:

$$\begin{array}{ccc}\hfill l\left(\gamma ,\beta |\overrightarrow{X}\right)& =& lnA+mln2k+mln\gamma -mln\beta -2\sum _{i=1}^{m}ln{x}_i-\sum _{i=1}^m{\displaystyle \frac{1}{\beta x_i}}\hfill \\ & & -\gamma k\sum _{i=1}^m(1+R_i)ln{Z}_{1,1}^X(\beta )-\sum _{i=1}^{m}ln\left[1-exp(-{\displaystyle \frac{2}{\beta x_i}})\right]\hfill \end{array}$$

(7)

The corresponding likelihood functions of $\gamma$ and $\beta$ are:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial l}{\partial \gamma }}={\displaystyle \frac{m}{\gamma }}-k\sum _{i=1}^m(1+R_i)ln{Z}_{1,1}^X(\beta )=0,\end{array}$$

(8)

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial l}{\partial \beta }}=-{\displaystyle \frac{m}{\beta }}+{\displaystyle \frac{1}{{\beta }^2}}\sum _{i=1}^m{\displaystyle \frac{1}{x_i}}-{\displaystyle \frac{2}{{\beta }^2}}\sum _{i=1}^m{\displaystyle \frac{W_{2,2}^X(\beta )}{x_i}}-{\displaystyle \frac{2k\gamma }{{\beta }^2}}\sum _{i=1}^m\left({\displaystyle \frac{1+R_i}{x_i}}\right)W_{1,2}^X(\beta )=0.\end{array}$$

(9)

where $Z_{a,b}^X(\beta )={\displaystyle \frac{1+exp(-{\displaystyle \frac{a}{\beta x_i}})}{1-exp(-{\displaystyle \frac{b}{\beta x_i}})}}$ and $W_{a,b}^X(\beta )={\displaystyle \frac{exp(-{\displaystyle \frac{a}{\beta x_i}})}{1-exp(-{\displaystyle \frac{b}{\beta x_i}})}}$.

Solving Equation (8), the MLE of $\gamma$ is:

$$\begin{array}{c}\hfill \widehat{\gamma }={\displaystyle \frac{m}{k{\sum }_{i=1}^m(R_i+1)ln{Z}_{1,1}^X(\beta )}}\end{array}$$

(10)

Putting Equation (10) into Equation (9), the MLE of $\beta$ can be written as:

$$\begin{array}{c}\hfill g(\beta )={\displaystyle \frac{1}{m}}\sum _{i=1}^m{\displaystyle \frac{1}{x_i}}\left[1-2W_{2,2}^X(\beta )\right]-{\displaystyle \frac{2{\sum }_{i=1}^m\left({\displaystyle \frac{R_i+1}{x_i}}\right)W_{1,2}^X(\beta )}{{\sum }_{i=1}^m\left(R_i+1\right)ln{Z}_{1,1}^X(\beta )}}\end{array}$$

Since $g(\beta )$ is related to $\widehat{\gamma }$ and $\beta$, the MLE of $\beta$ cannot be obtained directly. Using the “optimal” command in R software, the MLEs of $\widehat{\gamma }$ and $\widehat{\beta }$ can be solved easily.

Then, substituting $\widehat{\gamma }$ and $\widehat{\beta }$ into Equations (4) and (5), the MLEs of $R(t)$ and $H(t)$ are:

$$\begin{array}{c}\hfill \widehat{R}(t)={\left[{\displaystyle \frac{1-exp(-{\displaystyle \frac{1}{\widehat{\beta }t}})}{1+exp(-{\displaystyle \frac{1}{\widehat{\beta }t}})}}\right]}^{\widehat{\gamma }},\widehat{H}(t)={\displaystyle \frac{2\widehat{\gamma }}{\widehat{\beta }{t}^2}}exp\left(-{\displaystyle \frac{1}{\widehat{\beta }t}}\right){\left[1-exp\left(-{\displaystyle \frac{2}{\widehat{\beta }t}}\right)\right]}^{-1}\end{array}$$

Asymptotic Interval Estimation

We mainly present the asymptotic intervals for parameters in this subsection. Let $\theta =(\gamma ,\beta )$, the Fisher information matrix is shown below:

$$\begin{array}{c}\hfill I(\theta )=E\left[\begin{array}{cc}-{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial {\gamma }^2}}& -{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial \gamma \partial \beta }}\\ -{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial \beta \partial \gamma }}& -{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{{\partial }^2\beta }}\end{array}\right]\end{array}$$

The elements in the matrix are:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial {\gamma }^2}}& =-{\displaystyle \frac{m}{{\gamma }^2}},\hfill \\ \hfill {\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial \gamma \partial \beta }}& ={\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial \beta \partial \gamma }}=-{\displaystyle \frac{2k}{{\beta }^2}}\sum _{i=1}^m\left[\left({\displaystyle \frac{R_i+1}{x_i}}\right)W_{1,2}^X(\beta )\right],\hfill \\ \hfill {\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial {\beta }^2}}& ={\displaystyle \frac{m}{{\beta }^2}}-{\displaystyle \frac{2}{{\beta }^3}}\sum _{i=1}^m\left[{\displaystyle \frac{1}{x_i}}-{\displaystyle \frac{2k\gamma }{{\beta }^3}}\sum _{i=1}^n\left({\displaystyle \frac{Z_{2,2}^X(\beta )}{\beta x_i}}-2\right)\left({\displaystyle \frac{R_i+1}{x_i}}\right)W_{1,2}^X(\beta )\right]\hfill \\ \hfill & -{\displaystyle \frac{4}{{\beta }^3}}\sum _{i=1}^m\left[{\displaystyle \frac{W_{2,2}^X(\beta )}{x_i}}\left({\displaystyle \frac{W_{0,2}^X(\beta )}{\beta x_i}}-1\right)\right].\hfill \end{array}$$

Using $\widehat{\theta }=(\widehat{\gamma },\widehat{\beta })$ to represent the MLE of $\theta =(\gamma ,\beta )$. Then we write the observed Fisher information matrix as follows:

$$\begin{array}{c}\hfill I(\widehat{\theta })={\left[\begin{array}{cc}-{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial {\gamma }^2}}& -{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial \gamma \partial \beta }}\\ -{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{\partial \beta \partial \gamma }}& -{\displaystyle \frac{{\partial }^{2}l(\gamma ,\beta )}{{\partial }^2\beta }}\end{array}\right]}_{\theta =\widehat{\theta }}\end{array}$$

Thus, the observed variance-covariance matrix of $\widehat{\theta }$ can be obtained as:

$$\begin{array}{c}\hfill I^{-1}(\widehat{\theta })=\left[\begin{array}{cc}\widehat{V}ar(\widehat{\gamma })& \widehat{C}ov(\widehat{\gamma },\widehat{\beta })\\ \widehat{C}ov(\widehat{\beta },\widehat{\gamma })& \widehat{V}ar(\widehat{\beta })\end{array}\right]\end{array}$$

(11)

From the properties of MLE and normal distribution, we can deduce that $\widehat{\theta }$ approximately obey the bivariate normal distribution which have $\theta$ and $I^{-1}(\widehat{\theta })$ as the expectation and variance-covariance matrix. Then the two-sided equal tail asymptotic intervals of $\gamma$ and $\beta$ under confidence level $\alpha$ can be constructed as: $\left[\widehat{\gamma }\pm u_{\alpha /2}\sqrt{\widehat{V}ar(\widehat{\gamma })}\right]$ and $\left[\widehat{\beta }\pm u_{\alpha /2}\sqrt{\widehat{V}ar(\widehat{\beta })}\right]$. Here, $u_{\alpha /2}$ represents the upper $(\alpha /2)$-th percentile of standard normal distribution.

3. Bayesian Estimation

Under progressive first-failure censoring scheme, we mainly discuss the Bayesian estimations of the unknown parameters and reliability characteristics of IEHLD in this section.

In addition, three loss functions are used here. The first one is Squared Error loss function (SELF): $L(\alpha ,\widehat{\alpha })={\left(\widehat{\alpha }-\alpha \right)}^2$, which is a symmetric loss widely used in statistical research and practical cases. The second is Linex loss function (LLF): $L(\alpha ,\widehat{\alpha })=exp\left(h\left(\widehat{\alpha }-\alpha \right)\right)-h\left(\widehat{\alpha }-\alpha \right)-1$. It is an asymmetric loss first proposed by Klebanov [16], and its mathematical properties were studied by Klebanov and Rachev [17] and Zellner [18]. The last one is called General Entropy loss function (GELF): $L(\alpha ,\widehat{\alpha })={\left(\widehat{\alpha }/\alpha \right)}^q-qln\left(\widehat{\alpha }/\alpha \right)-1$, which is a generalization of the entropy loss and is also an asymmetric loss introduced by Calabria and Pulcini [19].

For our estimation, the prior distributions of unknown parameters need to be determined first. In this case, the joint conjugate prior of $\gamma$ and $\beta$ is not easy to obtain. Consider the gamma distribution has good flexibility and can be a prior distribution suitable for $\gamma$ and $\beta$, we assume that $\gamma$ and $\beta$ follows gamma distribution $G_1(a_1,b_1)$ and $G_2(a_2,b_2)$. They can be written as:

$$\begin{array}{cc}\hfill G_1(\gamma )={\displaystyle \frac{b_1^{a_1}}{\mathsf{\Gamma }(a_1)}}{\gamma }^{a_1-1}{e}^{-b_1\gamma },& \gamma >0,a_1,b_1>0,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill G_2(\beta )={\displaystyle \frac{b_2^{a_2}}{\mathsf{\Gamma }(a_2)}}{\beta }^{a_2-1}{e}^{-b_2\beta },& \beta >0,a_2,b_2>0.\hfill \end{array}$$

(13)

Then the joint prior distribution of $\gamma$ and $\beta$ reads:

$$\begin{array}{c}\hfill G(\gamma ,\beta )\propto {\gamma }^{a_1-1}{\beta }^{a_2-1}{e}^{-(b_1\gamma +b_2\beta )}\end{array}$$

(14)

Using Equations (6) and (14), the joint posterior distribution of $\gamma$ and $\beta$ can be obtained as:

$$\begin{array}{ccc}\hfill \pi (\gamma ,\beta |\overrightarrow{X})& =& {\displaystyle \frac{G(\gamma ,\beta )L(\overrightarrow{X}|\gamma ,\beta )}{{\int }_0^{\infty }{\int }_0^{\infty }G(\gamma ,\beta )L(\overrightarrow{X}|\gamma ,\beta )d\gamma d\beta }}\hfill \\ & =& D{\gamma }^{m+a_1-1}{\beta }^{-m+a_2-1}exp(-b_1\gamma -b_2\beta -\sum _{i=1}^m{\displaystyle \frac{1}{\beta x_i}})\hfill \\ & & \times \prod _{i=1}^m{\displaystyle \frac{{\left(1-exp(-{\displaystyle \frac{1}{\beta x_i}})\right)}^{\gamma k(1+R_i)-1}}{x_i{\left(1+exp(-{\displaystyle \frac{1}{\beta x_i}})\right)}^{\gamma k(1+R_i)+1}}}\hfill \end{array}$$

(15)

Where $D=1/({\int }_0^{\infty }{\int }_0^{\infty }G(\gamma ,\beta )L(\overrightarrow{X}|\gamma ,\beta )d\gamma d\beta )$ have no concern with $\gamma$ and $\beta$, is a normalizing constant.

Thus, using Equation (15), Bayesian estimation of any function of $\gamma$ and $\beta$ under different loss functions can be obtained. For example, let $\varphi (\gamma ,\beta )$ represents the function related to $\gamma$ and $\beta$, then the Bayesian estimation under SELF is the posterior expectation of $\varphi (\gamma ,\beta )$, say,

$$\begin{array}{c}\hfill E\left[\varphi (\gamma ,\beta )|\tilde{X}\right]={\displaystyle \frac{{\int }_0^{+\infty }{\int }_0^{+\infty }\varphi (\gamma ,\beta )G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }{{\int }_0^{+\infty }{\int }_0^{+\infty }G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }}\end{array}$$

(16)

Let $\varphi (\gamma ,\beta )=\gamma$ and $H(t)$, the Bayesian estimations of $\gamma$ and $H(t)$ under SELF are:

$$\begin{array}{cc}\hfill {\widehat{\gamma }}_{SELF}& ={\displaystyle \frac{{\int }_0^{+\infty }{\int }_0^{+\infty }\gamma G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }{{\int }_0^{+\infty }{\int }_0^{+\infty }G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }},\hfill \\ \hfill \widehat{H}{(t)}_{SELF}& ={\displaystyle \frac{{\int }_0^{+\infty }{\int }_0^{+\infty }H(t)G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }{{\int }_0^{+\infty }{\int }_0^{+\infty }G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }}.\hfill \end{array}$$

In the same way, the Bayesian estimations of $\beta$ and $R(t)$ under SELF can be obtained.

While under loss function LLF and GELF, the Bayesian estimations of $\alpha$ are:

$$\begin{array}{cc}\hfill {\widehat{\alpha }}_{LLF}& =-{\displaystyle \frac{1}{h}}ln{E}_{\alpha }[exp(-h\alpha )|\tilde{X}],\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\widehat{\alpha }}_{GELF}& ={\left(E_{\alpha }[{\alpha }^{-q}|\tilde{X}]\right)}^{-1/q}\hfill \end{array}$$

(18)

Replace $\alpha$ with $\gamma$ and $H(t)$, the Bayesian estimations under LLF, GELF can be considered to be:

$$\begin{array}{cc}\hfill {\widehat{\gamma }}_{LLF}& =-{\displaystyle \frac{1}{h}}ln\left[{\displaystyle \frac{{\int }_0^{+\infty }{\int }_0^{+\infty }{e}^{-h\gamma }G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }{{\int }_0^{+\infty }{\int }_0^{+\infty }G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }}\right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill \widehat{H}{(t)}_{LLF}& =-{\displaystyle \frac{1}{h}}ln\left[{\displaystyle \frac{{\int }_0^{+\infty }{\int }_0^{+\infty }{e}^{-hH(t)}G\left(\gamma ,\beta \right)L(\tilde{x}|\gamma ,\beta )d\gamma d\beta }{{\int }_0^{+\infty }{\int }_0^{+\infty }G\left(\gamma ,\beta \right)L(\tilde{x}|\gamma ,\beta )d\gamma d\beta }}\right]\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill {\widehat{\gamma }}_{GELF}& ={\left[{\displaystyle \frac{{\int }_0^{+\infty }{\int }_0^{+\infty }{\gamma }^{-q}G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }{{\int }_0^{+\infty }{\int }_0^{+\infty }G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }}\right]}^{-1/q},\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill \widehat{H}{(t)}_{GELF}& ={\left[{\displaystyle \frac{{\int }_0^{+\infty }{\int }_0^{+\infty }H{(t)}^{-q}G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }{{\int }_0^{+\infty }{\int }_0^{+\infty }G\left(\gamma ,\beta \right)L(\tilde{X}|\gamma ,\beta )d\gamma d\beta }}\right]}^{-1/q}\hfill \end{array}$$

(22)

the Bayesian estimations of $\beta$ and $R(t)$ can be obtained likewise.

From above, it can be seen that the form of Bayesian estimation is the ratio of two multiple integrals, thus it is unfeasible to obtain an explicit solution. In the next two subsections, we use Lindley approximation method and M-H method to approximate Bayesian estimation.

3.1. Lindley Approximation Method

In many studies, Bayesian estimation of distribution is usually in the form of multiple integral ratio, which is difficult to get analytical solution. The Lindley approximation method applied in this subsection was given by Lindley [20]. It is a point estimation which can be used to solve this problem.

