Statistical Inference of Two Gamma Distributions under the Joint Type-II Censoring Scheme

Abstract

The joint Type-II censoring scheme is a useful model when carrying out comparative lifecycle tests of units from various production lines. This article takes into account the estimation problem of the joint Type-II censoring data coming from two Gamma distributions with the same shape parameter but various scale parameters. The maximum likelihood estimators of the parameters from Gamma populations and asymptotic confidence intervals based on the observed Fisher information matrix are obtained. Bootstrap methods are also applied to construct confidence intervals. The Metropolis–Hastings algorithm is considered to draw Markov Chain Monte Carlo samples when computing Bayesian estimates as well as establishing the corresponding credible intervals. Monte Carlo simulations are adopted to compare the performance of the estimates. Finally, two real engineering datasets are analyzed.

1. Introduction

The modeling and analysis of lifetimes are of great significance in scientific and technological fields. Sometimes, it is impractical and inappropriate to study the complete sample because the products are becoming more and more reliable, and experimenters obtain longer lifetime data (see [1]). Under the considerations of time and cost savings, numerous censoring schemes for reliability and survival analysis have been introduced, among which the Type-II censoring scheme is generally applied by researchers.

A majority of conventional censoring schemes deal with the lifecycle tests of products coming from one single production line and have been well studied in the literature. In fact, in the industrial process, products are from various production lines, and comparative studies of the lifecycle of the units are needed. When experimenters decide to test the reliability of two populations that are being manufactured by two separate lines of production, joint censoring schemes are more appropriate. Reference [2] introduced the joint Type-II censoring scheme, in which statistical inferences for two exponential distributions are given.

Briefly, the joint Type-II censoring scheme is described as follows. Suppose that a random sample of size m is from a population, namely Popu-A. Let $(X_1,\cdots ,X_m)$ denote the lifetimes of these m units. Likewise, $(Y_1,\cdots ,Y_n)$ represent the lifetimes of the n units, which are from another population named Popu-B. They are placed simultaneously on a lifecycle testing experiment. Only r (r is a pre-fixed positive integer, and $r<m+n$) units could be completely observed until failure. It is also assumed that $W_1<\cdots <W_N$$(N=m+n)$ represent the order statistics of $(X_1,\cdots ,X_m;Y_1,\cdots ,Y_n)$. During the experiment, the successive failure times, along with the corresponding unit types, are recorded to be used in conducting the comparative life testing. When r failures occur, the lifecycle testing experiment will immediately be terminated. In this context, the censored data are recorded as $(\mathit{W},\mathit{Z})=\{(W_1,Z_1),\cdots ,(W_r,Z_r)\}$, where $\mathit{W}=(W_1,W_2,\cdots ,W_r)$, and $\mathit{Z}=(Z_1,Z_2,\cdots ,Z_r)$. Here, $w_i$ and $z_i$ denote the observed values of $W_i$ and $Z_i$, respectively. $z_i$ is equal to 1 or 0 accordingly as $w_i$ belongs to Popu-A or Popu-B. Let the total number of units withdrawn from Popu-A among $\mathit{W}$ be represented by $M_r={\sum }_{i=1}^r{Z}_i$. Likewise, $N_r={\sum }_{i=1}^r(1-Z_i)$ is the corresponding number of units withdrawn from Popu-B. See a brief description of this censoring scheme in Figure 1.

Comparative studies are much needed and practical in lifecycle testing. As a result, the joint censoring scheme has attracted considerable attention in recent years, and inferences for this censoring scheme have been extensively explored. Based on joint Type-II censoring samples from two exponential distributions, ref. [3] established the strong consistency as well as the asymptotic normality of the maximum likelihood estimators (MLEs) and provided large-sample confidence intervals. Under the square error, linear exponential, and generalized entropy loss functions, Bayes estimates for unknown parameters for two exponential distributions were given by [4]. On the basis of observed jointly censored data, the problem of predicting future failure times, including point and interval prediction, was also addressed. Ref. [5] dealt with statistical inferences for Weibull populations and compared the effectiveness of the approximate confidence intervals with two kinds of Bootstrap confidence intervals. Ref. [6] generalized the results of two samples to multi-sample cases for exponential populations. However, these studies all relied on two single-parameter populations.

Ref. [7] discussed the exact likelihood inference for two populations from two-parameter exponential distributions with the same location parameter and distinct scale parameters. They derived the exact distributions of the MLEs for three different parameters and their moment-generating functions under the joint Type-II censoring scheme. Recently, Ref. [8] studied two independent Lindley distributions under the same censoring scheme. The importance sampling method was applied under the consideration of complex posterior distribution for Bayesian estimations. Ref. [9] analyzed two Lomax distributions, in which the Expectation–Maximization algorithm and the Newton–Raphson procedure were utilized to estimate the parameters. To compare the results of various interval construction techniques, the coverage probabilities and average interval lengths were also taken into account.

The Gamma distribution is widely discussed when modeling lifetime distributions in reliability and survival analysis (see, for example, Refs. [10,11]), insurance (see [12,13]), and hydrology (see [14]). As with many other distributions, the Gamma distribution is long-tailed and positively skewed with an infinite terminus on the right. It is a particular form of a Pearson Type III distribution and is also considered an excellent alternative to the Weibull distribution for its wide applicability.

Suppose that X is a random variable that follows $Ga(\alpha ,\beta )$. Here, $Ga(\alpha ,\beta )$ represents a Gamma distribution with a shape parameter $\alpha$ as well as a scale parameter $1/\beta$. As a result, X has the following probability density function (PDF):

$$f(x;\alpha ,\beta )=\frac{{\beta }^{\alpha }}{\Gamma \left(\alpha \right)}{x}^{\alpha -1}{e}^{-\beta x},x>0,$$

(1)

where $\Gamma \left(\alpha \right)={\int }_0^{\infty }{u}^{\alpha -1}{e}^{-u}du$ is the complete gamma function and $f(x;\alpha ,\beta )=0$ for $x\le 0$. The cumulative distribution function (CDF) of X is written as:

$$F(x;\alpha ,\beta )=1-\frac{\Gamma (\alpha ,\beta x)}{\Gamma \left(\alpha \right)},x>0,$$

(2)

where $\Gamma (a,b)={\int }_b^{\infty }{u}^{a-1}{e}^{-u}du$ is the upper incomplete gamma function. The PDF of $Ga(\alpha ,\beta )$ is J-shaped when $\alpha \le 1$, and it is bell-shaped when $\alpha >1$. See the plot of the PDF in Figure 2.

The hazard function of $Ga(\alpha ,\beta )$ is expressed as:

$$h(x;\alpha ,\beta )=\frac{{\beta }^{\alpha }}{\Gamma (\alpha ,\beta x)}{x}^{\alpha -1}{e}^{-\beta x},x>0,$$

(3)

and its shape is up to $\alpha$. For $\alpha >1,\alpha <1$, or $\alpha =1$, the hazard function has an increasing, decreasing, or constant shape, respectively. See the plot of the hazard function in Figure 3.

The Gamma distribution is closely related to many other distributions. It is noticeable that when $\alpha =1$, the Gamma distribution is exponential with ordinate $\beta$ at $x=0$. If $\alpha =n/2$ and $\beta =1/2$, the Gamma distribution reduces to the chi-square distribution with a degree of freedom $df=n$.