Using Lindley approximation method, the Bayesian estimation of $\varphi (\gamma ,\beta )$ under SELF can be written as:

$$\begin{array}{ccc}\hfill {\widehat{\varphi }}_{LB}& =& \widehat{\varphi }+1/2\left[{\widehat{\sigma }}_{11}\left({\widehat{\varphi }}_{11}+2{\widehat{\varphi }}_1{\varrho }_1\right)+{\widehat{\sigma }}_{12}\left({\widehat{\varphi }}_{12}+2{\widehat{\varphi }}_1{\varrho }_2\right)+{\widehat{\sigma }}_{21}\left({\widehat{\varphi }}_{21}+2{\widehat{\varphi }}_2{\varrho }_1\right)\right.\hfill \\ & & +{\widehat{\sigma }}_{22}\left({\widehat{\varphi }}_{22}+2{\widehat{\varphi }}_2{\varrho }_2\right)+\left({\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{21}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{11}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{12}\right)\hfill \\ & & +\left.\left({\widehat{\ell }}_{21}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{12}{\widehat{\sigma }}_{21}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{21}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{22}\right)\right]\hfill \end{array}$$

(23)

where

$$\begin{array}{ccc}\hfill {\varphi }_1& =& {\displaystyle \frac{\partial \varphi }{\partial \gamma }},{\varphi }_2={\displaystyle \frac{\partial \varphi }{\partial \beta }},\hfill \\ \hfill {\varphi }_{11}& =& {\displaystyle \frac{{\partial }^2\varphi }{\partial {\gamma }^2}},{\varphi }_{12}={\displaystyle \frac{{\partial }^2\varphi }{\partial \gamma \partial \beta }},{\varphi }_{21}={\displaystyle \frac{{\partial }^2\varphi }{\partial \beta \partial \gamma }},{\varphi }_{22}={\displaystyle \frac{{\partial }^2\varphi }{\partial {\beta }^2}},\hfill \\ \hfill {\varrho }_1& =& {\displaystyle \frac{\partial \varrho }{\partial \gamma }}={\displaystyle \frac{(a_1-1)}{\gamma }}-b_1,{\varrho }_2={\displaystyle \frac{\partial \varrho }{\partial \beta }}={\displaystyle \frac{(a_2-1)}{\beta }}-b_2,\hfill \\ \hfill {\ell }_{30}& =& {\displaystyle \frac{{\partial }^3\ell (\gamma ,\beta )}{\partial {\gamma }^3}}={\displaystyle \frac{2m}{{\gamma }^3}},{\ell }_{21}=0,\hfill \\ \hfill {\ell }_{12}& =& {\displaystyle \frac{{\partial }^3\ell (\gamma ,\beta )}{\partial \gamma \partial {\beta }^2}}={\displaystyle \frac{2k}{{\beta }^3}}\sum _{i=1}^m\left[\left({\displaystyle \frac{1+R_i}{x_i}}\right)W_{1,2}^X(\beta )\left(2-{\displaystyle \frac{Z_{2,2}^X(\beta )}{\beta x_i}}\right)\right],\hfill \\ \hfill {\ell }_{03}& =& {\displaystyle \frac{{\partial }^3\ell (\gamma ,\beta )}{\partial {\beta }^3}}=-{\displaystyle \frac{2m}{{\beta }^3}}+{\displaystyle \frac{6}{{\beta }^4}}\sum _{i=1}^m{\displaystyle \frac{1}{x_i}}-{\displaystyle \frac{2k\gamma }{{\beta }^5}}\sum _{i=1}^m\left\{\left({\displaystyle \frac{1+R_i}{x_i}}\right)W_{1,2}^X(\beta )\right.\hfill \\ & & \times \left.\left[2W_{2,2}^X(\beta )\left({\displaystyle \frac{W_{0,2}^X(\beta )}{\beta x_i}}-1\right)+\left({\displaystyle \frac{Z_{2,2}^X(\beta )}{\beta x_i}}-2\right)\left(Z_{2,2}^X(\beta )-3\beta x_i\right)-1\right]\right\}\hfill \\ & & -{\displaystyle \frac{4}{{\beta }^4}}\sum _{i=1}^m\left[{\displaystyle \frac{W_{2,2}^X(\beta )W_{0,2}^X(\beta )}{x_i^2}}\left({\displaystyle \frac{2W_{2,2}^X(\beta )}{\beta x_i}}-1\right)\right]-{\displaystyle \frac{8}{{\beta }^3}}\sum _{i=1}^m\left[{\displaystyle \frac{W_{2,2}^X(\beta )}{x_i}}{\left({\displaystyle \frac{W_{0,2}^X(\beta )}{\beta x_i}}-1\right)}^2\right].\hfill \end{array}$$

Here $\varrho =lnG(\gamma ,\beta )=(a_1-1)ln\gamma +(a_2-1)ln\beta -b_1\gamma -b_2\beta$ and ${\widehat{\sigma }}_{ij}$ is the element in $ij$-th position of the corresponding $I^{-1}(\widehat{\theta })$ given in (11).

Then, let $\varphi (\gamma ,\beta )=\gamma$ and $\beta$, the Bayesian estimations of $\gamma$ and $\beta$ under SELF using lindley approximation method can be easily obtained as:

$$\begin{array}{ccc}\hfill {\widehat{\gamma }}_{SELF}& =& \widehat{\gamma }+\left({\widehat{\varrho }}_1{\widehat{\sigma }}_{11}+{\widehat{\varrho }}_2{\widehat{\sigma }}_{12}\right)+1/2\left[{\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}^2+{\widehat{\ell }}_{12}\left({\widehat{\sigma }}_{11}{\widehat{\sigma }}_{22}+2{\widehat{\sigma }}_{21}^2\right)+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{12}{\widehat{\sigma }}_{22}\right],\hfill \\ \hfill {\widehat{\beta }}_{SELF}& =& \widehat{\beta }+\left({\widehat{\varrho }}_1{\widehat{\sigma }}_{21}+{\widehat{\varrho }}_2{\widehat{\sigma }}_{22}\right)+1/2\left[{\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}{\widehat{\sigma }}_{12}+3{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}^2\right]\hfill \end{array}$$

(24)

In the same way, let $\varphi (\gamma ,\beta )=R(t)$, we have:

$$\begin{array}{ccc}\hfill {\varphi }_1& =& -{\left[Z_{1,1}^t(\beta )\right]}^{-\gamma }ln{Z}_{1,1}^t(\beta ),{\varphi }_2=-{\displaystyle \frac{\gamma }{{\beta }^{2}t}}{N}_{1,1}^t(\beta ){\left[Z_{1,1}^t(\beta )\right]}^{-\gamma +1}\left({\left[Z_{1,1}^t(\beta )\right]}^{-1}+1\right),\hfill \\ \hfill {\varphi }_{11}& =& {\left[Z_{1,1}^t(\beta )\right]}^{-\gamma }{\left[ln{Z}_{1,1}^t(\beta )\right]}^2,\hfill \\ \hfill {\varphi }_{12}& =& -{\displaystyle \frac{2}{{\beta }^{2}t}}{W}_{2,2}^t(\beta ){\left[Z_{1,1}^t(\beta )\right]}^{-\gamma }\left[1-\gamma ln{Z}_{1,1}^t(\beta )\right]={\varphi }_{21},\hfill \\ \hfill {\varphi }_{22}& =& {\displaystyle \frac{2\gamma }{{\beta }^{3}t}}{N}_{1,1}^t(\beta ){\left[Z_{1,1}^t(\beta )\right]}^{-\gamma }\left({\displaystyle \frac{1}{\beta t}}\left[2\left(\gamma -1\right)W_{1,2}^t(\beta )W_{0,1}^t(\beta )-N_{0,1}^t(\beta )\right]+Z_{1,1}^t(\beta )+{\displaystyle \frac{1}{t}}\right).\hfill \end{array}$$

where $N_{a,b}^t(\beta )=\left[exp(-{\displaystyle \frac{a}{\beta t}})\right]/{\left[1+exp(-{\displaystyle \frac{1}{\beta t}})\right]}^b$.

Next, for $H(t)$, we have:

$$\begin{array}{ccc}\hfill \varphi (\gamma ,\beta )& =& H(t),\hfill \\ \hfill {\varphi }_1& =& {\displaystyle \frac{2}{\beta t^2}}{W}_{1,2}^t(\beta ),{\varphi }_2={\displaystyle \frac{2\gamma }{{\beta }^2{t}^2}}{W}_{1,2}^t(\beta )\left[{\displaystyle \frac{Z_{2,2}^t(\beta )}{\beta t}}-1\right],{\varphi }_{11}=0,\hfill \\ \hfill {\varphi }_{12}& =& {\displaystyle \frac{2}{{\beta }^2{t}^2}}{W}_{1,2}^t(\beta )\left[{\displaystyle \frac{Z_{2,2}^t(\beta )}{\beta t}}-1\right]={\varphi }_{21},\hfill \\ \hfill {\varphi }_{22}& =& {\displaystyle \frac{2\gamma }{{\beta }^3{t}^2}}{W}_{1,2}^t(\beta )\left[{\displaystyle \frac{1}{{\beta }^2{t}^2}}\left(8{\left[W_{1,2}^t(\beta )\right]}^3+1\right)-{\displaystyle \frac{4W_{2,2}^t(\beta )}{\beta t}}\left(W_{1,2}^t(\beta )+1\right)+2\right].\hfill \end{array}$$

Thus, the Lindley approximations of reliability and failure functions say $R(t)$ and $H(t)$ under SELF are:

$$\begin{array}{ccc}\hfill \widehat{R}{(t)}_{SELF}& =& \widehat{R}(t)+1/2\left[{\widehat{\sigma }}_{11}\left({\widehat{\varphi }}_{11}+2{\widehat{\varphi }}_1{\varrho }_1\right)+{\widehat{\sigma }}_{12}\left({\widehat{\varphi }}_{12}+2{\widehat{\varphi }}_1{\varrho }_2\right)+{\widehat{\sigma }}_{21}\left({\widehat{\varphi }}_{21}+2{\widehat{\varphi }}_2{\varrho }_1\right)\right.\hfill \\ & & +{\widehat{\sigma }}_{22}\left({\widehat{\varphi }}_{22}+2{\widehat{\varphi }}_2{\varrho }_2\right)+\left({\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{21}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{11}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{12}\right)\hfill \\ & & +\left.\left({\widehat{\ell }}_{21}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{12}{\widehat{\sigma }}_{21}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{21}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{22}\right)\right],\hfill \\ \hfill \widehat{H}{(t)}_{SELF}& =& \widehat{H}(t)+1/2\left[{\widehat{\sigma }}_{11}\left({\widehat{\varphi }}_{11}+2{\widehat{\varphi }}_1{\varrho }_1\right)+{\widehat{\sigma }}_{12}\left({\widehat{\varphi }}_{12}+2{\widehat{\varphi }}_1{\varrho }_2\right)+{\widehat{\sigma }}_{21}\left({\widehat{\varphi }}_{21}+2{\widehat{\varphi }}_2{\varrho }_1\right)\right.\hfill \\ & & +{\widehat{\sigma }}_{22}\left({\widehat{\varphi }}_{22}+2{\widehat{\varphi }}_2{\varrho }_2\right)+\left({\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{21}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{11}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{12}\right)\hfill \\ & & +\left.\left({\widehat{\ell }}_{21}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{12}{\widehat{\sigma }}_{21}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{21}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{22}\right)\right].\hfill \end{array}$$

Similarly, the Lindley approximations of parameters and reliability characteristic under LLF are:

For $\gamma$:

$$\begin{array}{ccc}\hfill \varphi \left(\gamma ,\beta \right)& =& e^{-h\gamma },\hfill \\ \hfill {\varphi }_1& =& -h{e}^{-h\gamma },{\varphi }_{11}=h^2{e}^{-h\gamma },{\varphi }_2={\varphi }_{12}={\varphi }_{21}={\varphi }_{22}=0.\hfill \end{array}$$

Then the Lindley approximation of $\gamma$ under LLF can be written as:

$$\begin{array}{ccc}\hfill {\widehat{\gamma }}_{LLF}& =& -{\displaystyle \frac{1}{h}}ln\left\{e^{-h\gamma }+1/2\left[\left(h^2{e}^{-h\gamma }-2h{e}^{-h\gamma }{\varrho }_1\right){\widehat{\sigma }}_{11}-2h{e}^{-h\gamma }{\varrho }_2{\widehat{\sigma }}_{12}\right.\right.\hfill \\ & & -h{e}^{-h\gamma }{\widehat{\sigma }}_{11}\left({\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{21}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}\right)\hfill \\ & & \left.\left.-h{e}^{-h\gamma }{\widehat{\sigma }}_{21}\left({\widehat{\ell }}_{21}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{12}{\widehat{\sigma }}_{21}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}\right)\right]\right\}\hfill \end{array}$$

For $R(t)$, we have:

$$\begin{array}{ccc}\hfill \varphi \left(\gamma ,\beta \right)& =& e^{-hR(t)},\hfill \\ \hfill {\varphi }_1& =& {\displaystyle \frac{ln{Z}_{11}^t(\beta )}{{\left[Z_{11}^t(\beta )\right]}^{\gamma }}}h{e}^{-hR(t)},{\varphi }_2={\displaystyle \frac{2h\gamma N_{12}^t(\beta )}{{\beta }^{2}t{\left[Z_{11}^t(\beta )\right]}^{\gamma -1}}}{e}^{-hR(t)},\hfill \\ \hfill {\varphi }_{11}& =& {\displaystyle \frac{{\left[ln{Z}_{11}^t(\beta )\right]}^2}{{\left[Z_{11}^t(\beta )\right]}^{\gamma }}}\left[h{Z}_{11}^t{(\beta )}^{-\gamma }-1\right]h{e}^{-hR(t)},\hfill \\ \hfill {\varphi }_{12}& =& {\varphi }_{21}={\displaystyle \frac{2h{W}_{12}^t(\beta )}{{\beta }^{2}t{\left[Z_{11}^t(\beta )\right]}^{\gamma }}}\left[\gamma \left(h{\left[Z_{11}^t(\beta )\right]}^{-\gamma }-1\right)ln{Z}_{11}^t(\beta )+1\right]e^{-hR(t)},\hfill \\ \hfill {\varphi }_{22}& =& {\displaystyle \frac{2h\gamma W_{12}^t(\beta )}{{\beta }^4{t}^2{\left[Z_{11}^t(\beta )\right]}^{\gamma }}}\left[Z_{22}^t(\beta )+2\gamma W_{12}^t(\beta )\left(h{\left[Z_{11}^t(\beta )\right]}^{-\gamma }-1\right)-2\beta t\right]e^{-hR(t)}.\hfill \end{array}$$

For H(t):

$$\begin{array}{ccc}\hfill \varphi \left(\gamma ,\beta \right)& =& e^{-hH(t)},\hfill \\ \hfill {\varphi }_1& =& -{\displaystyle \frac{2h}{\beta t^2}}{W}_{12}^t(\beta )e^{-hH(t)},{\varphi }_2=-{\displaystyle \frac{2h\gamma }{{\beta }^2{t}^2}}{W}_{12}^t(\beta )\left[{\displaystyle \frac{Z_{22}^t(\beta )}{\beta t}}-1\right]e^{-hH(t)},\hfill \\ \hfill {\varphi }_{11}& =& {\left[{\displaystyle \frac{2h}{\beta t^2}}{W}_{12}^t(\beta )\right]}^2{e}^{-hH(t)},\hfill \\ \hfill {\varphi }_{12}& =& {\varphi }_{21}={\displaystyle \frac{2h}{{\beta }^2{t}^2}}{W}_{12}^t(\beta )\left[{\displaystyle \frac{1-2W_{22}^t(\beta )}{\beta t}}-{\displaystyle \frac{2h\gamma }{\beta t^2}}{W}_{12}^t(\beta )\left({\displaystyle \frac{W_{22}^t(\beta )-1}{\beta t}}+1\right)+1\right]e^{-hH(t)},\hfill \\ \hfill {\varphi }_{22}& =& {\displaystyle \frac{2h\gamma }{{\beta }^3{t}^2}}{W}_{12}^t(\beta )\left[{\displaystyle \frac{1}{{\beta }^2{t}^2}}\left(-8W_{22}^t{(\beta )}^2-8W_{22}^t(\beta )-W_{12}^t(\beta )\right)\right.\hfill \\ & & +\left.{\displaystyle \frac{4}{\beta t}}\left(4W_{22}^t(\beta )+W_{12}^t(\beta )\right)+{\displaystyle \frac{2h\gamma }{\beta t^2}}{W}_{12}^t(\beta ){\left({\displaystyle \frac{2W_{22}^t(\beta )-1}{\beta t}}+1\right)}^2-2\right]e^{-hH(t)}.\hfill \end{array}$$

Thus, the Lindley approximation of reliability function $R(t)$ under LLF is:

$$\begin{array}{ccc}\hfill \widehat{R}{(t)}_{LLF}& =& -{\displaystyle \frac{1}{h}}ln\left\{\widehat{R}(t)+1/2\left[{\widehat{\sigma }}_{11}\left({\widehat{\varphi }}_{11}+2{\widehat{\varphi }}_1{\varrho }_1\right)+{\widehat{\sigma }}_{12}\left({\widehat{\varphi }}_{12}+2{\widehat{\varphi }}_1{\varrho }_2\right)+{\widehat{\sigma }}_{21}\left({\widehat{\varphi }}_{21}+2{\widehat{\varphi }}_2{\varrho }_1\right)\right.\right.\hfill \\ & & +{\widehat{\sigma }}_{22}\left({\widehat{\varphi }}_{22}+2{\widehat{\varphi }}_2{\varrho }_2\right)+\left({\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{21}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{11}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{12}\right)\hfill \\ & & +\left.\left.\left({\widehat{\ell }}_{21}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{12}{\widehat{\sigma }}_{21}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{21}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{22}\right)\right]\right\}\hfill \end{array}$$

the ${\widehat{\beta }}_{LLF}$ and $\widehat{H}{(t)}_{LLF}$ can be obtain likewise.

Then, under loss function GELF, the Lindley approximation of $\gamma$ is:

$$\begin{array}{ccc}\hfill \varphi (\gamma ,\beta )& =& {\gamma }^{-q},\hfill \\ \hfill {\varphi }_1& =& -q{\gamma }^{-1-q},{\varphi }_{11}=q(q+1){\gamma }^{-2-q},{\varphi }_2={\varphi }_{12}={\varphi }_{21}={\varphi }_{22}=0.\hfill \end{array}$$

$$\begin{array}{ccc}{\widehat{\gamma }}_{GELF}\hfill & =& \left\{{\gamma }^{-q}+1/2\left[{\widehat{\sigma }}_{11}\left(q(q+1){\gamma }^{-2-q}-2q{\gamma }^{-1-q}{\varrho }_1\right)-2q{\gamma }^{-q-1}{\varrho }_2{\widehat{\sigma }}_{12}\right.\right.\hfill \\ & & -q{\gamma }^{-q-1}{\widehat{\sigma }}_{11}\left({\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{21}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}\right)\hfill \\ & & {\left.\left.-q{\gamma }^{-q-1}{\widehat{\sigma }}_{21}\left({\widehat{\ell }}_{21}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{12}{\widehat{\sigma }}_{21}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}\right)\right]\right\}}^{-1/q}\hfill \end{array}$$

For R(t), we have:

$$\begin{array}{ccc}\hfill \varphi \left(\gamma ,\beta \right)& =& {\left[Z_{11}^t(\beta )\right]}^{\gamma q},\hfill \\ \hfill {\varphi }_1& =& q{\left[Z_{11}^t(\beta )\right]}^{\gamma q}ln{Z}_{11}^t(\beta ),{\varphi }_2={\displaystyle \frac{2\gamma q}{{\beta }^{2}t}}{N}_{12}^t(\beta ){\left[Z_{11}^t(\beta )\right]}^{\gamma q+1},\hfill \\ \hfill {\varphi }_{11}& =& q^2{\left[Z_{11}^t(\beta )\right]}^{\gamma q}{\left[ln{Z}_{11}^t(\beta )\right]}^2,\hfill \\ \hfill {\varphi }_{12}& =& {\varphi }_{21}={\displaystyle \frac{2q}{{\beta }^{2}t}}{N}_{12}^t(\beta ){\left[Z_{11}^t(\beta )\right]}^{\gamma q+1}\left[\gamma qln{Z}_{11}^t(\beta )+1\right],\hfill \\ \hfill {\varphi }_{22}& =& {\displaystyle \frac{\gamma q}{{\beta }^{2}t}}{W}_{11}^t(\beta ){\left[Z_{11}^t(\beta )\right]}^{\gamma q-2}\left[{\displaystyle \frac{2{\left[Z_{11}^t(\beta )\right]}^2{W}_{01}^t(\beta )}{{\beta }^{2}t}}\right.\hfill \\ & & \left.-Z_{11}^t(\beta )\left(2Z_{11}^t(\beta )-\gamma q+3\right)+\gamma q-1\right].\hfill \end{array}$$

For H(t):

$$\begin{array}{ccc}\hfill \varphi \left(\gamma ,\beta \right)& =& {\left({\displaystyle \frac{2\gamma }{\beta t^2}}{W}_{12}^t(\beta )\right)}^{-q},\hfill \\ \hfill {\varphi }_1& =& {\displaystyle \frac{q}{\gamma }}{\left({\displaystyle \frac{2\gamma }{\beta t^2}}{W}_{12}^t(\beta )\right)}^{-q},{\varphi }_2=-{\displaystyle \frac{q}{\beta }}{\left({\displaystyle \frac{2\gamma }{\beta t^2}}{W}_{12}^t(\beta )\right)}^{-q}\left({\displaystyle \frac{Z_{22}^t(\beta )}{\beta t}}-1\right),\hfill \\ \hfill {\varphi }_{11}& =& -{\displaystyle \frac{q(q+1)}{{\gamma }^2}}{\left({\displaystyle \frac{2\gamma }{\beta t^2}}{W}_{12}^t(\beta )\right)}^{-q},\hfill \\ \hfill {\varphi }_{12}& =& {\varphi }_{21}={\displaystyle \frac{q}{\beta }}{\left({\displaystyle \frac{2}{\beta t^2}}{W}_{12}^t(\beta )\right)}^{-q}\left({\displaystyle \frac{Z_{22}^t(\beta )}{\beta t}}-1\right)\left[\left(q+1\right){\gamma }^{-1-q}-1\right],\hfill \\ \hfill {\varphi }_{22}& =& q{\left({\displaystyle \frac{2\gamma }{\beta t^2}}{W}_{12}^t(\beta )\right)}^{-q}\left[\left(q+1\right){\left({\displaystyle \frac{Z_{22}^t(\beta )}{\beta t}}-1\right)}^2-{\displaystyle \frac{1}{{\beta }^4{t}^2}}\left(8W_{22}^t{(\beta )}^2+W_{02}^t(\beta )\right)\right.\hfill \\ & & +\left.{\displaystyle \frac{4}{{\beta }^{4}t}}\left(4W_{22}^t(\beta )-1\right)-{\displaystyle \frac{2}{{\beta }^2}}\right].\hfill \end{array}$$

Thus, the Lindley approximation of reliability function $R(t)$ under GELF is:

$$\begin{array}{ccc}\hfill \widehat{R}{(t)}_{GELF}& =& \left\{\widehat{R}(t)+1/2\left[{\widehat{\sigma }}_{11}\left({\widehat{\varphi }}_{11}+2{\widehat{\varphi }}_1{\varrho }_1\right)+{\widehat{\sigma }}_{12}\left({\widehat{\varphi }}_{12}+2{\widehat{\varphi }}_1{\varrho }_2\right)+{\widehat{\sigma }}_{21}\left({\widehat{\varphi }}_{21}+2{\widehat{\varphi }}_2{\varrho }_1\right)\right.\right.\hfill \\ & & +{\widehat{\sigma }}_{22}\left({\widehat{\varphi }}_{22}+2{\widehat{\varphi }}_2{\varrho }_2\right)+\left({\widehat{\ell }}_{30}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{21}{\widehat{\sigma }}_{12}+{\widehat{\ell }}_{12}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{11}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{12}\right)\hfill \\ & & +{\left.\left.\left({\widehat{\ell }}_{21}{\widehat{\sigma }}_{11}+2{\widehat{\ell }}_{12}{\widehat{\sigma }}_{21}+{\widehat{\ell }}_{03}{\widehat{\sigma }}_{22}\right)\left({\widehat{\varphi }}_1{\widehat{\sigma }}_{21}+{\widehat{\varphi }}_2{\widehat{\sigma }}_{22}\right)\right]\right\}}^{-1/q}\hfill \end{array}$$

in the same way, ${\widehat{\beta }}_{GELF}$ and $\widehat{H}{(t)}_{GELF}$ can be obtained easily.

According to the above results, we only need to bring the corresponding parts into the corresponding equation, the required Bayesian estimations of each loss function can be obtained.

To better complete Bayesian estimation, we use the M-H method for interval estimation in the next subsection. Then, the Bayesian credible intervals of parameters are carried out.

3.2. Metropolis-Hastings Method

The M-H algorithm is a simulation algorithm based on MCMC technology, originally proposed by Metropolis and then generalized by Hasting. It is widely used in the field of stochastic process. Two important applications are sampling from given probability distributions, and estimating complex integrals by stochastic simulation. Nassar [21], Kohansal [22] and Panahi [14] generated the samples from Weibull distribution, Kumaraswamy distribution and inverted exponentiated Rayleigh distribution respectively under adaptive Type-II progressive hybrid censoring scheme using M-H algorithm with Gibbs sampler.

M-H algorithm uses the joint posterior density function and a proposal distribution to simulate samples. In addition, it uses the existing sample as the distribution parameter of the candidate sample and has an acceptance function to calculate the probability of deciding whether the candidate sample can be accepted. This may cause invalid samples, but it can make the sample distribution move faster towards the original distribution.

The joint posterior function of $\gamma$ and $\beta$ in (15) can be re expressed as:

$$\begin{array}{c}\hfill \pi (\gamma ,\beta |\overrightarrow{X})\propto {\pi }_1(\gamma |\beta ,\overrightarrow{X})\times {\pi }_2(\beta |\gamma ,\overrightarrow{X})\end{array}$$

(25)

where

$$\begin{array}{ccc}\hfill {\pi }_1(\gamma |\beta ,\overrightarrow{X})& =& {\gamma }^{m+a_1-1}exp\left[-\gamma \left(b_1-k\sum _{i=1}^m\left[\left(R_i+1\right)ln\left(Z_{1,1}^X(\beta )\right)\right]\right)\right]\hfill \\ \hfill {\pi }_2(\beta |\gamma ,\overrightarrow{X})& =& {\beta }^{-m+a_2-1}exp\left[-b_2\beta -\sum _{i=1}^m\left({\displaystyle \frac{1}{\beta x_i}}+ln\left[x_i\left(1-exp(-{\displaystyle \frac{2}{\beta x_i}})\right)\right]\right)\right]\hfill \end{array}$$

It is obviously that ${\pi }_1(\gamma |\beta ,\overrightarrow{X})$ is a gamma density function, thus we can generate the samples of $\gamma$ easily. However, the conditional posterior density ${\pi }_2(\beta |\gamma ,\overrightarrow{X})$ cannot be transformed into a well-known distribution through simplification and analysis, so it is hard to get samples by general methods. So we choose to apply the M-H algorithm within Gibbs sampling to generate samples subject to ${\pi }_1(\gamma |\beta ,\overrightarrow{x})$ and ${\pi }_2(\beta |\gamma ,\overrightarrow{X})$.

The steps are arranged in the following:

Step-1: Initialize the values of $\left({\gamma }^{(0)},{\beta }^{(0)}\right)$.

Step-2: Let $j=1$

Step-3: Generate ${\gamma }^{(j)}$ from $\mathbf{Gamma}\left(a_1+m,b_1-k{\sum }_{i=1}^m(1+R_i)ln(Z_{1,1}^X(\beta ))\right)$

under stage j and the progressive first-failure censoring data with given

censoring scheme.

Step-4: Get ${\beta }^{(j)}$ from ${\pi }_2\left({\beta }^{(j-1)}|{\gamma }^{(j)},data\right)$ using following steps:

Step-4.1: Generate a candidate sample ${\beta }^{\mathbf{c}}$ from $\mathbf{Normal}\left({\beta }^{(j-1)},Variance(\widehat{\beta })\right)$.

Step-4.2: Generate a data v from $\mathbf{Uniform}\left(0,1\right)$.

Step-4.3: Let $\alpha =min\left(1,{\displaystyle \frac{{\pi }_2({\beta }^{\mathbf{c}}|{\gamma }^{(j)},data)}{{\pi }_2({\beta }^{(j-1)}|{\gamma }^{(j)},data)}}\right)$. Put ${\beta }^{(j)}={\beta }^{\mathbf{c}}$ if $v\le \alpha$.

Else, put ${\beta }^{(j)}={\beta }^{(j-1)}$.

Step-5: Let $j=j+1$.

Step-6: Do step-3 to step-5 Q times, record the $\left({\gamma }^{(j)},{\beta }^{(j)}\right)$, $j=1,\cdots ,Q$

Then, the Bayesian estimation of $\gamma$ and $\lambda$ using M-H method under SELF, LLF, and GELF can be obtained as:

$$\begin{array}{c}{\tilde{\gamma }}_{M-SELF}={\displaystyle \frac{1}{Q-B}}\sum _{j=B+1}^Q{\gamma }^{(j)},{\tilde{\beta }}_{M-SELF}={\displaystyle \frac{1}{Q-B}}\sum _{j=B+1}^Q{\gamma }^{(j)}\hfill \\ {\tilde{\gamma }}_{M-LLF}=-{\displaystyle \frac{1}{h}}ln\left[{\displaystyle \frac{1}{Q-B}}\sum _{j=B+1}^{Q}exp\left(-h{\gamma }^{(j)}\right)\right],{\tilde{\beta }}_{M-LLF}=-{\displaystyle \frac{1}{h}}ln\left[{\displaystyle \frac{1}{Q-B}}\sum _{j=B+1}^{Q}exp\left(-h{\beta }^{(j)}\right)\right]\hfill \\ {\tilde{\gamma }}_{M-GELF}={\left[{\displaystyle \frac{1}{Q-B}}\sum _{j=B+1}^Q{\left({\gamma }^{(j)}\right)}^{-q}\right]}^{-1/q},{\tilde{\beta }}_{M-GELF}={\left[{\displaystyle \frac{1}{Q-B}}\sum _{j=B+1}^Q{\left({\beta }^{(j)}\right)}^{-q}\right]}^{-1/q}\hfill \end{array}$$

The constant B refers here is the burn-in-stage of Markov Chain which can eliminate the impact of choosing an initial value $({\gamma }^{(0)},{\beta }^{(0)})$ while ensuring the convergence of the algorithm.

One can use the algorithm proposed by Albert [23] with R package $LearnBayes$ to complete the estimation, which is very convenient.

3.3. Bayesian Credible Interval

Here we consider the Bayesian intervals of $\gamma$ and $\beta$ constructed by M-H method. Using the results of Section 3.2, let ${\gamma }_{(1)},{\gamma }_{(2)},\cdots ,{\gamma }_{(Q)}$ and ${\beta }_{(1)},{\beta }_{(2)},\cdots ,{\beta }_{(Q)}$ represent the order statistics of ${\gamma }^{(j)}$ and ${\beta }^{(j)}$, $j=1,\cdots ,Q$. The $100(1-\alpha )\%$ Bayesian credible intervals of parameters using MCMC technique can be constructed as:

$$\begin{array}{c}\hfill \left[{\gamma }_{\left(Q\alpha /2\right)},{\gamma }_{\left(Q(1-\alpha )/2\right)}\right]and\left[{\beta }_{\left(Q\alpha /2\right)},{\beta }_{\left(Q(1-\alpha )/2\right)}\right].\end{array}$$

4. Simulation Studies

We mainly analyze the effect of the estimators through a Monte Carlo simulation study in this section. Some examples are presented for illustration. First, we determine the true value of parameters and the combination of censoring parameters $(k,n,m,\Re )$. Then, applying the algorithm presented by [24] and the method of generating progressive first-failure samples mentioned in [25], we can generate the samples of IEHLD under progressive first-failure censoring. All analysis is done through software R.

First, take the parameter values as $\gamma =0.3$, $\beta =0.8$, and $t=1.5$ for the reliability and hazard functions, then take the hyper-parameters $(a_1,b_1,a_2,b_2)$ of the prior distributions $G_1(\gamma )$ and $G_2(\beta )$ as (1.06, 3.6, 16, 20). For the sake of comparing the effects of parameter values, the parameter values are changed to $\gamma =0.3$ and $\beta =1.5$ for MLE. While for Bayesian estimation, the hyper-parameters $(a_1,b_1,a_2,b_2)$ are changed to (0, 0, 0, 0) for noninformative estimates. It should be noted that the expectation of prior distribution should be equal to the parameter value which means that $a_1/b_1=\gamma$ and $a_2/b_2=\beta$. For the censoring parameters, we choose 3 and 5 for k, and determine two sets of combination for n and m say n=30, m=15, 20, 30; $n=50$, m=25, 30, 50 with different $R_i$.