The Gamma distribution has attracted a lot of interest due to its frequent application in lifecycle testing experiments under diverse conditions, and numerous studies have been developed using complete and censored samples. Ref. [15] obtained maximum likelihood estimators using the reparameterized distribution function together with the predictor–corrector method and showed the effectiveness of the proposed methods using a Monte Carlo simulation study. Ref. [16] calculated the approximate MLEs of the parameters and proposed a modified Expectation–Maximization algorithm to determine the MLEs for the Log–Gamma distribution on the basis of progressively Type-II censored data. The statistical inference of a three-parameter Weibull–Gamma distribution was given by [17], based on a progressively Type-II right censored sample. Ref. [18] took into account classical and Bayesian estimation methods using likelihood along with the maximum product of spacing functions for the two-parameter Gamma distribution when the progressive Type-II censoring scheme was conducted.

Although several statistical inference techniques for estimating parameters from different lifetime distributions related to the Gamma distribution have appeared in the literature, no work has been conducted on the inferences of the two-parameter Gamma distribution under the joint Type-II censoring scheme. The extension of this aspect is of great importance.

The main objective of this article is to conduct comparative lifecycle testing and analyze the two populations from the Gamma distribution under the joint Type-II censoring scheme. To start with, we compute MLEs and the mean square errors for the parameters. For deriving MLEs in explicit forms, the Newton–Raphson method is applied. Then, we obtain asymptotic confidence intervals together with bootstrap confidence intervals for the parameters. When computing the Bayes estimates of all parameters, the Metropolis–Hastings algorithm is considered. We calculate the Bayesian estimates based on the square error loss (SEL) function, as well as the linear exponential (LINEX) loss function under different priors. The effectiveness of all the estimation methods is compared after conducting Monte Carlo simulations. Finally, real engineering datasets are analyzed.

The remainder of this article is organized as follows. The Newton–Raphson method is applied to obtain the MLEs in Section 2. The asymptotic confidence intervals are presented in Section 3. The bootstrap methods are also used in Section 4 for interval construction. In Section 5, Bayesian estimations, along with credible intervals with independent Gamma priors and noninformative priors, are calculated. In Section 6, Monte Carlo simulations and analysis on real engineering data are demonstrated. In the end, some concluding remarks and open problems are reported in Section 7.

2. Maximum Likelihood Estimation

After sampling from Popu-A and Popu-B, we obtain the lifetimes of N products denoted as $(X_1,\cdots ,X_m;Y_1,\cdots ,Y_n)$, where $X_i$ follows $Ga(\alpha ,{\beta }_1)$ and $Y_i$ follows $Ga(\alpha ,{\beta }_2)$. We run the joint Type-II censoring scheme on the samples and obtain $(\mathit{W},\mathit{Z})=\{(w_1,z_1),(w_2,z_2),\cdots ,(w_r,z_r)\}$. Hence, the likelihood function for $\alpha$, ${\beta }_1$, and ${\beta }_2$ is defined as:

$$L\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)=C\prod _{i=1}^r{\left\{f\left(w_i\right)\right\}}^{z_i}{\left\{g\left(w_i\right)\right\}}^{1-z_i}{\left\{\overline{F}\left(w_r\right)\right\}}^{m-m_r}{\left\{\overline{G}\left(w_r\right)\right\}}^{n-n_r},$$

(4)

where $\overline{F}\left(x\right)=1-F\left(x\right)$ and $\overline{G}\left(x\right)=1-G\left(x\right)$ are represented as the survival functions of $(X_1,\cdots ,X_m;Y_1,\cdots ,Y_n)$ and $C=\frac{m!n!}{\left(m-m_r\right)!\left(n-n_r\right)!}$.

The likelihood function for $\alpha$, ${\beta }_1$, and ${\beta }_2$ can be derived with the combination of Functions (1) and (4):

$$\begin{array}{cc}\hfill L\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)& \hfill =C{\beta }_1^{\alpha m_r}{\beta }_2^{\alpha n_r}\left\{\prod _{i=1}^r{w}_i^{\alpha -1}\right\}{e}^{-{\beta }_1{\sum }_{i=1}^r{w}_i{z}_i}{e}^{-{\beta }_2{\sum }_{i=1}^r{w}_i\left(1-z_i\right)}\hfill \\ \hfill & \hfill \times \frac{\Gamma {\left(\alpha ,{\beta }_1{w}_r\right)}^{m-m_r}\Gamma {\left(\alpha ,{\beta }_2{w}_r\right)}^{n-n_r}}{\Gamma {\left(\alpha \right)}^{m+n}}.\hfill \end{array}$$

(5)

Taking the logarithm of Function (5) and removing the constant C, we obtain the log-likelihood function as:

$$\begin{array}{cc}\hfill l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)& =\alpha m_{r}log{\beta }_1+\alpha n_{r}log{\beta }_2+(\alpha -1)\sum _{i=1}^{r}log{w}_i-{\beta }_1\sum _{i=1}^r{w}_i{z}_i\hfill \\ \hfill & -{\beta }_2\sum _{i=1}^r{w}_i(1-z_i)-(m+n)log\Gamma \left(\alpha \right)+\left(m-m_r\right)log\Gamma \left(\alpha ,{\beta }_1{w}_r\right)\hfill \\ \hfill & +\left(n-n_r\right)log\Gamma \left(\alpha ,{\beta }_2{w}_r\right).\hfill \end{array}$$

(6)

We differentiate the log-likelihood function in Equation (6) with respect to $\alpha$, ${\beta }_1$, and ${\beta }_2$ before equating it to zero. $\widehat{\alpha }$, $\widehat{{\beta }_1}$, and $\widehat{{\beta }_2}$, which are the MLEs of $\alpha$, ${\beta }_1$, and ${\beta }_2$, respectively, can be obtained after solving the normal equations in the following forms:

$$\frac{\partial l}{\partial \alpha }=m_{r}log{\beta }_1+n_{r}log{\beta }_2+\sum _{i=1}^{r}log{w}_i-(m+n){\psi }_1\left(\alpha \right)+\left(m-m_r\right){\varphi }_1+\left(n-n_r\right){\varphi }_2=0,$$

(7)

$$\frac{\partial l}{\partial {\beta }_1}=\frac{\alpha m_r}{{\beta }_1}-\sum _{i=1}^r{w}_i{z}_i+\left(m-m_r\right){\Omega }_1=0,$$

(8)

$$\frac{\partial l}{\partial {\beta }_2}=\frac{\alpha n_r}{{\beta }_2}-\sum _{i=1}^r{w}_i\left(1-z_i\right)+\left(n-n_r\right){\Omega }_2=0,$$

(9)

where ${\psi }_1\left(\alpha \right)=\partial log\Gamma \left(\alpha \right)/\partial \alpha$ and ${\varphi }_i,{\Omega }_i,i=1,2$ are defined by:

$$\begin{array}{c}\hfill {\varphi }_i=\varphi \left(\alpha ,{\beta }_i\right)=\frac{\partial log\Gamma \left(\alpha ,{\beta }_i{w}_r\right)}{\partial \alpha }=\frac{1}{\Gamma \left(\alpha ,{\beta }_i{w}_r\right)}{\int }_{{\beta }_i{w}_r}^{\infty }{t}^{(\alpha -1)}{e}^{-t}logtdt,i=1,2,\end{array}$$

(10)

and

$$\begin{array}{cc}\hfill {\Omega }_i=\Omega \left(\alpha ,{\beta }_i\right)& \hfill =\frac{\partial log\Gamma \left(\alpha ,{\beta }_i{w}_r\right)}{\partial {\beta }_i}=\frac{{\beta }_i^{\alpha -1}{w}_r^{\alpha }{e}^{-{\beta }_i{w}_r}}{\Gamma \left(\alpha ,{\beta }_i{w}_r\right)},i=1,2.\hfill \end{array}$$

(11)

Remark 1. 