For convenience, when expressing different censoring schemes, the $(1*5)$ represents (1, 1, 1, 1, 1) and ((1, 0)*3) represents (1, 0, 1, 0, 1, 0).

(1) The MLEs are tabulated in

Table 1. Maximum likelihood estimates and MSEs of parameters and reliability and hazard functions when $\gamma =0.3$, $\beta =1.5$, $t=1.5$, $R(t)=0.6337$, $H(t)=0.1936$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{ML}}$		${\widehat{\mathit{\beta }}}_{\mathbf{ML}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{ML}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{ML}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	15	(15,0*14)	0.3619	0.0342	1.4697	0.1403	0.6082	0.0091	0.2304	0.0124
			(0*6,10,5,0*7)	0.3763	0.0413	1.4531	0.1186	0.6017	0.0103	0.2390	0.0149
			((2,0)*7,1)	0.3840	0.0593	1.4451	0.1285	0.6013	0.0110	0.2430	0.0194
		20	(10,0*19)	0.3402	0.0135	1.4734	0.1189	0.6172	0.0053	0.2173	0.0050
			((2,0)*2,(1,0)*6,0*4)	0.3480	0.0198	1.4622	0.1146	0.6153	0.0062	0.2217	0.0071
			((0*3,1*4)*2,1,1,0*4)	0.3543	0.0208	1.4477	0.1045	0.6114	0.0064	0.2257	0.0075
		30	(0*30)	0.3184	0.0064	1.4746	0.0773	0.6303	0.0029	0.2040	0.0024
	50	25	(25,0*24)	0.3322	0.0090	1.4719	0.0850	0.6197	0.0040	0.2127	0.0034
			(0*8,2,3*7,2,0*8)	0.3405	0.0166	1.4777	0.0758	0.6158	0.0057	0.2179	0.0063
			(3*6,0*12,1*7)	0.3418	0.0141	1.4710	0.0816	0.6143	0.0050	0.2187	0.0053
		30	(20,0*29)	0.3224	0.0075	1.4854	0.0752	0.6254	0.0033	0.2067	0.0028
			((2,0,0)*10)	0.3350	0.0104	1.4737	0.0684	0.6169	0.0040	0.2147	0.0039
			(0*29,20)	0.3436	0.0166	1.4734	0.0807	0.6141	0.0051	0.2197	0.0061
		50	(0*50)	0.3154	0.0037	1.4811	0.0538	0.6284	0.0018	0.2026	0.0014
5	30	15	(15,0*14)	0.3709	0.0452	1.4658	0.1099	0.6044	0.0102	0.2359	0.0163
			(0*6,10,5,0*7)	0.4012	0.0647	1.4465	0.1042	0.5883	0.0143	0.2547	0.0235
			((2,0)*7,1)	0.4429	0.1922	1.4273	0.1170	0.5777	0.0188	0.2785	0.0607
		20	(10,0*19)	0.3482	0.0196	1.4694	0.0836	0.6120	0.0067	0.2226	0.0073
			((2,0)*2,(1,0)*6,0*4)	0.3647	0.0298	1.4480	0.0872	0.6049	0.0084	0.2325	0.0110
			((0*3,1*4)*2,1,1,0*4)	0.3644	0.0324	1.4589	0.0874	0.6050	0.0089	0.2325	0.0121
		30	(0*30)	0.3308	0.0099	1.4790	0.0694	0.6203	0.0038	0.2119	0.0037
	50	25	(25,0*24)	0.3401	0.0147	1.4683	0.0705	0.6158	0.0055	0.2177	0.0056
			(0*8,2,3*7,2,0*8)	0.3491	0.0222	1.4812	0.0615	0.6098	0.0075	0.2235	0.0084
			(3*6,0*12,1*7)	0.3474	0.0188	1.4680	0.0649	0.6115	0.0062	0.2222	0.0070
		30	(20,0*29)	0.3280	0.0100	1.4833	0.0671	0.6221	0.0039	0.2103	0.0038
			((2,0,0)*10)	0.3450	0.0145	1.4634	0.0521	0.6103	0.0054	0.2211	0.0055
			(0*29,20)	0.3609	0.0251	1.4530	0.0715	0.6053	0.0076	0.2305	0.0094
		50	(0*50)	0.3127	0.0045	1.4872	0.04174	0.6302	0.0021	0.2010	0.0017

and

Table 2. Maximum likelihood estimates and MSEs of parameters and reliability and hazard functions when $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{ML}}$		${\widehat{\mathit{\beta }}}_{\mathbf{ML}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{ML}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{ML}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	15	(15,0*14)	0.3521	0.0260	0.7766	0.0377	0.7480	0.0038	0.2006	0.0057
			(0*6,10,5,0*7)	0.3832	0.0531	0.7729	0.0388	0.7390	0.0051	0.2152	0.0111
			((2,0)*7,1)	0.3899	0.0576	0.7708	0.0341	0.7351	0.0050	0.2190	0.0121
		20	(10,0*19)	0.3410	0.0169	0.7849	0.0326	0.7480	0.0029	0.1965	0.0040
			((2,0)*2,(1,0)*6,0*4)	0.3463	0.0213	0.7744	0.0278	0.7484	0.0027	0.1985	0.0044
			((0*3,1*4)*2,1,1,0*4)	0.3517	0.0215	0.7845	0.0289	0.7434	0.0031	0.2021	0.0050
		30	(0*30)	0.3250	0.0085	0.7861	0.0253	0.6395	0.0034	0.1885	0.0021
	50	25	(25,0*24)	0.3266	0.0089	0.7988	0.0250	0.7495	0.0021	0.1905	0.0023
			(0*8,2,3*7,2,0*8)	0.3384	0.0120	0.7752	0.0300	0.7504	0.0024	0.1949	0.0028
			(3*6,0*12,1*7)	0.3343	0.0123	0.7832	0.0207	0.7492	0.0022	0.1942	0.0032
		30	(20,0*29)	0.3253	0.0087	0.7798	0.0220	0.6407	0.0028	0.1891	0.0023
			((2,0,0)*10)	0.3296	0.0094	0.7779	0.0195	0.6409	0.0027	0.1913	0.0023
			(0*29,20)	0.3403	0.0149	0.7825	0.0222	0.6400	0.0030	0.1967	0.0035
		50	(0*50)	0.3135	0.0038	0.7938	0.0134	0.6335	0.0017	0.1844	0.0011
5	30	15	(15,0*14)	0.3735	0.0495	0.7845	0.0318	0.7366	0.0051	0.2127	0.0104
			(0*6,10,5,0*7)	0.4006	0.0608	0.7640	0.0289	0.7292	0.0060	0.2262	0.0137
			((2,0)*7,1)	0.4381	0.1288	0.7615	0.0318	0.7208	0.0092	0.2434	0.0278
		20	(10,0*19)	0.3505	0.0202	0.7774	0.0251	0.7442	0.0032	0.2021	0.0051
			((2,0)*2,(1,0)*6,0*4)	0.3604	0.0237	0.7718	0.0242	0.7409	0.0035	0.2071	0.0059
			((0*3,1*4)*2,1,1,0*4)	0.3729	0.0508	0.7698	0.0259	0.7392	0.0040	0.2120	0.0100
		30	(0*30)	0.3264	0.0093	0.7878	0.0197	0.6373	0.0025	0.1902	0.0024
	50	25	(25,0*24)	0.3377	0.0129	0.7890	0.0189	0.7451	0.0023	0.1968	0.0034
			(0*8,2,3*7,2,0*8)	0.3449	0.0184	0.7783	0.0226	0.7468	0.0026	0.1989	0.0044
			(3*6,0*12,1*7)	0.3507	0.0209	0.7857	0.0199	0.7414	0.0032	0.2032	0.0054
		30	(20,0*29)	0.3303	0.0101	0.7876	0.0172	0.6368	0.0023	0.1926	0.0026
			((2,0,0)*10)	0.3502	0.0199	0.7773	0.0169	0.6404	0.0023	0.2028	0.0049
			(0*29,20)	0.3490	0.0208	0.7834	0.0185	0.6387	0.0025	0.2020	0.0053
		50	(0*50)	0.3212	0.0048	0.7841	0.0120	0.6366	0.0016	0.1884	0.0013

, These tables show:

• When k and n are fixed but m increases, the $MSE$s of $\gamma$, $\beta$, $R(t)$ and $H(t)$ decrease.
• When k is fixed but n increases, the $MSE$s decrease.
• When n and m are fixed but k increases, the $MSE$s decrease.
• When $\gamma$ is fixed but $\beta$ increases, the $MSE$s increase.

(2) The Bayesian estimations using Lindley approximation method under SELF, LLF and GELF are tabulated in

Table 3. Lindley approximation values and MSEs of parameters and reliability and hazard functions under SELF using noninformative priors when $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=0$, $b_1=0$, $a_2=0$, $b_2=0$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3318	0.0126	0.8374	0.0371	0.7496	0.0025	0.2044	0.0038
			((2,0)*2,(1,0)*6,0*4)	0.3502	0.0233	0.8261	0.0390	0.7457	0.0030	0.2124	0.0062
			((0*3,1*4)*2,1,1,0*4)	0.3472	0.0243	0.8445	0.0402	0.7435	0.0028	0.2116	0.0064
		30	(0*30)	0.3217	0.0070	0.8298	0.0294	0.7518	0.0017	0.1970	0.0022
	50	25	(25,0*24)	0.3266	0.0096	0.8127	0.0244	0.7525	0.0027	0.1990	0.0031
			(0*8,2,3*7,2,0*8)	0.3455	0.0161	0.8187	0.0245	0.7432	0.0023	0.2082	0.0047
			(3*6,0*12,1*7)	0.3329	0.0135	0.8257	0.0268	0.7489	0.0022	0.2013	0.0039
		30	(20,0*29)	0.3212	0.0070	0.8250	0.0244	0.7512	0.0018	0.1959	0.0022
			((2,0,0)*10)	0.3288	0.0096	0.8262	0.0229	0.7485	0.0017	0.1993	0.0028
			(0*29,20)	0.3379	0.0149	0.8216	0.0250	0.7479	0.0020	0.2040	0.0043
		50	(0*50)	0.3095	0.0033	0.8170	0.0163	0.7556	0.0010	0.1876	0.0010
5	30	20	(10,0*19)	0.3550	0.0283	0.8218	0.0303	0.7420	0.0031	0.2143	0.0074
			((2,0)*2,(1,0)*6,0*4)	0.3620	0.0320	0.8272	0.0317	0.7392	0.0035	0.2175	0.0084
			((0*3,1*4)*2,1,1,0*4)	0.3690	0.0374	0.8201	0.0295	0.7388	0.0039	0.2208	0.0101
		30	(0*30)	0.3271	0.0093	0.8298	0.0221	0.7476	0.0019	0.1988	0.0029
	50	25	(25,0*24)	0.3385	0.0124	0.8115	0.0211	0.7465	0.0020	0.2041	0.0037
			(0*8,2,3*7,2,0*8)	0.3737	0.0331	0.8075	0.0212	0.7347	0.0038	0.2223	0.0093
			(3*6,0*12,1*7)	0.3954	0.0690	0.8255	0.0285	0.7312	0.0052	0.2335	0.0173
		30	(20,0*29)	0.3258	0.0087	0.8202	0.0188	0.7495	0.0017	0.1970	0.0027
			((2,0,0)*10)	0.3399	0.0160	0.8186	0.0194	0.7452	0.0023	0.2039	0.0046
			(0*29,20)	0.3729	0.0395	0.8174	0.0255	0.7360	0.0036	0.2213	0.0105
		50	(0*50)	0.3157	0.0044	0.8159	0.0117	0.7511	0.0010	0.1904	0.0013

,

Table 4. Lindley approximation values and MSEs of parameters and reliability and hazard functions under SELF when $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3089	0.0040	0.8028	0.0055	0.7495	0.0021	0.1991	0.0020
			((2,0)*2,(1,0)*6,0*4)	0.3076	0.0040	0.8122	0.0067	0.7481	0.0022	0.1979	0.0020
			((0*3,1*4)*2,1,1,0*4)	0.3099	0.0056	0.8197	0.0044	0.7432	0.0024	0.2008	0.0022
		30	(0*30)	0.3096	0.0032	0.8132	0.0053	0.7512	0.0014	0.1942	0.0014
	50	25	(25,0*24)	0.3119	0.0039	0.8105	0.0040	0.7509	0.0018	0.1948	0.0017
			(0*8,2,3*7,2,0*8)	0.3150	0.0041	0.8102	0.0051	0.7491	0.0017	0.1957	0.0018
			(3*6,0*12,1*7)	0.3112	0.0041	0.8161	0.0054	0.7493	0.0019	0.1945	0.0018
		30	(20,0*29)	0.3151	0.0039	0.8125	0.0047	0.7488	0.0017	0.1954	0.0016
			((2,0,0)*10)	0.3106	0.0034	0.8171	0.0055	0.7510	0.0014	0.1922	0.0014
			(0*29,20)	0.3091	0.0031	0.8187	0.0054	0.7503	0.0014	0.1933	0.0014
		50	(0*50)	0.3040	0.0023	0.8148	0.0063	0.7516	0.0009	0.1894	0.0009
5	30	20	(10,0*19)	0.3091	0.0039	0.8152	0.0049	0.7486	0.0020	0.1960	0.0018
			((2,0)*2,(1,0)*6,0*4)	0.3018	0.0059	0.8178	0.0047	0.7480	0.0019	0.1938	0.0017
			((0*3,1*4)*2,1,1,0*4)	0.2907	0.0370	0.8174	0.0049	0.7513	0.0018	0.1899	0.0024
		30	(0*30)	0.3126	0.0034	0.8134	0.0056	0.7505	0.0014	0.1933	0.0014
	50	25	(25,0*24)	0.3161	0.0044	0.8147	0.0048	0.7484	0.0018	0.1953	0.0018
			(0*8,2,3*7,2,0*8)	0.3032	0.0264	0.8153	0.0054	0.7503	0.0016	0.1895	0.0053
			(3*6,0*12,1*7)	0.3109	0.0058	0.8166	0.0051	0.7496	0.0016	0.1929	0.0018
		30	(20,0*29)	0.3188	0.0045	0.8131	0.0060	0.7474	0.0016	0.1956	0.0018
			((2,0,0)*10)	0.3146	0.0054	0.8163	0.0058	0.7481	0.0014	0.1937	0.0018
			(0*29,20)	0.3020	0.0080	0.8176	0.0047	0.7505	0.0012	0.1904	0.0017
		50	(0*50)	0.3097	0.0027	0.8159	0.0061	0.7520	0.0009	0.1879	0.0009