We only consider the case that $0<m_r<r$. With the derivation of ${\beta }_1$, if $m_r$ = 0, Equation (8) reduces to:

$$\frac{\partial l}{\partial {\beta }_1}=\left(m-m_r\right){\Omega }_1=\left(m-m_r\right)\frac{\partial log\Gamma \left(\alpha ,{\beta }_1{w}_r\right)}{\partial {\beta }_1}=0.$$

(12)

$log\Gamma \left(\alpha ,{\beta }_1{w}_r\right)$ monotonically increases when ${\beta }_1$ tends to 0. However, the parameter β in the Gamma distribution is greater than 0. Thus, we come to the conclusion that the MLE of ${\beta }_1$ will not exist if $m_r$ = 0. Likewise, if $n_r$ = 0, $\widehat{{\beta }_2}$ will not exist.

It is noticeable that Equations (7)–(9) are nonlinear; thus, the explicit solutions of the MLEs are unavailable. Here, an iterative method, namely the Newton–Raphson method, is chosen to compute $\widehat{\alpha }$, $\widehat{{\beta }_1}$, and $\widehat{{\beta }_2}$. For more details of MLEs, readers can refer to [19].

3. The Asymptotic Confidence Interval

Proposition 1. 

The $(1-\tau )\times 100\%$ asymptotic confidence intervals for α, ${\beta }_1$, and ${\beta }_2$ are derived as follows:

$$(\widehat{\alpha }-z_{\tau /2}\sqrt{var\left(\widehat{\alpha }\right)},\widehat{\alpha }+z_{\tau /2}\sqrt{var\left(\widehat{\alpha }\right)}),$$

(13)

$$(\widehat{{\beta }_1}-z_{\tau /2}\sqrt{var\left(\widehat{{\beta }_1}\right)},\widehat{{\beta }_1}+z_{\tau /2}\sqrt{var\left(\widehat{{\beta }_1}\right)}),$$

(14)

$$(\widehat{{\beta }_2}-z_{\tau /2}\sqrt{var\left(\widehat{{\beta }_2}\right)},\widehat{{\beta }_2}+z_{\tau /2}\sqrt{var\left(\widehat{{\beta }_2}\right)}),$$

(15)

where $\widehat{\alpha }$, $\widehat{{\beta }_1}$, and $\widehat{{\beta }_2}$ denote the MLEs of α, ${\beta }_1$, and ${\beta }_2$, respectively. $var\left(\widehat{\alpha }\right)$, $var\left(\widehat{{\beta }_1}\right)$, and $var\left(\widehat{{\beta }_2}\right)$ are the asymptotic variances of the MLEs. $z_{\tau /2}$ denotes the upper $\tau /2$$\times 100\%$ point of the standard normal distribution, $i.e.$, here, $z_{\tau /2}$ is satisfied with $\tau /2={\int }_{z_{\tau /2}}^{\infty }(1/\sqrt{2\pi })e^{-1/2u^2}du$.

Proof. 

Since the asymptotic normality of $\widehat{\Phi }=(\widehat{\alpha },\widehat{{\beta }_1},\widehat{{\beta }_2})$, $\widehat{\Phi }$ is distributed as the multivariate normal distribution $N_3(\Phi ,Cov(\widehat{\alpha },\widehat{{\beta }_1},\widehat{{\beta }_2}))$, where the vector of unknown parameters ($\alpha$, ${\beta }_1$, ${\beta }_2$) is represented as $\Phi$, and the approximate asymptotic variance–covariance matrix for MLEs is denoted by $Cov(\widehat{\alpha },\widehat{{\beta }_1},\widehat{{\beta }_2})$. Then, the $(1-\tau )\times 100\%$ asymptotic confidence intervals of parameters $\alpha$, ${\beta }_1$, and ${\beta }_2$ shown in Proposition 1 can be derived.    □

$Cov(\widehat{\alpha },\widehat{{\beta }_1},\widehat{{\beta }_2})$ can be calculated by inverting the Fisher information matrix with the elements that are the negative expectation of the second-order derivatives of log-likelihood functions. Here, the observed Fisher information matrix is considered an alternative (see more details in [20]). It is computed by replacing the expected value, for example, $E\left(\frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{W},\mathit{Z}\right)}{\partial {\alpha }^2}\right)$, with $\frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{W},\mathit{Z}\right)}{\partial {\alpha }^2}{|}_{\alpha =\widehat{\alpha }}$.

The observed Fisher information matrix $I(\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2)$ is presented as:

$$I(\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2)=-{\left[\begin{array}{ccc}\frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial {\alpha }^2}& \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial \alpha \partial {\beta }_1}& \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial \alpha \partial {\beta }_2}\\ \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial {\beta }_1\partial \alpha }& \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial {\beta }_1^2}& \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial {\beta }_1\partial {\beta }_2}\\ \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial {\beta }_2\partial \alpha }& \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial {\beta }_2\partial {\beta }_1}& \frac{{\partial }^{2}l\left(\alpha ,{\beta }_1,{\beta }_2;\mathit{w},\mathit{z}\right)}{\partial {\beta }_2^2}\end{array}\right]}_{\alpha =\widehat{\alpha },{\beta }_1={\widehat{\beta }}_1,{\beta }_2={\widehat{\beta }}_2}.$$

For the elements in the matrix, see Appendix A. We invert $I(\widehat{\alpha },{\widehat{\beta }}_1{\widehat{\beta }}_2)$, and obtain the approximate asymptotic variance–covariance matrix for $\widehat{\alpha }$, $\widehat{{\beta }_1}$, and $\widehat{{\beta }_2}$:

$$Cov(\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2)=I^{-1}(\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2)=\left[\begin{array}{ccc}var\left(\widehat{\alpha }\right)& cov(\widehat{\alpha },\widehat{{\beta }_1})& cov(\widehat{\alpha },\widehat{{\beta }_2})\\ cov(\widehat{{\beta }_1},\widehat{\alpha })& var\left(\widehat{{\beta }_1}\right)& cov(\widehat{{\beta }_1},\widehat{{\beta }_2})\\ cov(\widehat{{\beta }_2},\widehat{\alpha })& cov(\widehat{{\beta }_2},\widehat{{\beta }_1})& var\left(\widehat{{\beta }_2}\right)\end{array}\right].$$

4. Bootstrap Methods

A large sample is an essential precondition for the accuracy of the asymptotic confidence interval method introduced in Section 3. If we obtain a small sample size, the MLE for the parameter will not be satisfied with the asymptotic normality. Thus, the asymptotic confidence interval will lose its precision. To overcome this problem, bootstrap methods were introduced (see more details in [21,22]), which are more suitable for small samples. The following steps given in Algorithms 1 and 2 are used to construct $(1-\tau )\times 100\%$ Bootstrap-p and Bootstrap-t confidence intervals for $\alpha$, ${\beta }_1$, and ${\beta }_2$.

Algorithm 1 Generation process of Bootstrap-p confidence intervals.

Step 1  Sample from $Ga(\alpha ,{\beta }_1)$ and $Ga(\alpha ,{\beta }_2)$, respectively. Step 2  Set r as a suitable number of failures. Step 3  Use the procedures described in Section 1 to obtain the observed censored data $(\mathit{w},\mathit{z})$ and compute the MLEs $\widehat{\Phi }=(\widehat{\alpha },\widehat{{\beta }_1},\widehat{{\beta }_2})$. Step 4  Use $\widehat{\Phi }$ to generate new bootstrap samples $(w_1^{*},\cdots ,w_r^{*};z_1^{*},\cdots ,z_r^{*})$ under the same censoring scheme. Step 5  After calculation, obtain new MLEs of the parameters, which are currently denoted by $\widehat{{\alpha }^{*}}$, $\widehat{{\beta }_1^{*}}$, and $\widehat{{\beta }_2^{*}}$. Step 6  Repeat Steps 4–5 M times before deriving $\{{\widehat{\alpha }}_1^{*},\cdots ,{\widehat{\alpha }}_M^{*}\}$, $\{{\widehat{\beta }}_{11}^{*},\cdots ,{\widehat{\beta }}_{1M}^{*}\}$, and $\{{\widehat{\beta }}_{21}^{*},\cdots ,{\widehat{\beta }}_{2M}^{*}\}$. Sort the bootstrap estimates in ascending order; then, we obtain $\{{\widehat{\alpha }}_{\left(1\right)}^{*},\cdots ,{\widehat{\alpha }}_{\left(M\right)}^{*}\}$, $\{{\widehat{\beta }}_{\left(11\right)}^{*},\cdots ,{\widehat{\beta }}_{\left(1M\right)}^{*}\}$, and $\{{\widehat{\beta }}_{\left(21\right)}^{*},\cdots ,{\widehat{\beta }}_{\left(2M\right)}^{*}\}$. Step 7  The $(1-\tau )\times 100\%$ Bootstrap-p confidence intervals for the three parameters are presented by: $$\left({\widehat{\alpha }}_{\left[M\times (\tau /2)\right]}^{*},{\widehat{\alpha }}_{\left[M\times (1-(\tau /2\left)\right)\right]}^{*}\right),$$ (16) $$\left({\widehat{\beta }}_{\left(i\left[M\times (\tau /2)\right]\right)}^{*},{\widehat{\beta }}_{\left(i\left[M\times (1-(\tau /2\left)\right)\right]\right)}^{*}\right),i=1,2,$$ (17) where $[\Psi ]$ denotes the integral part of $\Psi$.

Algorithm 2 Generation process of Bootstrap-t confidence intervals.

Step 1  Repeat Steps 1 to 5 in Algorithm 1. Step 2  After obtaining the estimates $\widehat{{\alpha }_i^{*}}$, $\widehat{{\beta }_{1i}^{*}}$, and $\widehat{{\beta }_{2i}^{*}}$, we compute their approximate asymptotic variance–covariance matrix, namely $Cov({\widehat{\alpha }}_i^{*},{\widehat{\beta }}_{1i}^{*},{\widehat{\beta }}_{2i}^{*})$. $Cov({\widehat{\alpha }}_i^{*},{\widehat{\beta }}_{1i}^{*},{\widehat{\beta }}_{2i}^{*})$ is given by: $$Cov({\widehat{\alpha }}_i^{*},{\widehat{\beta }}_{1i}^{*},{\widehat{\beta }}_{2i}^{*})=I^{-1}({\widehat{\alpha }}_i^{*},{\widehat{\beta }}_{1i}^{*},{\widehat{\beta }}_{2i}^{*}),$$ (18) where $I(\widehat{\alpha },{\widehat{\beta }}_1,{\widehat{\beta }}_2)$ is given in Section 3. Step 3  Record $var\left({\widehat{\alpha }}_i^{*}\right)$, $var\left({\widehat{\beta }}_{1i}^{*}\right)$, and $var\left({\widehat{\beta }}_{2i}^{*}\right)$ from $Cov({\widehat{\alpha }}_i^{*},{\widehat{\beta }}_{1i}^{*},{\widehat{\beta }}_{2i}^{*})$. Step 4  Compute the Bootstrap-t statistics $T_{1i}^{*}$, $T_{2i}^{*}$, and $T_{3i}^{*}$ given by: $$T_{1i}^{*}=\frac{\left({\widehat{\alpha }}_i^{*}-{\widehat{\alpha }}_i\right)}{\sqrt{var\left({\widehat{\alpha }}_i^{*}\right)}},$$ (19) $$T_{2i}^{*}=\frac{\left({\widehat{\beta }}_{1i}^{*}-{\widehat{\beta }}_{1i}\right)}{\sqrt{var\left({\widehat{\beta }}_{1i}^{*}\right)}},\mathrm{and}{T}_{3i}^{*}=\frac{\left({\widehat{\beta }}_{2i}^{*}-{\widehat{\beta }}_{2i}\right)}{\sqrt{var\left({\widehat{\beta }}_{2i}^{*}\right)}}.$$ (20) Step 5  Sort $(T_{11}^{*},\cdots ,T_{1M}^{*})$, $(T_{21}^{*},\cdots ,T_{2M}^{*})$, and $(T_{31}^{*},\cdots ,T_{3M}^{*})$ in ascending order and obtain $(T_{\left(11\right)}^{*},\cdots ,T_{\left(1M\right)}^{*})$, $(T_{\left(21\right)}^{*},\cdots ,T_{\left(2M\right)}^{*})$, and $(T_{\left(31\right)}^{*},\cdots ,T_{\left(3M\right)}^{*})$. Step 6  The $(1-\tau )\times 100\%$ Bootstrap-t confidence intervals for $\alpha$, ${\beta }_1$, and ${\beta }_2$ are given as: $$\left(\widehat{\alpha }+\sqrt{var\left(\widehat{\alpha }\right)}{T}_{\left(1\right[M\times (\tau /2)\left]\right]}^{*},\widehat{\alpha }+\sqrt{\left.var\left(\widehat{\alpha }\right.\right)}{T}_{\left(1\right[M\times (1-(\tau /2\left)\right)\left]\right)}^{*}\right),$$ (21) $$\left({\widehat{\beta }}_1+\sqrt{var\left({\widehat{\beta }}_1\right)}{T}_{\left(2\right[M\times (\tau /2)\left]\right]}^{*},{\widehat{\beta }}_1+\sqrt{\left.var\left({\widehat{\beta }}_1\right.\right)}{T}_{\left(2\right[M\times (1-(\tau /2\left)\right)\left]\right)}^{*}\right),$$ (22) and $$\left({\widehat{\beta }}_2+\sqrt{var\left({\widehat{\beta }}_2\right)}{T}_{\left(3\right[M\times (\tau /2)\left]\right]}^{*},{\widehat{\beta }}_2+\sqrt{\left.var\left({\widehat{\beta }}_2\right.\right)}{T}_{\left(3\right[M\times (1-(\tau /2\left)\right)\left]\right)}^{*}\right),$$ (23) where $[\Psi ]$ denotes the integral part of $\Psi$.

5. Bayesian Inference

The Bayesian inference of $\alpha$, ${\beta }_1$, and ${\beta }_2$ is studied in this section. Unlike the frequentist statistical methods, Bayesian estimation is more objective and reasonable for its flexibility to incorporate prior information into the analysis, and it requires fewer sample data. Thus, the Bayesian inference is particularly valuable, since the limited availability of data is one of the major problems in terms of reliability analysis.