,

Table 5. Lindley approximation values and MSEs of parameters and reliability and hazard functions under LLF when $h=-0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3122	0.0043	0.8008	0.0183	0.7564	0.0019	0.2071	0.0029
			((2,0)*2,(1,0)*6,0*4)	0.3060	0.0057	0.8088	0.0112	0.7595	0.0018	0.2066	0.0029
			((0*3,1*4)*2,1,1,0*4)	0.3025	0.0104	0.8140	0.0212	0.7589	0.0019	0.2084	0.0030
		30	(0*30)	0.3115	0.0033	0.8165	0.0061	0.7566	0.0013	0.1985	0.0018
	50	25	(25,0*24)	0.3157	0.0041	0.8103	0.0055	0.7565	0.0016	0.2010	0.0022
			(0*8,2,3*7,2,0*8)	0.3159	0.0039	0.8181	0.0056	0.7580	0.0014	0.2045	0.0024
			(3*6,0*12,1*7)	0.3168	0.0041	0.8187	0.0059	0.7558	0.0014	0.2047	0.0025
		30	(20,0*29)	0.3114	0.0034	0.8165	0.0053	0.7573	0.0014	0.1958	0.0016
			((2,0,0)*10)	0.3161	0.0047	0.8179	0.0061	0.7555	0.0014	0.2019	0.0022
			(0*29,20)	0.3138	0.0033	0.8252	0.0060	0.7562	0.0012	0.2057	0.0024
		50	(0*50)	0.3096	0.0024	0.8090	0.0062	0.7579	0.0009	0.1904	0.0010
5	30	20	(10,0*19)	0.3062	0.0107	0.8154	0.0065	0.7612	0.0017	0.2063	0.0028
			((2,0)*2,(1,0)*6,0*4)	0.3006	0.0206	0.8173	0.0054	0.7629	0.0018	0.2039	0.0146
			((0*3,1*4)*2,1,1,0*4)	0.2985	0.0096	0.8200	0.0054	0.7643	0.0015	0.2082	0.0027
		30	(0*30)	0.3153	0.0037	0.8154	0.0065	0.7571	0.0012	0.2013	0.0021
	50	25	(25,0*24)	0.3171	0.0056	0.8126	0.0066	0.7570	0.0016	0.2038	0.0027
			(0*8,2,3*7,2,0*8)	0.3068	0.0196	0.8180	0.0056	0.7606	0.0014	0.2066	0.0030
			(3*6,0*12,1*7)	0.3132	0.0043	0.8202	0.0057	0.7612	0.0013	0.2043	0.0026
		30	(20,0*29)	0.3167	0.0040	0.8138	0.0058	0.7572	0.0013	0.1995	0.0020
			((2,0,0)*10)	0.3163	0.0041	0.8164	0.0059	0.7589	0.0011	0.2033	0.0024
			(0*29,20)	0.3083	0.0064	0.8194	0.0050	0.7629	0.0010	0.2086	0.0027
		50	(0*50)	0.3120	0.0030	0.8158	0.0065	0.7560	0.0008	0.1921	0.0012

,

Table 6. Lindley approximation values and MSEs of parameters and reliability and hazard functions under LLF when $h=0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3107	0.0042	0.8070	0.0045	0.7521	0.0021	0.2046	0.0027
			((2,0)*2,(1,0)*6,0*4)	0.3107	0.0042	0.8070	0.0045	0.7521	0.0021	0.2046	0.0027
			((0*3,1*4)*2,1,1,0*4)	0.3033	0.0034	0.8139	0.0044	0.7555	0.0017	0.2048	0.0025
		30	(0*30)	0.3056	0.0030	0.8109	0.0045	0.7562	0.0014	0.1946	0.0015
	50	25	(25,0*24)	0.3130	0.0040	0.8076	0.0042	0.7527	0.0018	0.1994	0.0021
			(0*8,2,3*7,2,0*8)	0.3166	0.0041	0.8116	0.0047	0.7516	0.0016	0.2055	0.0026
			(3*6,0*12,1*7)	0.3151	0.0043	0.8129	0.0049	0.7518	0.0017	0.2033	0.0025
		30	(20,0*29)	0.3113	0.0036	0.8096	0.0048	0.7536	0.0015	0.1958	0.0017
			((2,0,0)*10)	0.3112	0.0034	0.8119	0.0058	0.7545	0.0012	0.1983	0.0019
			(0*29,20)	0.3065	0.0031	0.8209	0.0046	0.7557	0.0013	0.1998	0.0020
		50	(0*50)	0.3077	0.0022	0.8087	0.0057	0.75527	0.0008	0.1895	0.0009
5	30	20	(10,0*19)	0.3097	0.0041	0.8128	0.0042	0.7542	0.0019	0.2059	0.0028
			((2,0)*2,(1,0)*6,0*4)	0.3036	0.0049	0.8155	0.0042	0.7571	0.0020	0.2056	0.0029
			((0*3,1*4)*2,1,1,0*4)	0.2972	0.0059	0.8155	0.0042	0.7610	0.0022	0.2036	0.0030
		30	(0*30)	0.3101	0.0033	0.8156	0.0055	0.7543	0.0013	0.1978	0.0018
	50	25	(25,0*24)	0.3142	0.0040	0.8105	0.0047	0.7533	0.0016	0.2010	0.0023
			(0*8,2,3*7,2,0*8)	0.3085	0.0038	0.8152	0.0050	0.7575	0.0014	0.2020	0.0023
			(3*6,0*12,1*7)	0.3102	0.0041	0.8115	0.0050	0.7568	0.0015	0.2031	0.0027
		30	(20,0*29)	0.3135	0.0039	0.8068	0.0053	0.7545	0.0014	0.1979	0.0019
			((2,0,0)*10)	0.3150	0.0037	0.8158	0.0050	0.7527	0.0013	0.2019	0.0022
			(0*29,20)	0.3075	0.0036	0.8140	0.0045	0.7581	0.0012	0.2059	0.0025
		50	(0*50)	0.3083	0.0027	0.8073	0.0054	0.7561	0.0008	0.1900	0.0011

,

Table 7. Lindley approximation values and MSEs of parameters and reliability and hazard functions under LLF when $h=1$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3072	0.0038	0.8009	0.0032	0.7526	0.0020	0.2025	0.0024
			((2,0)*2,(1,0)*6,0*4)	0.3094	0.0038	0.8085	0.0031	0.7503	0.0019	0.2061	0.0027
			((0*3,1*4)*2,1,1,0*4)	0.3032	0.0036	0.8102	0.0033	0.7554	0.0041	0.2032	0.0026
		30	(0*30)	0.3063	0.0028	0.8084	0.0035	0.7537	0.0014	0.1950	0.0014
	50	25	(25,0*24)	0.3132	0.0041	0.8001	0.0039	0.7515	0.0018	0.1995	0.0021
			(0*8,2,3*7,2,0*8)	0.3110	0.0037	0.8059	0.0046	0.7532	0.0015	0.2018	0.0023
			(3*6,0*12,1*7)	0.3081	0.0036	0.8062	0.0045	0.7545	0.0015	0.1987	0.0021
		30	(20,0*29)	0.3100	0.0036	0.8059	0.0042	0.7525	0.0016	0.1952	0.0017
			((2,0,0)*10)	0.3100	0.0036	0.8070	0.0049	0.7535	0.0014	0.1976	0.0019
			(0*29,20)	0.3054	0.0029	0.8141	0.0050	0.7545	0.0013	0.1995	0.0019
		50	(0*50)	0.3068	0.0022	0.8078	0.0058	0.7541	0.0009	0.1891	0.0009
5	30	20	(10,0*19)	0.3049	0.0035	0.8136	0.0031	0.7537	0.0018	0.2014	0.0023
			((2,0)*2,(1,0)*6,0*4)	0.3052	0.0035	0.8143	0.0033	0.7540	0.0018	0.2046	0.0026
			((0*3,1*4)*2,1,1,0*4)	0.3026	0.0029	0.8158	0.0033	0.7555	0.0018	0.2044	0.0022
		30	(0*30)	0.3081	0.0033	0.8124	0.0049	0.7533	0.0014	0.1964	0.0018
	50	25	(25,0*24)	0.3121	0.0038	0.8064	0.0048	0.7522	0.0015	0.1995	0.0021
			(0*8,2,3*7,2,0*8)	0.3075	0.0037	0.8139	0.0049	0.7552	0.0017	0.2006	0.0023
			(3*6,0*12,1*7)	0.3115	0.0036	0.8043	0.0046	0.7553	0.0022	0.2028	0.0025
		30	(20,0*29)	0.3098	0.0037	0.8073	0.0053	0.7539	0.0014	0.1957	0.0018
			((2,0,0)*10)	0.3138	0.0039	0.8067	0.0051	0.7523	0.0013	0.2014	0.0023
			(0*29,20)	0.3057	0.0029	0.8136	0.0039	0.7566	0.0017	0.2030	0.0023
		50	(0*50)	0.3112	0.0028	0.8053	0.0057	0.7526	0.0009	0.1919	0.0011

,

Table 8. Lindley approximation values and MSEs of parameters and reliability and hazard functions under GELF when $q=-0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3064	0.0040	0.8071	0.0033	0.7510	0.0021	0.1820	0.0014
			((2,0)*2,(1,0)*6,0*4)	0.3043	0.0040	0.8117	0.0038	0.7509	0.0021	0.1813	0.0014
			((0*3,1*4)*2,1,1,0*4)	0.3018	0.0034	0.8155	0.0041	0.7516	0.0020	0.1802	0.0012
		30	(0*30)	0.3041	0.0029	0.8116	0.0039	0.7539	0.0014	0.1800	0.0010
	50	25	(25,0*24)	0.3078	0.0037	0.8056	0.0039	0.7530	0.0018	0.1818	0.0013
			(0*8,2,3*7,2,0*8)	0.3074	0.0037	0.8037	0.0046	0.7517	0.0019	0.1825	0.0013
			(3*6,0*12,1*7)	0.3082	0.0042	0.8112	0.0049	0.7523	0.0019	0.1822	0.0014
		30	(20,0*29)	0.3103	0.0034	0.8024	0.0047	0.7529	0.0014	0.1829	0.0011
			((2,0,0)*10)	0.3074	0.0035	0.8138	0.0051	0.7526	0.0014	0.1817	0.0011
			(0*29,20)	0.3082	0.0033	0.8138	0.0048	0.7514	0.0015	0.1825	0.0011
		50	(0*50)	0.3055	0.0020	0.8088	0.0061	0.7545	0.0009	0.1804	0.0006
5	30	20	(10,0*19)	0.3048	0.0037	0.8148	0.0038	0.7510	0.0020	0.1814	0.0013
			((2,0)*2,(1,0)*6,0*4)	0.3069	0.0050	0.8151	0.0040	0.7514	0.0027	0.1818	0.0013
			((0*3,1*4)*2,1,1,0*4)	0.2985	0.0035	0.8216	0.0036	0.7535	0.0019	0.1781	0.0012
		30	(0*30)	0.3078	0.0034	0.8152	0.0053	0.7521	0.0014	0.1820	0.0011
	50	25	(25,0*24)	0.3118	0.0044	0.8108	0.0044	0.7503	0.0019	0.1842	0.0015
			(0*8,2,3*7,2,0*8)	0.3055	0.0036	0.8142	0.0045	0.7526	0.0017	0.1810	0.0012
			(3*6,0*12,1*7)	0.3102	0.0039	0.8131	0.0047	0.7505	0.0016	0.1836	0.0013
		30	(20,0*29)	0.3084	0.0036	0.8096	0.0050	0.7533	0.0014	0.1820	0.0012
			((2,0,0)*10)	0.3113	0.0037	0.8090	0.0052	0.7516	0.0013	0.1838	0.0012
			(0*29,20)	0.3045	0.0029	0.8152	0.0041	0.7538	0.0013	0.1804	0.0010
		50	(0*50)	0.3064	0.0027	0.8065	0.0063	0.7554	0.0008	0.1807	0.0008

,

Table 9. Lindley approximation values and MSEs of parameters and reliability and hazard functions under GELF when $q=0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.2997	0.0041	0.8012	0.0022	0.7495	0.0024	0.1781	0.0015
			((2,0)*2,(1,0)*6,0*4)	0.2997	0.0037	0.8044	0.0024	0.7495	0.0020	0.1781	0.0013
			((0*3,1*4)*2,1,1,0*4)	0.2972	0.0039	0.8092	0.0021	0.7503	0.0021	0.1768	0.0014
		30	(0*30)	0.3030	0.0031	0.8022	0.0034	0.7499	0.0015	0.1796	0.0010
	50	25	(25,0*24)	0.3036	0.0036	0.7973	0.0034	0.7495	0.0019	0.1798	0.0012
			(0*8,2,3*7,2,0*8)	0.3009	0.0034	0.8015	0.0026	0.7503	0.0018	0.1786	0.0012
			(3*6,0*12,1*7)	0.3042	0.0039	0.8006	0.0043	0.7489	0.0018	0.1802	0.0013
		30	(20,0*29)	0.3003	0.0030	0.8000	0.0040	0.7523	0.0015	0.1779	0.0010
			((2,0,0)*10)	0.3001	0.0033	0.8028	0.0050	0.7522	0.0015	0.1778	0.0011
			(0*29,20)	0.2988	0.0031	0.8073	0.0038	0.7522	0.0014	0.1774	0.0010
		50	(0*50)	0.3032	0.0022	0.8025	0.0056	0.7524	0.0009	0.1795	0.0007
5	30	20	(10,0*19)	0.2957	0.0038	0.8058	0.0030	0.7527	0.0022	0.1757	0.0013
			((2,0)*2,(1,0)*6,0*4)	0.2976	0.0037	0.8078	0.0030	0.7546	0.0101	0.1769	0.0013
			((0*3,1*4)*2,1,1,0*4)	0.2942	0.0035	0.8120	0.0028	0.7566	0.0061	0.1751	0.0012
		30	(0*30)	0.3016	0.0033	0.8064	0.0046	0.7502	0.0014	0.1789	0.0011
	50	25	(25,0*24)	0.3012	0.0039	0.8043	0.0042	0.7499	0.0017	0.1787	0.0013
			(0*8,2,3*7,2,0*8)	0.2971	0.0037	0.8052	0.0036	0.7526	0.0018	0.1764	0.0013
			(3*6,0*12,1*7)	0.3036	0.0040	0.8034	0.0041	0.7485	0.0018	0.1802	0.0013
		30	(20,0*29)	0.3007	0.0036	0.8005	0.0049	0.7526	0.0014	0.1781	0.0011
			((2,0,0)*10)	0.3031	0.0042	0.8053	0.0051	0.7503	0.0017	0.1797	0.0013
			(0*29,20)	0.2968	0.0031	0.8090	0.0073	0.7528	0.0019	0.1764	0.0010
		50	(0*50)	0.3025	0.0025	0.8029	0.0053	0.7528	0.0009	0.1792	0.0008

and

Table 10. Lindley approximation values and MSEs of parameters and reliability and hazard functions under GELF when $q=1$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{LB}}$		${\widehat{\mathit{\beta }}}_{\mathbf{LB}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{LB}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{LB}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.2970	0.0040	0.7960	0.0016	0.7498	0.0022	0.1762	0.0014
			((2,0)*2,(1,0)*6,0*4)	0.2955	0.0037	0.8023	0.0019	0.7490	0.0020	0.1755	0.0013
			((0*3,1*4)*2,1,1,0*4)	0.2918	0.0039	0.8062	0.0017	0.7512	0.0020	0.1735	0.0013
		30	(0*30)	0.2999	0.0034	0.7947	0.0034	0.7507	0.0016	0.1777	0.0011
	50	25	(25,0*24)	0.3010	0.0034	0.7922	0.0035	0.7488	0.0018	0.1783	0.0011
			(0*8,2,3*7,2,0*8)	0.2989	0.0038	0.8010	0.0019	0.7482	0.0021	0.1776	0.0013
			(3*6,0*12,1*7)	0.2994	0.0039	0.7986	0.0036	0.7487	0.0017	0.1777	0.0013
		30	(20,0*29)	0.3026	0.0033	0.7903	0.0045	0.7500	0.0016	0.1790	0.0011
			((2,0,0)*10)	0.3015	0.0037	0.7957	0.0044	0.7494	0.0015	0.1787	0.0012
			(0*29,20)	0.2950	0.0032	0.8076	0.0036	0.7502	0.0015	0.1757	0.0010
		50	(0*50)	0.2990	0.0020	0.8022	0.0052	0.7523	0.0009	0.1775	0.0006
5	30	20	(10,0*19)	0.2942	0.0043	0.8010	0.0026	0.7509	0.0020	0.1747	0.0014
			((2,0)*2,(1,0)*6,0*4)	0.2934	0.0042	0.8060	0.0026	0.7500	0.0020	0.1745	0.0014
			((0*3,1*4)*2,1,1,0*4)	0.2958	0.0040	0.8118	0.0034	0.7506	0.0034	0.1760	0.0013
		30	(0*30)	0.2989	0.0032	0.7970	0.0046	0.7506	0.0014	0.1773	0.0010
	50	25	(25,0*24)	0.2983	0.0042	0.7967	0.0043	0.7502	0.0017	0.1769	0.0014
			(0*8,2,3*7,2,0*8)	0.2971	0.0040	0.8052	0.0034	0.7489	0.0019	0.1767	0.0013
			(3*6,0*12,1*7)	0.2963	0.0041	0.8007	0.0041	0.7496	0.0017	0.1761	0.0013
		30	(20,0*29)	0.3014	0.0040	0.7948	0.0051	0.7498	0.0017	0.1786	0.0013
			((2,0,0)*10)	0.2972	0.0040	0.7949	0.0048	0.7522	0.0014	0.1762	0.0013
			(0*29,20)	0.2957	0.0036	0.8027	0.0035	0.7544	0.0041	0.1757	0.0011
		50	(0*50)	0.3003	0.0026	0.7971	0.0053	0.7530	0.0009	0.1781	0.0008

. While under LLF, we choose $h=-0.5$, $0.5$ and 1. While under GELF, we consider $q=-0.5$, $0.5$ and 1. The tables show that:

• When k and n are fixed but m increases, the $MSE$s of $\gamma$, $\beta$, $R(t)$ and $H(t)$ decrease.
• When k is fixed but n increases, the $MSE$s decrease slightly.
• When n and m are fixed but k increases, the $MSE$s have no obvious trend on the whole.
• The $MSE$s when using noninformative priors are bigger than using the informative priors under SELF.
• While under LLF, $h=1$ is the best mode with the smallest $MSE$s, and $h=-0.5$ is a little bit worse relatively.
• While under GELF, there is no significant difference in $MSE$s among the three modes, the estimation effect seems better when $q=0.5$.
• Among the three loss functions, there is little difference between the $MSE$s under LLF and GELF, the estimation effect of GELF is slightly better, while SELF is the worst of the three.