5.1. Prior and Posterior Distribution

Assume that $\alpha$, ${\beta }_1$, and ${\beta }_2$ have independent Gamma priors with corresponding hyperparameters $(a,b)$, $(c_1,d_1)$, and $(c_2,d_2)$. Specifically, the prior functions are defined by:

$${\pi }_0\left(\alpha \right)=\frac{b^a}{\Gamma \left(a\right)}{\alpha }^{a-1}{e}^{-b\alpha },\alpha >0,a>0,b>0,$$

(24)

$${\pi }_0\left({\beta }_1\right)=\frac{d_1^{c_1}}{\Gamma \left(c_1\right)}{\beta }_1^{c_1-1}{e}^{-d_1{\beta }_1},{\beta }_1>0,c_1>0,d_1>0,$$

(25)

and

$${\pi }_0\left({\beta }_2\right)=\frac{d_2^{c_2}}{\Gamma \left(c_2\right)}{\beta }_2^{c_2-1}{e}^{-d_2{\beta }_2},{\beta }_2>0,c_2>0,d_2>0.$$

(26)

Thus, the joint density function of $\alpha$, ${\beta }_1$, and ${\beta }_2$ is as follows:

$${\pi }_0\left(\alpha ,{\beta }_1,{\beta }_2\right)\propto {\alpha }^{a-1}{\beta }_1^{c_1-1}{\beta }_2^{c_2-1}{e}^{-b\alpha -d_1{\beta }_1-d_2{\beta }_2}.$$

(27)

The joint posterior distribution of the three unknown parameters denoted by $\pi \left(\alpha ,{\beta }_1,{\beta }_2\mid \mathit{w},\mathit{z}\right)$ can be derived according to the combination of the likelihood function in Equation (5) and the joint density function in Equation (27) as:

$$\begin{array}{cc}\hfill \pi \left(\alpha ,{\beta }_1,{\beta }_2\mid \mathit{w},\mathit{z}\right)& \propto {\pi }_0\left(\alpha ,{\beta }_1,{\beta }_2\right)L\left(\alpha ,{\beta }_1,{\beta }_2\mid \mathit{w},\mathit{z}\right)\hfill \\ \hfill & \propto {\alpha }^{a-1}{\beta }_1^{\alpha m_r+c_1-1}{\beta }_2^{\alpha n_r+c_2-1}{e}^{-b\alpha +(\alpha -1){\sum }_{r=1}^{n}log{w}_i}{e}^{-{\beta }_1{\sum }_{i=1}^r{w}_i{z}_i-d_1{\beta }_1}\hfill \\ \hfill & \times e^{-{\beta }_2{\sum }_{i=1}^r\left(1-z_i\right)w_i-d_2{\beta }_2}\frac{\Gamma {\left(\alpha ,{\beta }_1{w}_r\right)}^{m-m_r}\Gamma {\left({\alpha }_2,{\beta }_2{w}_r\right)}^{n-n_r}}{\Gamma {\left(\alpha \right)}^{m+n}}.\hfill \end{array}$$

(28)

5.2. Loss Functions

The choice of loss functions is of great importance for parameter estimation in Bayesian inference. In this subsection, we adopt the symmetric square error loss (SEL) function as well as the asymmetric linear exponential (LINEX) loss function.

Suppose that $\widehat{\kappa }(\Phi )$ is an arbitrary estimate of $\kappa (\Phi )$. Here, $\kappa (\Phi )$ means any function of $\Phi =(\alpha ,{\beta }_1,{\beta }_2)$. The SEL function is expressed as follows:

$$L_{SEL}(\widehat{\kappa }(\Phi ),\kappa (\Phi ))={(\widehat{\kappa }(\Phi )-\kappa (\Phi ))}^2.$$

(29)

Let $\kappa {(\Phi )}_{SEL}^{*}$ denote the Bayesian estimate of $\kappa (\Phi )$ under the SEL function, and it can be given as follows:

$$\kappa {(\Phi )}_{SEL}^{*}=E\left[\kappa (\Phi )\mid \mathit{w},\mathit{z}\right].$$

(30)

The estimate can be calculated by:

$$\kappa {(\Phi )}_{SEL}^{*}={\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\kappa (\Phi )\pi \left(\alpha ,{\beta }_1,{\beta }_2\mid \mathit{w},\mathit{z}\right)d\alpha d{\beta }_{1}d{\beta }_2.$$

(31)

Overestimation and underestimation are assigned equal significance by the SEL function. However, the symmetric loss function is not that suitable under the condition that overestimation and underestimation do not have equal weight. To resolve this problem, we adopt the LINEX loss function, as follows:

$$L_{LI}(\widehat{\kappa }(\Phi ),\kappa (\Phi ))=\zeta (e^{h(\widehat{\kappa }(\Phi )-\kappa (\Phi ))}-h(\widehat{\kappa }(\Phi )-\kappa (\Phi ))-1),h\ne 0,\zeta >0.$$

(32)

Without loss of generality, we assume that $\zeta$ in Function (32) equals 1. The shape of the LINEX loss function is up to the value of h. An overestimation weighs more than an underestimation when $h>0$, and for $h<0$, underestimation is more important than the overestimation. The corresponding Bayes estimate $\kappa {(\Phi )}_{LI}^{*}$ is obtained to be:

$$\kappa {(\Phi )}_{LI}^{*}=-\frac{1}{h}log\left[E\left(e^{-h\kappa (\Phi )}\mid \mathit{w},\mathit{z}\right)\right].$$

(33)

It is calculated by:

$$\kappa {(\Phi )}_{LI}^{*}=-\frac{1}{h}log\left({\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }{e}^{-h\kappa (\Phi )}\pi \left(\alpha ,{\beta }_1,{\beta }_2\mid \mathit{w},\mathit{z}\right)d\alpha d{\beta }_{1}d{\beta }_2\right).$$

(34)

It is analytically difficult to obtain explicit solutions to the Bayesian estimates. To solve this problem, the Metropolis–Hastings algorithm is applied to obtain the approximate explicit forms of $\kappa {(\Phi )}_{SEL}^{*}$ and $\kappa {(\Phi )}_{LI}^{*}$ and then construct the corresponding credible intervals.

5.3. Metropolis–Hastings Algorithm

The Markov Chain Monte Carlo (MCMC) technique and the Metropolis–Hastings algorithm are adopted to obtain the explicit forms of Bayesian estimates; thereafter, we establish the highest posterior density (HPD) credible intervals. A detailed discussion of the Metropolis–Hastings algorithm can be found in [23].

Proposition 2. 

$\kappa {(\Phi )}_{SEL}^{*}$ can be calculated by:

$${\alpha }_{SEL}^{*}=\frac{1}{M-M_0}\sum _{j=M_0+1}^M{\alpha }^{\left(j\right)},$$

(35)

$${\beta }_{1SEL}^{*}=\frac{1}{M-M_0}\sum _{j=M_0+1}^M{\beta }_1^{\left(j\right)},$$

(36)

and

$${\beta }_{2SEL}^{*}=\frac{1}{M-M_0}\sum _{j=M_0+1}^M{\beta }_2^{\left(j\right)},$$

(37)

when $\kappa (\Phi )$= α, ${\beta }_1$, and ${\beta }_2$, respectively.

Likewise, $\kappa {(\Phi )}_{LI}^{*}$ can be expressed as:

$${\alpha }_{LI}^{*}=-\frac{1}{h}log\left\{\frac{1}{M-M_0}\sum _{j=M_0+1}^M{e}^{-h{\alpha }^{\left(j\right)}}\right\},$$

(38)

$${\beta }_{1LI}^{*}=-\frac{1}{h}log\left\{\frac{1}{M-M_0}\sum _{j=M_0+1}^M{e}^{-h{\beta }_1^{\left(j\right)}}\right\},$$

(39)

and

$${\beta }_{2LI}^{*}=-\frac{1}{h}log\left\{\frac{1}{M-M_0}\sum _{j=M_0+1}^M{e}^{-h{\beta }_2^{\left(j\right)}}\right\},$$

(40)

where M is the size of the MCMC sample and $M_0$ is the number of burn-in iterative values.

Proof. 

Based on the joint prior distribution in (27), the full conditional distributions of $\alpha$, ${\beta }_1$, and ${\beta }_2$ are expressed as:

$$\begin{array}{cc}\hfill {\pi }_1\left(\alpha \mid {\beta }_1,{\beta }_2,\mathit{w},\mathit{z}\right)& \hfill ={\alpha }^{a-1}{\beta }_1^{\alpha m_r+c_1-1}{\beta }_2^{\alpha n_r+c_2-1}{e}^{-b\alpha +(\alpha -1){\sum }_{i=1}^{n}ln{w}_i}\hfill \\ \hfill & \hfill \times \frac{\Gamma {\left(\alpha ,{\beta }_1{w}_r\right)}^{m-m_r}\Gamma {\left(\alpha ,{\beta }_2{w}_r\right)}^{n-n_r}}{\Gamma {\left(\alpha \right)}^{m+n}},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill {\pi }_2\left({\beta }_1\mid \alpha ,{\beta }_2,\mathit{w},\mathit{z}\right)& \hfill ={\beta }_1^{\alpha m_r+c_1-1}{e}^{-{\beta }_1{\sum }_{i=1}^r{w}_i{z}_i-d_1{\beta }_1}\Gamma {\left(\alpha ,{\beta }_1{w}_r\right)}^{m-m_r},\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {\pi }_3\left({\beta }_2\mid \alpha ,{\beta }_1,\mathit{w},\mathit{z}\right)& \hfill ={\beta }_2^{\alpha n_r+c_2-1}{e}^{-{\beta }_2{\sum }_{i=1}^r{w}_i(1-z_i)-d_2{\beta }_2}\Gamma {\left(\alpha ,{\beta }_2{w}_r\right)}^{n-n_r}.\hfill \end{array}$$

(43)

From Functions (41)–(43), it is observed that these full conditional distributions do not follow any well-known distribution, respectively. For deriving Bayesian estimates, the Metropolis–Hastings algorithm is applied. The proposal distribution for $\Phi =(\alpha ,{\beta }_1,{\beta }_2)$ is supposed to be the multivariate normal distribution and MCMC samples are generated by the Metropolis–Hastings technique described in Algorithm 3.

Algorithm 3 Steps for generating samples from the posterior distribution in (27).

Step 1 Set ${\Phi }_0=({\alpha }^{\left(0\right)},{\beta }_1^{\left(0\right)},{\beta }_2^{\left(0\right)})$ to be the initial values of parameters. Step 2 Let $j=1$. Step 3 Generate a proposal ${\Phi }^{*}=({\alpha }^{*},{\beta }_1^{*},{\beta }_2^{*})$ from the multivariate normal distribution $N_3({\Phi }_{j-1},Cov(\widehat{\alpha },\widehat{{\beta }_1},\widehat{{\beta }_2}))$, where ${\Phi }_{j-1}=({\alpha }^{(j-1)},{\beta }_1^{(j-1)},{\beta }_2^{(j-1)})$. Step 4 Obtain $\upsilon =min\left\{\frac{\pi \left({\Phi }^{*}\mid \mathit{w},\mathit{z}\right)}{\pi \left({\Phi }_{j-1}\mid \mathit{w},\mathit{z}\right)},1\right\}$ and q from $U(0,1)$, the Uniform distribution. Step 5 If $q<\upsilon$, let ${\Phi }_j={\Phi }^{*}$; else, let ${\Phi }_j={\Phi }_{j-1}$. Step 6 Set $j=j+1$. Step 7 Obtain $\{{\Phi }_1,{\Phi }_2,\cdots ,{\Phi }_M\}$ after repeating Steps 3–6 M times.

Removing the first $M_0$ burn-in iterative values, we obtain $\kappa {(\Phi )}_{SEL}^{*}$ and $\kappa {(\Phi )}_{LI}^{*}$ in Proposition 2.    □

Next, we calculate the $(1-\tau )\times 100\%$ HPD credible intervals for $\alpha$, ${\beta }_1$, and ${\beta }_2$. For more details of HPD credible intervals, see [24]. We sort the remaining $M-M_0$ values in ascending order to be $\{{\alpha }_{\left(1\right)},\cdots ,{\alpha }_{\left(M-M_0\right)}\}$, $\{{\beta }_{\left(11\right)},\cdots ,{\beta }_{\left(1,M-M_0\right)}\}$, and $\{{\beta }_{\left(21\right)},\cdots ,{\beta }_{\left(2,M-M_0\right)}\}$. The $(1-\tau )\times 100\%$ HPD credible intervals for $\alpha$, ${\beta }_1$ and ${\beta }_2$ can be given as:

$$\left({\alpha }_{\left(s^{*}\right)},{\alpha }_{\left(s^{*}+(1-\tau )\times \left(M-M_0\right)\right)}\right)\mathrm{and}\left({\beta }_{\left(i{t}_i^{*}\right)},{\beta }_{\left(i\left[t_i^{*}+(1-\tau )\times \left(M-M_0\right)\right]\right)}\right),i=1,2,$$

(44)

where $s^{*},t_i^{*}$, $i=1,2$ are selected if:

$${\alpha }_{\left(s^{*}+\left[(1-\tau )\times \left(M-M_0\right)\right]\right)}-{\alpha }_{\left(s^{*}\right)}=\underset{1\le s\le \left(M-M_0\right)-\left[(1-\tau )\times \left(M-M_0\right)\right]}{min}\left({\alpha }_{\left(s+\left[(1-\tau )\times \left(M-M_0\right)\right]\right)}-{\alpha }_{\left(s\right)}\right)$$

(45)

and

$${\beta }_{\left(i,t_i^{*}+\left[(1-\tau )\times \left(M-M_0\right)\right]\right)}-{\beta }_{\left(i{t}_i^{*}\right)}=\underset{1\le t_i\le \left(M-M_0\right)-\left[(1-\tau )\times \left(M-M_0\right)\right]}{min}\left({\beta }_{\left(i,t_i^{*}+\left[(1-\tau )\times \left(M-M_0\right)\right]\right)}-{\beta }_{\left(i{t}_i\right)}\right).$$

(46)

Likewise, $\left[(1-\tau )\times (M-M_0)\right]$ denotes the integral part of $(1-\tau )\times (M-M_0)$.

6. Simulations and Real Data Analysis

6.1. Monte Carlo Simulations

To investigate the effectiveness of frequentist and Bayesian estimations produced in the preceding sections, Monte Carlo simulations were carried out. To begin with, we sampled from two populations, namely Popu-A and Popu-B, following $Ga(\alpha ,{\beta }_1)$, and $Ga(\alpha ,{\beta }_2)$, respectively. $\alpha =0.50$, ${\beta }_2=0.25$, and ${\beta }_2=0.40$ were set as true values of the parameters. For the sample sizes $(m,n)$, we chose $(40,40),(60,65),$ and $(80,80)$, and for the number of failures, we chose $r=65,70,75,85,100,115,125,150$. The process of generating the joint Type-II censoring data is shown in Algorithm 4.

Algorithm 4 Steps for generating jointly Type-II censored datasets.