(3) The Bayesian estimations using M-H method under SELF, LLF and GELF are tabulated in

Table 11. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method and noninformative priors under SELF when $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=0$, $b_1=0$, $a_2=0$, $b_2=0$, $R(t)=0$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3515	0.0182	0.8602	0.0469	0.7388	0.0033	0.2009	0.0042
			((2,0)*2,(1,0)*6,0*4)	0.3670	0.0392	0.8460	0.0429	0.7387	0.0030	0.2063	0.0062
			((0*3,1*4)*2,1,1,0*4)	0.3686	0.0297	0.8408	0.0451	0.7387	0.0034	0.2079	0.0062
		30	(0*30)	0.3306	0.0086	0.8417	0.0340	0.7476	0.0019	0.1908	0.0020
	50	25	(25,0*24)	0.3345	0.0094	0.8360	0.0303	0.7456	0.0022	0.1934	0.0024
			(0*8,2,3*7,2,0*8)	0.3550	0.0187	0.8247	0.0267	0.7407	0.0024	0.2036	0.0044
			(3*6,0*12,1*7)	0.3617	0.0214	0.8155	0.0280	0.7400	0.0024	0.2060	0.0047
		30	(20,0*29)	0.3299	0.0075	0.8283	0.0267	0.7477	0.0017	0.1912	0.0018
			((2,0,0)*10)	0.3474	0.0128	0.8132	0.0234	0.7439	0.0019	0.1995	0.0030
			(0*29,20)	0.3525	0.0170	0.8182	0.0260	0.7442	0.0021	0.2012	0.0038
		50	(0*50)	0.3185	0.0040	0.8183	0.0182	0.7517	0.0011	0.1860	0.0010
5	30	20	(10,0*19)	0.3791	0.0407	0.8226	0.0338	0.7359	0.0038	0.2135	0.0083
			((2,0)*2,(1,0)*6,0*4)	0.3817	0.0377	0.8248	0.0319	0.7344	0.0039	0.2154	0.0083
			((0*3,1*4)*2,1,1,0*4)	0.3992	0.0552	0.8230	0.0340	0.7303	0.0046	0.2233	0.0115
		30	(0*30)	0.3491	0.0130	0.8124	0.0236	0.7431	0.0019	0.2006	0.0031
	50	25	(25,0*24)	0.3559	0.0187	0.8157	0.0215	0.7394	0.0028	0.2047	0.0047
			(0*8,2,3*7,2,0*8)	0.3875	0.0630	0.8060	0.0200	0.7317	0.0040	0.2197	0.0120
			(3*6,0*12,1*7)	0.3839	0.0350	0.8024	0.0201	0.7323	0.0036	0.2180	0.0080
		30	(20,0*29)	0.3455	0.0144	0.8175	0.0204	0.7424	0.0020	0.1995	0.0034
			((2,0,0)*10)	0.3649	0.0220	0.8039	0.0190	0.7383	0.0028	0.2090	0.0054
			(0*29,20)	0.4122	0.0523	0.7870	0.0222	0.7272	0.0042	0.2305	0.0113
		50	(0*50)	0.3200	0.0052	0.8138	0.01417	0.7518	0.0010	0.1871	0.0013

,

Table 12. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method under SELF when $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3529	0.0192	0.8509	0.0475	0.7416	0.0029	0.2002	0.0040
			((2,0)*2,(1,0)*6,0*4)	0.3668	0.0230	0.8338	0.0423	0.7378	0.0035	0.2077	0.0053
			((0*3,1*4)*2,1,1,0*4)	0.3700	0.0240	0.8340	0.0420	0.7376	0.0033	0.2091	0.0054
		30	(0*30)	0.3287	0.0081	0.8515	0.0338	0.7457	0.0019	0.1907	0.0020
	50	25	(25,0*24)	0.3367	0.0116	0.8394	0.0278	0.7437	0.0022	0.1948	0.0028
			(0*8,2,3*7,2,0*8)	0.3515	0.0176	0.8222	0.0242	0.7424	0.0027	0.2019	0.0043
			(3*6,0*12,1*7)	0.3586	0.0187	0.8157	0.0248	0.7394	0.0026	0.2054	0.0045
		30	(20,0*29)	0.3317	0.0078	0.8334	0.0248	0.7445	0.0020	0.1930	0.0020
			((2,0,0)*10)	0.3428	0.0125	0.8165	0.0234	0.7457	0.0018	0.1973	0.0029
			(0*29,20)	0.3593	0.0244	0.8266	0.0302	0.7412	0.0023	0.2042	0.0051
		50	(0*50)	0.3170	0.0038	0.8269	0.0181	0.7500	0.0011	0.1858	0.0010
5	30	20	(10,0*19)	0.3743	0.0299	0.8153	0.0328	0.7374	0.0034	0.2115	0.0065
			((2,0)*2,(1,0)*6,0*4)	0.3823	0.0394	0.8216	0.0298	0.7341	0.0039	0.2158	0.0086
			((0*3,1*4)*2,1,1,0*4)	0.3922	0.0485	0.8182	0.0312	0.7327	0.0041	0.2199	0.0098
		30	(0*30)	0.3479	0.0125	0.8115	0.0232	0.7438	0.0018	0.1999	0.0030
	50	25	(25,0*24)	0.3568	0.0224	0.8126	0.0221	0.7404	0.0030	0.2047	0.0053
			(0*8,2,3*7,2,0*8)	0.3764	0.0372	0.8092	0.0206	0.7353	0.0039	0.2147	0.0087
			(3*6,0*12,1*7)	0.3904	0.0452	0.7992	0.0227	0.7325	0.0040	0.2203	0.0099
		30	(20,0*29)	0.3427	0.0128	0.8164	0.0192	0.7437	0.0020	0.1982	0.0031
			((2,0,0)*10)	0.3604	0.0204	0.8019	0.0178	0.7405	0.0025	0.2066	0.0048
			(0*29,20)	0.3935	0.0749	0.8003	0.0223	0.7350	0.0037	0.2202	0.0129
		50	(0*50)	0.3245	0.0054	0.8121	0.0129	0.7487	0.0011	0.1899	0.0014

,

Table 13. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method under LLF when $h=-0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3572	0.0182	0.8618	0.0474	0.7396	0.0030	0.2029	0.0041
			((2,0)*2,(1,0)*6,0*4)	0.3591	0.0220	0.8562	0.0459	0.7418	0.0028	0.2031	0.0048
			((0*3,1*4)*2,1,1,0*4)	0.3760	0.0330	0.8498	0.0447	0.7376	0.0033	0.2103	0.0064
		30	(0*30)	0.3319	0.0093	0.8537	0.0348	0.7469	0.0019	0.1915	0.0022
	50	25	(25,0*24)	0.3443	0.0129	0.8417	0.0317	0.7423	0.0025	0.1983	0.0032
			(0*8,2,3*7,2,0*8)	0.3630	0.0213	0.8228	0.0269	0.7411	0.0024	0.2062	0.0047
			(3*6,0*12,1*7)	0.3592	0.0199	0.8142	0.0263	0.7444	0.0022	0.2035	0.0042
		30	(20,0*29)	0.3350	0.0090	0.8347	0.0238	0.7454	0.0019	0.1939	0.0022
			((2,0,0)*10)	0.3481	0.0128	0.8180	0.0248	0.7451	0.0018	0.1994	0.0030
			(0*29,20)	0.3647	0.0215	0.8224	0.0268	0.7406	0.0021	0.2064	0.0044
		50	(0*50)	0.3187	0.0042	0.8274	0.0195	0.7513	0.0010	0.1862	0.0011
5	30	20	(10,0*19)	0.3745	0.0308	0.8347	0.0332	0.7380	0.0032	0.2105	0.0063
			((2,0)*2,(1,0)*6,0*4)	0.4034	0.0702	0.8254	0.0337	0.7324	0.0044	0.2228	0.0119
			((0*3,1*4)*2,1,1,0*4)	0.3992	0.0537	0.8281	0.0317	0.7329	0.0041	0.2213	0.0101
		30	(0*30)	0.3455	0.0127	0.8285	0.0234	0.7440	0.0019	0.1987	0.0030
	50	25	(25,0*24)	0.3644	0.0363	0.8193	0.0233	0.7392	0.0028	0.2073	0.0065
			(0*8,2,3*7,2,0*8)	0.3990	0.0601	0.8065	0.0218	0.7314	0.0041	0.2233	0.0115
			(3*6,0*12,1*7)	0.3875	0.0410	0.8112	0.0234	0.7344	0.0036	0.2178	0.0086
		30	(20,0*29)	0.3381	0.0117	0.8288	0.0204	0.7459	0.0019	0.1956	0.0029
			((2,0,0)*10)	0.3760	0.0308	0.8057	0.0200	0.7362	0.0030	0.2133	0.0065
			(0*29,20)	0.4172	0.0785	0.7994	0.0221	0.7301	0.0040	0.2288	0.0126
		50	(0*50)	0.3227	0.0050	0.8186	0.0128	0.7497	0.0010	0.1887	0.0013

,

Table 14. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method under LLF when $h=0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3505	0.0155	0.8326	0.0400	0.7399	0.0030	0.2006	0.0036
			((2,0)*2,(1,0)*6,0*4)	0.3589	0.0232	0.8229	0.0374	0.7411	0.0030	0.2039	0.0052
			((0*3,1*4)*2,1,1,0*4)	0.3607	0.0218	0.8284	0.0356	0.7380	0.0032	0.2057	0.0051
		30	(0*30)	0.3344	0.0088	0.8299	0.0309	0.7441	0.0020	0.1936	0.0022
	50	25	(25,0*24)	0.3376	0.0111	0.8291	0.0283	0.7425	0.0023	0.1957	0.0028
			(0*8,2,3*7,2,0*8)	0.3577	0.0197	0.8159	0.0268	0.7385	0.0028	0.2056	0.0048
			(3*6,0*12,1*7)	0.3540	0.0163	0.8084	0.0226	0.7394	0.0024	0.2036	0.0038
		30	(20,0*29)	0.3231	0.0073	0.8375	0.0270	0.7470	0.0018	0.1887	0.0019
			((2,0,0)*10)	0.3419	0.0132	0.8160	0.0233	0.7436	0.0021	0.1976	0.0032
			(0*29,20)	0.3617	0.0225	0.8042	0.0274	0.7405	0.0024	0.2062	0.0049
		50	(0*50)	0.3171	0.0041	0.8190	0.0157	0.7500	0.0010	0.1861	0.0011
5	30	20	(10,0*19)	0.3729	0.0322	0.8110	0.0314	0.7359	0.0037	0.2116	0.0072
			((2,0)*2,(1,0)*6,0*4)	0.3897	0.0480	0.8070	0.0295	0.7307	0.0046	0.2207	0.0110
			((0*3,1*4)*2,1,1,0*4)	0.3986	0.0581	0.8014	0.0298	0.7291	0.0050	0.2248	0.0131
		30	(0*30)	0.3412	0.0110	0.8118	0.0214	0.7439	0.0020	0.1974	0.0028
	50	25	(25,0*24)	0.3502	0.0164	0.8093	0.0208	0.7402	0.0027	0.2024	0.0042
			(0*8,2,3*7,2,0*8)	0.3667	0.0264	0.8033	0.0190	0.7361	0.0032	0.2110	0.0066
			(3*6,0*12,1*7)	0.3788	0.0321	0.8009	0.0212	0.7314	0.0038	0.2169	0.0079
		30	(20,0*29)	0.3347	0.0118	0.8094	0.0185	0.7473	0.0019	0.1942	0.0029
			((2,0,0)*10)	0.3539	0.0189	0.8066	0.0181	0.7395	0.0026	0.2047	0.0049
			(0*29,20)	0.3940	0.0426	0.7879	0.0216	0.7306	0.0038	0.2232	0.0097
		50	(0*50)	0.3193	0.0048	0.8150	0.0130	0.7496	0.0010	0.1875	0.0013

,

Table 15. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method under LLF when $h=1$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3433	0.0156	0.8313	0.0361	0.7401	0.0028	0.1978	0.0037
			((2,0)*2,(1,0)*6,0*4)	0.3469	0.0173	0.8295	0.0350	0.7401	0.0029	0.1994	0.0040
			((0*3,1*4)*2,1,1,0*4)	0.3637	0.0254	0.8106	0.0328	0.7366	0.0035	0.2078	0.0061
		30	(0*30)	0.3301	0.0077	0.8207	0.0278	0.7453	0.0019	0.1916	0.0019
	50	25	(25,0*24)	0.3326	0.0092	0.8180	0.0272	0.7443	0.0022	0.1933	0.0024
			(0*8,2,3*7,2,0*8)	0.3504	0.0174	0.8015	0.0207	0.7407	0.0026	0.2027	0.0045
			(3*6,0*12,1*7)	0.3549	0.0192	0.7933	0.0237	0.7410	0.0026	0.2041	0.0047
		30	(20,0*29)	0.3241	0.0072	0.8266	0.0247	0.7461	0.0018	0.1894	0.0019
			((2,0,0)*10)	0.3406	0.0127	0.8094	0.0213	0.7422	0.0022	0.1979	0.0032
			(0*29,20)	0.3517	0.0178	0.7997	0.0233	0.7419	0.0022	0.2025	0.0043
		50	(0*50)	0.3177	0.0038	0.8060	0.0150	0.7509	0.0010	0.1862	0.0010
5	30	20	(10,0*19)	0.3594	0.0222	0.8117	0.0288	0.7361	0.0035	0.2071	0.0056
			((2,0)*2,(1,0)*6,0*4)	0.3766	0.0336	0.7987	0.0283	0.7327	0.0043	0.2157	0.0085
			((0*3,1*4)*2,1,1,0*4)	0.3806	0.0378	0.8026	0.0291	0.7312	0.0047	0.2180	0.0098
		30	(0*30)	0.3386	0.0116	0.8096	0.0235	0.7442	0.0019	0.1963	0.0029
	50	25	(25,0*24)	0.3483	0.0158	0.8038	0.0203	0.7395	0.0027	0.2022	0.0041
			(0*8,2,3*7,2,0*8)	0.3684	0.0274	0.7987	0.0187	0.7325	0.0038	0.2133	0.0073
			(3*6,0*12,1*7)	0.3699	0.0265	0.7896	0.0187	0.7331	0.0038	0.2137	0.0071
		30	(20,0*29)	0.3410	0.0118	0.8100	0.0208	0.7407	0.0021	0.1988	0.0031
			((2,0,0)*10)	0.3530	0.0162	0.7967	0.0163	0.7379	0.0026	0.2051	0.0044
			(0*29,20)	0.3846	0.0348	0.7856	0.0219	0.7304	0.0038	0.2205	0.0088
		50	(0*50)	0.3236	0.0052	0.8045	0.0128	0.7479	0.0010	0.1899	0.0014