Step 1  Generate two samples from $Ga(\alpha ,{\beta }_1)$ and $Ga(\alpha ,{\beta }_2)$, respectively. Thus, we obtain $x_1,\cdots ,x_m$ from Popu-A and $y_1,\cdots ,y_n$ from Popu-B. Step 2  Combine the two samples we obtained in Step 1 as $(\mathbf{x};\mathbf{y})=(x_1,\cdots ,x_m;y_1,\cdots ,y_n)$. Sort $(\mathbf{x};\mathbf{y})$ in ascending order and denote them as $w_1<w_2<\cdots <w_N$, where $N=m+n$. Step 3  Choose a proper number of failures, namely r. Retain the first r values, and we obtain $w_1,w_2,\cdots ,w_r$. Step 4  Calculate $z_1,z_2,\cdots ,z_r$ where $z_i=\left\{\begin{array}{ccc}1;\hfill & \mathrm{if}{w}_i\in \mathrm{Popu}-\mathrm{A},0;\hfill & \mathrm{else}.\end{array}\right.$ Step 5  Record the joint censoring data as $(\mathit{W},\mathit{Z})=\{(w_1,z_1),(w_2,z_2),\cdots ,(w_r,z_r)\}$.

After obtaining the censoring data, the MLEs $\widehat{\alpha }$, $\widehat{{\beta }_1}$, and $\widehat{{\beta }_2}$ are calculated using the likelihood function in Equation (5) and Newton–Raphson method; then, we presented the mean square errors (MSEs) of the MLEs. In terms of the Bayesian estimations, the estimates as well as the MSEs under the SEL and LINEX loss functions with both the Gamma priors and the noninformative priors are obtained. For interval estimates, we constructed asymptotic confidence intervals (ACIs), bootstrap confidence intervals (Bootstrap CIs), and HPD credible intervals. After that, the average lengths (ALs) and coverage probabilities (CPs) of these intervals were also given based on 1000 repetitions.

For comparing the performance of the Bayesian estimates, $(a,b,c_1,c_2,d_1,d_2)=(1,2,1,2,4,5)$ were chosen as the hyperparameters of informative priors so that the means of the prior density equaled the true parameter values. For the noninformative priors, we put $(a,b,c_1,c_2,d_1,d_2)=(0,0,0,0,0,0)$. Additionally, $h=2$ was taken into account when calculating $\kappa {(\Phi )}_{LI}^{*}$. Based on the Metropolis–Hastings algorithm, $M=5000$ MCMC samples were generated, and the first $M_0=1000$ iterative values were taken as a burn-in period.

The statistical software R was utilized for calculation in this article. For deriving the MLEs of the parameters in the Newton–Raphson approach, we chose the real values to be the initial values in order to maximize the likelihood function.

Table 1. The averages and MSEs of MLEs and Bayes estimates with the Gamma priors.

$(\mathit{m},\mathit{n})$	r	Parameter	MLE (MSE)	Bayes Estimates
				SEL (MSE)	LINEX (MSE)
$(40,40)$	65	$\alpha$	0.5234 (0.0068)	0.5066 (0.0074)	0.5057 (0.0161)
		${\beta }_1$	0.2809 (0.0103)	0.2658 (0.0389)	0.2596 (0.0954)
		${\beta }_2$	0.4446 (0.0211)	0.4241 (0.1223)	0.4088 (0.0224)
	70	$\alpha$	0.5206 (0.0058)	0.5064 (0.0117)	0.5037 (0.0215)
		${\beta }_1$	0.2783 (0.0087)	0.2663 (0.0081)	0.2611 (0.0151)
		${\beta }_2$	0.4401 (0.0192)	0.4221 (0.0314)	0.4110 (0.0552)
	75	$\alpha$	0.5198 (0.0054)	0.5063 (0.0035)	0.5046 (0.0083)
		${\beta }_1$	0.2722 (0.0061)	0.2626 (0.0117)	0.2584 (0.0262)
		${\beta }_2$	0.4329 (0.0153)	0.4181 (0.0077)	0.4084 (0.0159)
$(60,65)$	85	$\alpha$	0.5152 (0.0042)	0.5088 (0.0056)	0.5046 (0.0103)
		${\beta }_1$	0.2741 (0.0089)	0.2692 (0.0041)	0.2635 (0.0077)
		${\beta }_2$	0.4375 (0.0175)	0.4174 (0.0128)	0.4075 (0.0236)
	100	$\alpha$	0.5140 (0.0038)	0.5062 (0.0055)	0.5039 (0.0105)
		${\beta }_1$	0.2730 (0.0064)	0.2663 (0.0042)	0.2624 (0.0080)
		${\beta }_2$	0.4260 (0.0121)	0.4159 (0.0059)	0.4083 (0.0118)
	115	$\alpha$	0.5076 (0.0031)	0.5008 (0.0039)	0.5010 (0.0081)
		${\beta }_1$	0.2639 (0.0043)	0.2584 (0.0017)	0.2557 (0.0035)
		${\beta }_2$	0.4215 (0.0095)	0.4130 (0.0050)	0.4070 (0.0105)
$(80,80)$	100	$\alpha$	0.5124 (0.0035)	0.5064 (0.0033)	0.5054 (0.0074)
		${\beta }_1$	0.2727 (0.0073)	0.2622 (0.0136)	0.2615 (0.0420)
		${\beta }_2$	0.4315 (0.0154)	0.4170 (0.0109)	0.4077 (0.0232)
	125	$\alpha$	0.5120 (0.0030)	0.5050 (0.0008)	0.5034 (0.0017)
		${\beta }_1$	0.2701 (0.0052)	0.2638 (0.0024)	0.2608 (0.0046)
		${\beta }_2$	0.4229 (0.0097)	0.4134 (0.0048)	0.4073 (0.0094)
	150	$\alpha$	0.5103 (0.0025)	0.5050 (0.0052)	0.5037 (0.0110)
		${\beta }_1$	0.2628 (0.0029)	0.2591 (0.0013)	0.2574 (0.0027)
		${\beta }_2$	0.4182 (0.0070)	0.4131 (0.0028)	0.4076 (0.0055)

and

Table 2. The averages and MSEs of the MLEs and Bayes estimates with the noninformative priors.

$(\mathit{m},\mathit{n})$	r	Parameter	MLE (MSE)	Bayes Estimates
				SEL (MSE)	LINEX (MSE)
$(40,40)$	65	$\alpha$	0.5215 (0.0059)	0.5120 (0.0456)	0.5078 (0.0500)
		${\beta }_1$	0.2790 (0.0098)	0.2735 (0.0865)	0.2665 (0.0611)
		${\beta }_2$	0.4407 (0.0204)	0.4331 (0.1236)	0.4183 (0.0520)
	70	$\alpha$	0.5185 (0.0062)	0.5083 (0.0041)	0.5044 (0.0081)
		${\beta }_1$	0.2780 (0.0082)	0.2712 (0.0063)	0.2652 (0.0134)
		${\beta }_2$	0.4360 (0.0188)	0.4272 (0.0148)	0.4141 (0.0635)
	75	$\alpha$	0.5105 (0.0062)	0.5089 (0.0021)	0.5032 (0.0040)
		${\beta }_1$	0.2756 (0.0068)	0.2684 (0.0022)	0.2637 (0.0045)
		${\beta }_2$	0.4373 (0.0084)	0.4281 (0.0158)	0.4135 (0.0740)
$(60,65)$	85	$\alpha$	0.5113 (0.0039)	0.5032 (0.0139)	0.5010 (0.0253)
		${\beta }_1$	0.2760 (0.0089)	0.2706 (0.0098)	0.2643 (0.0180)
		${\beta }_2$	0.4264 (0.0144)	0.4191 (0.0125)	0.4076 (0.0237)
	100	$\alpha$	0.5111 (0.0036)	0.5013 (0.0060)	0.4991 (0.0130)
		${\beta }_1$	0.2668 (0.0064)	0.2606 (0.0040)	0.2566 (0.0076)
		${\beta }_2$	0.4252 (0.0120)	0.4146 (0.0168)	0.4064 (0.0392)
	115	$\alpha$	0.5093 (0.0033)	0.5025 (0.0108)	0.5006 (0.0236)
		${\beta }_1$	0.2644 (0.0042)	0.2595 (0.0142)	0.2568 (0.0325)
		${\beta }_2$	0.4222 (0.0097)	0.4169 (0.0174)	0.4053 (0.0435)
$(80,80)$	100	$\alpha$	0.5131 (0.0037)	0.5049 (0.0041)	0.5048 (0.0082)
		${\beta }_1$	0.2701 (0.0071)	0.2646 (0.0076)	0.2595 (0.0167)
		${\beta }_2$	0.4325 (0.0155)	0.4235 (0.0183)	0.4133 (0.0364)
	125	$\alpha$	0.5114 (0.0028)	0.5068 (0.0129)	0.5032 (0.0287)
		${\beta }_1$	0.2646 (0.0041)	0.2615 (0.0101)	0.2586 (0.0222)
		${\beta }_2$	0.4222 (0.0092)	0.4192 (0.0489)	0.4125 (0.1201)
	150	$\alpha$	0.5088 (0.0024)	0.5037 (0.0030)	0.5023 (0.0064)
		${\beta }_1$	0.2624 (0.0032)	0.2593 (0.0017)	0.2575 (0.0035)
		${\beta }_2$	0.4167 (0.0071)	0.4139 (0.0199)	0.4092 (0.0456)