,

Table 16. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method under GELF when $q=-0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3438	0.0166	0.8378	0.0421	0.7407	0.0032	0.1967	0.0038
			((2,0)*2,(1,0)*6,0*4)	0.3568	0.0209	0.8225	0.0368	0.7377	0.0033	0.2029	0.0046
			((0*3,1*4)*2,1,1,0*4)	0.3457	0.0192	0.8397	0.0403	0.7412	0.0031	0.1976	0.0044
		30	(0*30)	0.3247	0.0077	0.8351	0.0324	0.7470	0.0018	0.1882	0.0018
	50	25	(25,0*24)	0.3334	0.0103	0.8379	0.0319	0.7407	0.0025	0.1936	0.0027
			(0*8,2,3*7,2,0*8)	0.3491	0.0197	0.8108	0.0245	0.7413	0.0027	0.2004	0.0046
			(3*6,0*12,1*7)	0.3517	0.0170	0.8077	0.0252	0.7394	0.0025	0.2018	0.0039
		30	(20,0*29)	0.3263	0.0083	0.8232	0.0251	0.7469	0.0019	0.1895	0.0021
			((2,0,0)*10)	0.3371	0.0119	0.8126	0.0225	0.7444	0.0020	0.1950	0.0029
			(0*29,20)	0.3533	0.0176	0.8041	0.0239	0.7406	0.0025	0.2024	0.0042
		50	(0*50)	0.3112	0.0034	0.8234	0.01706	0.7515	0.0010	0.1829	0.0009
5	30	20	(10,0*19)	0.3637	0.0267	0.8040	0.0295	0.7381	0.0035	0.2065	0.0059
			((2,0)*2,(1,0)*6,0*4)	0.3733	0.0426	0.8171	0.0282	0.7320	0.0044	0.2121	0.0091
			((0*3,1*4)*2,1,1,0*4)	0.3886	0.0522	0.8033	0.0308	0.7297	0.0045	0.2183	0.0101
		30	(0*30)	0.3358	0.0119	0.8219	0.0247	0.7436	0.0020	0.1945	0.0029
	50	25	(25,0*24)	0.3478	0.0170	0.8100	0.0203	0.7384	0.0027	0.2010	0.0041
			(0*8,2,3*7,2,0*8)	0.3703	0.0290	0.7973	0.0189	0.7322	0.0036	0.2121	0.0068
			(3*6,0*12,1*7)	0.3632	0.0239	0.7981	0.0193	0.7361	0.0032	0.2081	0.0058
		30	(20,0*29)	0.3367	0.0120	0.8088	0.0190	0.7441	0.0019	0.1952	0.0030
			((2,0,0)*10)	0.3561	0.0188	0.8001	0.0189	0.7373	0.0027	0.2051	0.0046
			(0*29,20)	0.3809	0.0439	0.7887	0.0223	0.7345	0.0038	0.2158	0.0096
		50	(0*50)	0.3183	0.0051	0.8112	0.0128	0.7498	0.0011	0.1867	0.0013

,

Table 17. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method under GELF when $q=0.5$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3320	0.0133	0.8181	0.0360	0.7360	0.0034	0.1923	0.0033
			((2,0)*2,(1,0)*6,0*4)	0.3281	0.0130	0.8207	0.0376	0.7381	0.0033	0.1903	0.0034
			((0*3,1*4)*2,1,1,0*4)	0.3489	0.0249	0.7999	0.0372	0.7340	0.0038	0.1995	0.0054
		30	(0*30)	0.3152	0.0063	0.8121	0.0281	0.7471	0.0019	0.1837	0.0016
	50	25	(25,0*24)	0.3196	0.0080	0.8163	0.0251	0.7419	0.0023	0.1870	0.0021
			(0*8,2,3*7,2,0*8)	0.3404	0.0162	0.7910	0.0236	0.7369	0.0031	0.1972	0.0041
			(3*6,0*12,1*7)	0.3297	0.0138	0.8038	0.0241	0.7409	0.0025	0.1915	0.0034
		30	(20,0*29)	0.3166	0.0064	0.8060	0.0231	0.7464	0.0019	0.1850	0.0017
			((2,0,0)*10)	0.3220	0.0085	0.8061	0.0208	0.7430	0.0020	0.1884	0.0022
			(0*29,20)	0.3413	0.0157	0.7908	0.0237	0.7379	0.0024	0.1973	0.0037
		50	(0*50)	0.3106	0.0032	0.8042	0.0156	0.7498	0.0011	0.1827	0.0009
5	30	20	(10,0*19)	0.3413	0.0196	0.8061	0.0316	0.7343	0.0037	0.1972	0.0049
			((2,0)*2,(1,0)*6,0*4)	0.3477	0.0224	0.8008	0.0253	0.7293	0.0042	0.2013	0.0056
			((0*3,1*4)*2,1,1,0*4)	0.3645	0.0371	0.7855	0.0284	0.7270	0.0052	0.2083	0.0083
		30	(0*30)	0.3243	0.0088	0.8011	0.0210	0.7433	0.0018	0.1892	0.0022
	50	25	(25,0*24)	0.3421	0.0201	0.7964	0.0224	0.7349	0.0032	0.1982	0.0046
			(0*8,2,3*7,2,0*8)	0.3504	0.0275	0.7890	0.0197	0.7307	0.0042	0.2028	0.0066
			(3*6,0*12,1*7)	0.3476	0.0241	0.7902	0.0225	0.7331	0.0038	0.2011	0.0060
		30	(20,0*29)	0.3305	0.0101	0.7886	0.0177	0.7418	0.0021	0.1925	0.0026
			((2,0,0)*10)	0.3332	0.0133	0.7909	0.0168	0.7395	0.0026	0.1942	0.0035
			(0*29,20)	0.3601	0.0319	0.7818	0.0220	0.7291	0.0041	0.2070	0.0073
		50	(0*50)	0.3133	0.0043	0.8076	0.0127	0.7472	0.0011	0.1848	0.0012

and

Table 18. Estimation values and MSEs of parameters, reliability and hazard functions using Metropolis-Hastings method under GELF when $q=1$, $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, $R(t)=0.7563$, $H(t)=0.1786$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{MH}}$		${\widehat{\mathit{\beta }}}_{\mathbf{MH}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{MH}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{MH}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	20	(10,0*19)	0.3249	0.0134	0.8033	0.0359	0.7377	0.0034	0.1883	0.0034
			((2,0)*2,(1,0)*6,0*4)	0.3312	0.0138	0.7939	0.0319	0.7342	0.0035	0.1921	0.0035
			((0*3,1*4)*2,1,1,0*4)	0.3346	0.0208	0.7905	0.0319	0.7348	0.0039	0.1931	0.0049
		30	(0*30)	0.3182	0.0072	0.8048	0.0287	0.7422	0.0021	0.1858	0.0018
	50	25	(25,0*24)	0.3231	0.0105	0.8010	0.0268	0.7394	0.0026	0.1883	0.0026
			(0*8,2,3*7,2,0*8)	0.3364	0.0174	0.7870	0.0221	0.7334	0.0031	0.1956	0.0042
			(3*6,0*12,1*7)	0.3278	0.0127	0.7936	0.0233	0.7369	0.0028	0.1913	0.0032
		30	(20,0*29)	0.3158	0.0067	0.8066	0.0226	0.7420	0.0021	0.1854	0.0018
			((2,0,0)*10)	0.3203	0.0098	0.7941	0.0227	0.7436	0.0019	0.1869	0.0024
			(0*29,20)	0.3304	0.0139	0.7880	0.0242	0.7390	0.0024	0.1920	0.0032
		50	(0*50)	0.3103	0.0032	0.8023	0.0149	0.7472	0.0012	0.1830	0.0009
5	30	20	(10,0*19)	0.3324	0.0178	0.7895	0.0263	0.7336	0.0039	0.1933	0.0045
			((2,0)*2,(1,0)*6,0*4)	0.3431	0.0233	0.7828	0.0261	0.7284	0.0047	0.1989	0.0059
			((0*3,1*4)*2,1,1,0*4)	0.3477	0.0531	0.7854	0.0293	0.7290	0.0053	0.2000	0.0109
		30	(0*30)	0.3147	0.0081	0.8038	0.0210	0.7432	0.0019	0.1847	0.0021
	50	25	(25,0*24)	0.3276	0.0113	0.7929	0.0212	0.7365	0.0028	0.1916	0.0031
			(0*8,2,3*7,2,0*8)	0.3457	0.0275	0.7798	0.0197	0.7282	0.0047	0.2007	0.0069
			(3*6,0*12,1*7)	0.3388	0.0179	0.7792	0.0190	0.7310	0.0037	0.1974	0.0046
		30	(20,0*29)	0.3212	0.0088	0.7928	0.0160	0.7407	0.0022	0.1886	0.0024
			((2,0,0)*10)	0.3327	0.0163	0.7837	0.0177	0.7357	0.0032	0.1944	0.0044
			(0*29,20)	0.3454	0.0224	0.7734	0.0210	0.7295	0.0040	0.2003	0.0057
		50	(0*50)	0.3112	0.0034	0.8234	0.0171	0.7515	0.0010	0.1829	0.0009

. While under LLF, we choose $h=-0.5$, 0.5 and 1. While under GELF, we consider $q=-0.5$, 0.5 and 1. The tables show that:

• When k and n are fixed but m increases, the $MSE$s of $\gamma$, $\beta$, $R(t)$ and $H(t)$ decrease.
• When k is fixed but n increases, the $MSE$s decrease.
• When n and m are fixed but k increases, the $MSE$s incraese.
• The $MSE$s when using noninformative priors are bigger than using the informative priors.
• While under LLF, $h=1$ is the best mode with the smallest $MSE$s, and the mode $h=-0.5$ has the biggest $MSE$s among the three.
• While under GELF, the estimation effect when $q=1$ is a little bit better than $q=0.5$, and the mode $q=-0.5$ is a little bit worse relatively.
• Among the three loss functions, the $MSE$s under SELF are smallest on the whole, while the $MSE$s under GELF are a little bit bigger than under LLF.

(4) Between the Bayesian estimation methods in this paper, the estimation effect of Lindley approximation method seems a little better than M-H method overall. Between the interval estimations in this paper, the MLE performs a little better than M-H method as a whole. In

Table 19. Average length and coverage probability of 95% asymptotic confidence/Bayesian credible interval for parameters under SELF when $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=1.08$, $b_1=3.6$, $a_2=16$, $b_2=20$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{ML}}$		${\widehat{\mathit{\beta }}}_{\mathbf{ML}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{ML}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{ML}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	15	(15,0*14)	0.4707	0.974	0.7516	0.904	0.4978	0.932	0.9052	0.936
			(0*6,10,5,0*7)	0.5912	0.958	0.6902	0.878	0.6254	0.910	0.8724	0.924
			((2,0)*7,1)	0.6093	0.968	0.7196	0.896	0.6860	0.920	0.8956	0.934
	30	20	(10,0*19)	0.3883	0.980	0.6826	0.890	0.3983	0.940	0.8227	0.936
			((2,0)*2,(1,0)*6,0*4)	0.4104	0.956	0.6686	0.886	0.4248	0.926	0.8135	0.928
			((0*3,1*4)*2,1,1,0*4)	0.4383	0.964	0.6645	0.900	0.4563	0.924	0.7992	0.936
		30	(0*30)	0.2968	0.976	0.6071	0.916	0.2971	0.948	0.7175	0.962
	50	25	(25,0*24)	0.3295	0.964	0.5966	0.928	0.3326	0.926	0.6525	0.946
			(0*8,2,3*7,2,0*8)	0.3896	0.958	0.5602	0.926	0.4029	0.922	0.6291	0.946
			(3*6,0*12,1*7)	0.3876	0.974	0.5550	0.880	0.4011	0.944	0.6229	0.920
		30	(20,0*29)	0.2905	0.97	0.5682	0.914	0.2930	0.936	0.6228	0.940
			((2,0,0)*10)	0.3335	0.97	0.5450	0.894	0.3402	0.934	0.6031	0.932
			(0*29,20)	0.3830	0.97	0.5716	0.904	0.4137	0.932	0.6423	0.944
		50	(0*50)	0.2155	0.960	0.4810	0.942	0.2124	0.952	0.5215	0.950
5	30	15	(15,0*14)	0.5232	0.974	0.6766	0.918	0.5790	0.956	0.7666	0.958
			(0*6,10,5,0*7)	0.6486	0.960	0.6355	0.900	0.7333	0.920	0.7483	0.930
			((2,0)*7,1)	0.8226	0.962	0.6677	0.902	1.0002	0.902	0.7834	0.916
	30	20	(10,0*19)	0.4349	0.966	0.6256	0.920	0.4667	0.914	0.7103	0.928
			((2,0)*2,(1,0)*6,0*4)	0.5159	0.976	0.5943	0.902	0.5598	0.924	0.6779	0.930
			((0*3,1*4)*2,1,1,0*4)	0.5229	0.972	0.5934	0.894	0.5692	0.944	0.6780	0.918
		30	(0*30)	0.3355	0.968	0.5457	0.930	0.3423	0.952	0.6046	0.950
	50	25	(25,0*24)	0.3839	0.976	0.5355	0.908	0.3986	0.934	0.5755	0.922
			(0*8,2,3*7,2,0*8)	0.4622	0.954	0.5046	0.914	0.4885	0.942	0.5513	0.954
			(3*6,0*12,1*7)	0.4586	0.964	0.5142	0.906	0.5132	0.916	0.5587	0.934
		30	(20,0*29)	0.3368	0.968	0.5040	0.900	0.3473	0.936	0.5376	0.936
			((2,0,0)*10)	0.4105	0.964	0.4887	0.906	0.4358	0.910	0.5305	0.918
			(0*29,20)	0.4885	0.962	0.5440	0.928	0.5804	0.932	0.5856	0.944
		50	(0*50)	0.2433	0.962	0.4239	0.944	0.2463	0.944	0.4527	0.958

and

Table 20. Average length and coverage probability of 95% asymptotic confidence/Bayesian credible interval for parameters under SELF using noninformative priors when $\gamma =0.3$, $\beta =0.8$, $t=1.5$, $a_1=0$, $b_1=0$, $a_2=0$, $b_2=0$, N = 1000.

k	n	m	Scheme	${\widehat{\mathit{\gamma }}}_{\mathbf{ML}}$		${\widehat{\mathit{\beta }}}_{\mathbf{ML}}$		$\widehat{\mathit{R}}{(\mathit{t})}_{\mathbf{ML}}$		$\widehat{\mathit{H}}{(\mathit{t})}_{\mathbf{ML}}$
				EV	MSE	EV	MSE	EV	MSE	EV	MSE
3	30	15	(15,0*14)	0.4502	0.948	0.7333	0.892	0.4825	0.946	0.9200	0.934
			(0*6,10,5,0*7)	0.5913	0.956	0.6894	0.886	0.6748	0.930	0.9020	0.932
			((2,0)*7,1)	0.6047	0.968	0.7386	0.886	0.6687	0.926	0.9346	0.934
	30	20	(10,0*19)	0.3713	0.974	0.6827	0.890	0.3793	0.934	0.8317	0.938
			((2,0)*2,(1,0)*6,0*4)	0.4152	0.960	0.6724	0.898	0.4323	0.924	0.8341	0.916
			((0*3,1*4)*2,1,1,0*4)	0.4293	0.962	0.6595	0.906	0.4538	0.942	0.8334	0.936
		30	(0*30)	0.2999	0.962	0.5989	0.912	0.3009	0.958	0.7166	0.942
	50	25	(25,0*24)	0.3226	0.936	0.5753	0.886	0.3319	0.942	0.6519	0.920
			(0*8,2,3*7,2,0*8)	0.4071	0.966	0.5528	0.920	0.4235	0.932	0.6264	0.936
			(3*6,0*12,1*7)	0.3839	0.972	0.5522	0.904	0.4045	0.944	0.6250	0.956
		30	(20,0*29)	0.2923	0.954	0.5637	0.926	0.2951	0.936	0.6277	0.938
			((2,0,0)*10)	0.3295	0.958	0.5415	0.908	0.3368	0.944	0.6090	0.934
			(0*29,20)	0.3900	0.954	0.5725	0.896	0.4253	0.914	0.6520	0.930
		50	(0*50)	0.2189	0.954	0.4723	0.922	0.2191	0.960	0.5170	0.950
5	30	15	(15,0*14)	0.5569	0.972	0.6786	0.900	0.6172	0.934	0.7805	0.932
			(0*6,10,5,0*7)	0.7654	0.962	0.6269	0.878	0.8917	0.904	0.7415	0.916
			((2,0)*7,1)	0.7470	0.946	0.6753	0.884	0.9281	0.934	0.8098	0.916
	30	20	(10,0*19)	0.4440	0.972	0.6215	0.910	0.4718	0.928	0.7083	0.914
			((2,0)*2,(1,0)*6,0*4)	0.5222	0.968	0.6035	0.904	0.5671	0.924	0.7082	0.936
			((0*3,1*4)*2,1,1,0*4)	0.5133	0.974	0.6059	0.910	0.5541	0.930	0.6961	0.932
		30	(0*30)	0.3434	0.966	0.5381	0.914	0.3530	0.944	0.6063	0.928
	50	25	(25,0*24)	0.3709	0.978	0.5377	0.932	0.3888	0.944	0.5787	0.950
			(0*8,2,3*7,2,0*8)	0.4859	0.960	0.4974	0.902	0.5187	0.926	0.5491	0.946
			(3*6,0*12,1*7)	0.4520	0.962	0.5189	0.904	0.5015	0.926	0.5685	0.912
		30	(20,0*29)	0.3364	0.960	0.5027	0.926	0.3481	0.928	0.5379	0.920
			((2,0,0)*10)	0.3948	0.948	0.4994	0.904	0.4185	0.926	0.5479	0.916
			(0*29,20)	0.4739	0.958	0.5306	0.924	0.5614	0.942	0.5837	0.944
		50	(0*50)	0.2443	0.962	0.4238	0.950	0.2447	0.950	0.4486	0.962

, the Bayesian credible intervals and asymptotic confidence intervals of parameters are presented in the form of the average length and the coverage probability.

Please note that the $EV$ and $MSE$ in the table represent the estimated value and the mean square error between the estimated value and the real value respectively.

5. Practical Data Analysis

We mainly apply the practical data sets prestented in Nichols and Padgett [26] for analysis and illustration in this section. The data reports the observations of tensile strength of 100 carbon fibers, which are tabulated in

Table 21. The practical data sets of tensile strength of carbon fibers.

$\left[1\right]$	3.7	3.11	4.42	3.28	3.75	2.96	3.39	3.31	3.15	2.81	1.41	2.76	3.19
$\left[\mathbf{14}\right]$	1.59	2.17	3.51	1.84	1.61	1.57	1.89	2.75	3.27	2.41	3.09	2.43	2.53
$\left[\mathbf{27}\right]$	2.81	3.31	2.35	2.77	2.68	4.91	1.57	2.00	1.17	2.17	0.39	2.79	1.08
$\left[\mathbf{40}\right]$	2.88	2.73	2.87	3.19	1.87	2.95	2.67	4.20	2.85	2.55	2.17	2.97	3.68
$\left[\mathbf{53}\right]$	0.81	1.22	5.08	1.69	3.68	4.70	2.03	2.82	2.50	1.47	3.22	3.15	2.97
$\left[\mathbf{66}\right]$	2.93	3.33	2.56	2.59	2.83	1.36	1.84	5.56	1.12	2.48	1.25	2.48	2.03
$\left[\mathbf{79}\right]$	1.61	2.05	3.60	3.11	1.69	4.90	3.39	3.22	2.55	3.56	2.38	1.92	0.98
$\left[\mathbf{92}\right]$	1.59	1.73	1.71	1.18	4.38	0.85	1.80	2.12	3.65

.

First, we fit IEHLD and the following five reliability models to the data to compare the fitness:

$$\begin{array}{c}Exponential:f(x;\theta )=\theta exp\left(-\theta x\right)\hfill \\ HLD:f(x;\theta )={\displaystyle \frac{2exp\left(-x/\theta \right)}{\theta {\left(1+exp\left(-x/\theta \right)\right)}^2}}\hfill \\ Weibull:f(x;\gamma ,\beta )=\gamma \beta x^{\beta -1}exp\left(-\gamma x^{\beta }\right)\hfill \\ IWD:f(x;\gamma ,\beta )=\gamma \beta exp\left(-\beta x^{-\gamma }\right)x^{-\gamma -1}\hfill \\ GIRD:f(x;\gamma ,\beta )={\displaystyle \frac{2\gamma }{{\beta }^2{x}^3}}exp\left(-{\displaystyle \frac{1}{{\beta }^2{x}^2}}\right){\left[1-exp\left(-{\displaystyle \frac{1}{{\beta }^2{x}^2}}\right)\right]}^{\gamma -1}\hfill \end{array}$$

Among them, the Weibull distribution is the original model of the practical data we use, and the IWD and GIRD are two similarly shaped hazard models with IEHLD. The Q-Q plot of fitting these models are shown in Figure 2 for illustration. Then, we use Bayesian information criterion (BIC), Akaike information criterion (AIC), and Kolmogorov–Smirnov (KS) test statistics to present the numerical results based on the MLE. The results are tabulated in

Table 22. Results of fitting different functions to the practical data.

N.O.P	Distribution	MLEs	−$ln\mathit{L}$	AIC	BIC	K-S
1	Exponential	$\widehat{\theta }=0.3829$	195.9925	393.9850	396.5901	0.3218
1	HLD	$\widehat{\theta }=1.7442$	181.1867	364.3733	366.9785	0.2919
2	Weibull	$\widehat{\gamma }=0.0496$	140.9977	285.9954	291.2057	0.0632
		$\widehat{\beta }=2.7924$
2	IWD	$\widehat{\gamma }=1.7737$	172.5033	349.0067	354.2170	0.5232
		$\widehat{\beta }=3.0856$
2	GIRD	$\widehat{\gamma }=1.1500$	174.0566	352.1132	357.3235	0.2012
		$\widehat{\beta }=0.5271$
2	IEHLD	$\widehat{\gamma }=5.5581$	151.9385	307.8769	313.0873	0.1358
		$\widehat{\beta }=0.1523$

Note: The N.O.P represents the number of parameters in each distribution.

. It can be seen that IEHLD has relatively good fitting degree.

Next, randomly group the data into 50 groups with 2 units in each group. That means $n=50$, $k=2$. After that, choose the minimum value in each group, using schemes ${\Re }_1$, ${\Re }_2$ and ${\Re }_3$ when $m=35$ to generate the progressive first-failure samples. The grouped data are tabulated in

Table 23. The grouped data.

Group Iteam	1	2	3	4	5	6	7	8	9	10	11	12	13
1	3.39	2.35	1.8	2.05	4.90	2.03	0.81	1.17	2.96	2.48	2.68	2.17	3.27
2	2.75	1.47	3.6	1.22	2.17	2.73	3.33	3.39	0.85	1.36	2.41	2.88	1.59
Group Iteam	14	15	16	17	18	19	20	21	22	23	24	25	26
1	2.48	4.70	3.09	3.28	2.17	5.56	1.18	4.42	2.81	3.15	3.22	2.93	3.22
2	2.81	2.67	1.08	2.97	4.91	2.43	1.61	3.19	3.19	3.31	2.12	3.11	1.84
Group Iteam	27	28	29	30	31	32	33	34	35	36	37	38	39
1	5.08	1.57	4.20	2.03	2.77	3.56	1.12	2.50	1.71	3.65	1.61	3.11	1.92
2	1.25	2.56	2.53	3.68	0.39	1.89	2.87	1.84	3.51	2.95	3.31	2.82	2.55
Group Iteam	40	41	42	43	44	45	46	47	48	49	50
1	2.85	2.79	1.73	4.38	3.15	1.69	1.41	1.69	1.59	2.76	2.97
2	2.59	2.55	1.87	1.57	2.00	3.70	2.38	0.98	3.68	2.83	3.75

and the minimum values in each group are displayed as follows: 0.39, 0.81, 0.85, 0.98, 1.08, 1.12, 1.17, 1.18, 1.22, 1.25, 1.36, 1.41, 1.47, 1.57, 1.57, 1.59, 1.59, 1.61, 1.69, 1.71, 1.73, 1.80, 1.84, 1.84, 1.89, 1.92, 2.00, 2.03, 2.03, 2.12, 2.17, 2.17, 2.17, 2.41, 2.43, 2.48, 2.53, 2.55, 2.59, 2.67, 2.75, 2.76, 2.81, 2.82, 2.93, 2.95, 2.97, 2.97, 3.15, 3.19.

Then, using the generated progressive first-failure censored data listed in

Table 24. The progressive first-failure censored samples generated from the practical data sets.

$\left(\mathit{k},\mathit{m},\mathit{n}\right)$	Censoring Scheme	Progressive First-Failure Censored Sample
$\left(2,50,35\right)$	${\Re }_1$ = (1*15, 0*20)	0.39, 0.85, 1.08, 1.17, 1.22, 1.36, 1.47, 1.57, 1.59
		1.69, 1.73, 1.84, 1.89, 2.00, 2.03, 2.17, 2.17, 2.17
		2.41, 2.43, 2.48, 2.53, 2.55, 2.59, 2.67, 2.75, 2.76
		2.81, 2.82, 2.93, 2.95, 2.97, 2.97, 3.15, 3.19
	${\Re }_2$ = (0*15, 3*5, 0*15)	0.39, 0.81, 0.85, 0.98, 1.08, 1.12, 1.17, 1.18, 1.22
		1.25, 1.36, 1.41, 1.47, 1.57, 1.57, 1.59, 1.71, 1.84
		2.03, 2.17, 2.48, 2.53, 2.55, 2.59, 2.67, 2.75, 2.76
		2.81, 2.82, 2.93, 2.95, 2.97, 2.97, 3.15, 3.19
	${\Re }_3$ = (0*34, 15)	0.39, 0.81, 0.85, 0.98, 1.08, 1.12, 1.17, 1.18, 1.22
		1.25, 1.36, 1.41, 1.47, 1.57, 1.57, 1.59, 1.59, 1.61
		1.69, 1.71, 1.73, 1.80, 1.84, 1.84, 1.89, 1.92, 2.00
		2.03, 2.03, 2.12, 2.17, 2.17, 2.17, 2.41, 2.43

, the MLEs and Bayesian estimations of $\gamma$, $\beta$, $H(t)$ and $R(t)$ are conducted. Simultaneously, the Bayesian credible intervals and asymptotic confidence intervals of parameters are also presented. Take $t=10$, the corresponding results are listed in

Table 25. Maximum Likelihood and Bayesian Estimates under different schemes.

Estimation	Censoring Scheme
	${\Re }_{\mathbf{1}}$	${\Re }_{\mathbf{2}}$	${\Re }_{\mathbf{3}}$
${\widehat{\gamma }}_{ML}$	3.7306 [ 0.6542,6.8100 ]	1.8196 [ 0.5276,3.1116 ]	1.6307 [ 0.4579,2.8035 ]
${\widehat{\gamma }}_{LB}$	3.9320	1.8655	1.6837
${\widehat{\gamma }}_{MH}$	1.3597 [ 0.2560,4.1397 ]	1.9471 [ 0.2569,3.1614 ]	1.7058 [ 0.2512,4.3786 ]
${\widehat{\beta }}_{ML}$	0.1555 [ 0.1115,0.1996 ]	0.2127 [ 0.1490,0.2765 ]	0.2356 [ 0.1630,0.3082 ]
${\widehat{\beta }}_{LB}$	0.1597	0.2198	0.2433
${\widehat{\beta }}_{MH}$	0.3211 [ 0.1479,0.7694 ]	0.2120 [ 0.1758,0.7935 ]	0.2400 [ 0.1731,0.8261 ]
$\widehat{R}{(t)}_{ML}$	0.0128	0.0694	0.0779
$\widehat{R}{(t)}_{LB}$	0.0278	0.0907	0.0990
$\widehat{R}{(t)}_{MH}$	0.2027	0.0777	0.0902
$\widehat{H}{(t)}_{ML}$	0.3486	0.1754	0.1583
$\widehat{H}{(t)}_{LB}$	0.3732	0.1831	0.1664
$\widehat{H}{(t)}_{MH}$	0.1296	0.1868	0.1650

.

6. Optimal Censoring Mode

In this part, the optimal censoring mode of the progressive first-failure censoring schemes presented before is considered. Here we use four criteria mentioned in [27] to evaluate the optimal censoring scheme.

$\left[\mathbf{1}\right]$: Minimize the determinant of the corresponding $I^{-1}(\widehat{\theta })$, say $det(I^{-1}(\widehat{\theta }))$.

$\left[\mathbf{2}\right]$: Minimize the trace of the corresponding $I^{-1}(\widehat{\theta })$, say $trace(I^{-1}(\widehat{\theta }))$.

$\left[\mathbf{3}\right]$: Minimize $Var[ln{\widehat{\alpha }}_p]$, the variance of the logarithmic of the MLE of p-th quantile ${\alpha }_p$. In IEHLD, ${\alpha }_p=-1/\left[\beta ln{\displaystyle \frac{1-{\left(1-p\right)}^{1/\gamma }}{1+{\left(1-p\right)}^{1/\gamma }}}\right]$, $Var[ln{\widehat{\alpha }}_p]={\left[▽ln{\widehat{\alpha }}_p\right]}^T\left[Var(\widehat{\theta })\right]\left[▽ln{\widehat{\alpha }}_p\right]$. Please note that ${\left[▽ln{\widehat{\alpha }}_p\right]}^T$ is the gradients of $ln{\alpha }_p$ when $\gamma$ and $\beta$ are obtained as the MLEs $\widehat{\gamma }$ and $\widehat{\beta }$. We take p = 0.05 and p = 0.95 for this criterion.

$\left[\mathbf{4}\right]$: Minimize the integral ${\int }_0^{1}Var[ln{\widehat{\alpha }}_p]\mho (p)dp$ with the weight function $\mho (p)=1$.

The corresponding results are tabulated in

Table 26. Comparision of censoring schemes using different criteria.

Censoring Scheme	Different Criteria
	$\mathit{d}\mathit{e}\mathit{t}({\mathit{I}}^{-\mathbf{1}}(\widehat{\mathit{\theta }}))$	$\mathit{t}\mathit{r}\mathit{a}\mathit{c}\mathit{e}({\mathit{I}}^{-\mathbf{1}}(\widehat{\mathit{\theta }}))$	$\mathit{V}\mathit{a}\mathit{r}\left[\mathbf{ln}{\widehat{\mathit{\alpha }}}_{\mathbf{0.05}}\right]$	$\mathit{V}\mathit{a}\mathit{r}\left[\mathbf{ln}{\widehat{\mathbf{\alpha }}}_{\mathbf{0.95}}\right]$	${\int }_{\mathbf{0}}^{\mathbf{1}}\mathit{V}\mathit{a}\mathit{r}\left[\mathbf{ln}{\widehat{\mathit{\alpha }}}_{\mathit{p}}\right]\mathit{d}\mathit{p}$
${\Re }_1$ = (1*15, 0*20)	$2.009\times {10}^{-4}$	2.4690	4.8676	60.6790	24.1327
${\Re }_2$ = (0*15, 3*5, 0*15)	$9.996\times {10}^{-5}$	0.4356	0.1813	5.2060	1.7082
${\Re }_3$ = (0*34, 15)	$1.043\times {10}^{-4}$	0.3594	0.1096	3.9038	1.2315

. The minimum value under each criterion is bold for a clearer expression. Through the table we can see, while using criterion $\left[\mathbf{1}\right]$, ${\Re }_2$ is the best scheme, but when using the other three criteria, ${\Re }_3$ is the best scheme.

7. Concluding Remarks

We mainly propose the MLEs and Bayesian estimations of the parameters and the reliability and failure functions of IEHLD using progressive first-failure censored samples. For MLE, we use two sets of parameter values to compare the difference, the asymptotic intervals of parameters are presented at the same time. For Bayesian estimation, Lindley approximation method and Metropolis-Hastings method are conducted as point and interval estimation under three loss functions say GELF, LLF and SELF. The Bayesian intervals of parameters are also obtained. Then we use practical data sets to illustrate the effect of different models and compare different estimation methods. Finally, using four criteria, the optimal censoring plan of the progressive first-failure censoring schemes is discussed.

There are still a lot to be done in the study of censoring mode. In order to improve the experimental efficiency, it is very important to optimize the censoring mode. Now, many new censoring modes have been put forward such as generalized Type-II progressive hybrid censoring, adaptive progressive Type-II censoring. One can also combine existing censoring modes to achieve better results.