present the averages of the MLEs based on 5000 repetitions of the Newton–Raphson method. The Bayes estimates with Gamma priors under the SEL and LINEX loss functions are shown in Table 1, whereas Table 2 presents the estimates for the noninformative priors. It is noted that all the estimates of the parameters were substantially unbiased, with fairly low MSEs. Mostly, $\widehat{\alpha }$ owned much smaller MSEs, indicating that $\alpha$ obtained better estimates. The result makes sense given that $\alpha$ is considered the same in the two Gamma populations. Mostly, the MSEs of the Bayes estimates under the LINEX loss function were the largest. However, the averages of these estimates were closest to the true values.

The comparison of the MLEs and Bayes estimates under the same number of samples with different r is shown in Figure 4, Figure 5 and Figure 6 (taking $m=60,n=65,r=85,100,115$ as an example). The estimates of the parameters moved closer to the real values as the failure information increased, that is, as the total number of units from two production lines N and the number of failures r increased. This makes sense, since the larger observed sample sizes allowed for the more effective utilization of the datasets’ available information and the generation of more accurate estimations. Frequently, we noted that the Bayes estimates with Gamma priors performed better.

Table 3. The average lengths of the ACIs, Bootstrap CIs, and HPD credible intervals for $\alpha$ along with their CPs in parentheses (the nominal level is set to be 95%).

$(\mathit{m},\mathit{n})$	r	ACI (CP)	Boot-p (CP)	Boot-t (CP)	HPD Credible Interval
					Info. (CP)	Non-Info. (CP)
$(40,40)$	65	0.2926 (0.952)	0.3135 (0.917)	0.2853 (0.946)	0.2537 (0.856)	0.2950 (0.825)
	70	0.2827 (0.959)	0.3011 (0.924)	0.2757 (0.945)	0.1312 (0.861)	0.2249 (0.816)
	75	0.2746 (0.955)	0.2922 (0.926)	0.2684 (0.951)	0.1216 (0.843)	0.1347 (0.813)
$(60,65)$	85	0.2455 (0.950)	0.2583 (0.912)	0.2413 (0.945)	0.1482 (0.846)	0.1742 (0.793)
	100	0.2307 (0.953)	0.2423 (0.931)	0.2281 (0.948)	0.1451 (0.813)	0.1433 (0.800)
	115	0.2179 (0.952)	0.2272 (0.921)	0.2152 (0.949)	0.1554 (0.794)	0.1434 (0.786)
$(80,80)$	100	0.2234 (0.961)	0.2332 (0.927)	0.2204 (0.950)	0.1454 (0.803)	0.2162 (0.770)
	125	0.2048 (0.942)	0.2108 (0.930)	0.2012 (0.942)	0.1088 (0.794)	0.1864 (0.759)
	150	0.1902 (0.945)	0.1959 (0.939)	0.1880 (0.950)	0.1570 (0.728)	0.1575 (0.741)

,

Table 4. The average lengths of the ACIs, Bootstrap CIs, and HPD credible intervals for ${\beta }_1$ along with their CPs in parentheses (the nominal level is set to be 95%).

$(\mathit{m},\mathit{n})$	r	ACI (CP)	Boot-p (CP)	Boot-t (CP)	HPD Credible Interval
					info. (CP)	non-info. (CP)
$(40,40)$	65	0.3597 (0.954)	0.4060 (0.923)	0.3573 (0.946)	0.3268 (0.865)	0.3217 (0.864)
	70	0.3304 (0.954)	0.3686 (0.921)	0.3242 (0.936)	0.3922 (0.853)	0.2763 (0.841)
	75	0.2990 (0.967)	0.3359 (0.929)	0.2949 (0.944)	0.2705 (0.861)	0.1919 (0.853)
$(60,65)$	85	0.3498 (0.948)	0.3769 (0.929)	0.3453 (0.944)	0.2642 (0.878)	0.2309 (0.833)
	100	0.2918 (0.949)	0.3143 (0.933)	0.2888 (0.948)	0.2832 (0.856)	0.2321 (0.824)
	115	0.2430 (0.958)	0.2628 (0.934)	0.2420 (0.948)	0.2134 (0.821)	0.2920 (0.783)
$(80,80)$	100	0.3239 (0.957)	0.3541 (0.931)	0.3288 (0.931)	0.3220 (0.845)	0.2677 (0.833)
	125	0.2579 (0.947)	0.2690 (0.927)	0.2518 (0.938)	0.1350 (0.832)	0.2132 (0.816)
	150	0.2057 (0.956)	0.2171 (0.940)	0.2036 (0.959)	0.1162 (0.777)	0.1485 (0.746)

and

Table 5. The average lengths of the ACIs, Bootstrap CIs, and HPD credible intervals for ${\beta }_2$ along with their CPs in parentheses (the nominal level is set to be 95%).

$(\mathit{m},\mathit{n})$	r	ACI (CP)	Boot-p (CP)	Boot-t (CP)	HPD Credible Interval
					Info. (CP)	Non-Info. (CP)
$(40,40)$	65	0.5152 (0.958)	0.5765 (0.917)	0.5099 (0.938)	0.3101 (0.916)	0.6916 (0.875)
	70	0.4808 (0.945)	0.5353 (0.929)	0.4735 (0.950)	0.3672 (0.899)	0.2490 (0.901)
	75	0.4493 (0.959)	0.5140 (0.921)	0.4515 (0.951)	0.3367 (0.900)	0.5647 (0.886)
$(60,65)$	85	0.4684 (0.959)	0.5142 (0.921)	0.4713 (0.944)	0.5108 (0.909)	0.3763 (0.870)
	100	0.4000 (0.949)	0.4331 (0.928)	0.4000 (0.950)	0.3020 (0.879)	0.3499 (0.867)
	115	0.3560 (0.947)	0.3824 (0.915)	0.3529 (0.946)	0.2405 (0.867)	0.4426 (0.848)
$(80,80)$	100	0.4486 (0.953)	0.4827 (0.938)	0.4494 (0.947)	0.3777 (0.803)	0.5090 (0.853)
	125	0.3630 (0.952)	0.3820 (0.936)	0.3590 (0.950)	0.2601 (0.872)	0.3878 (0.863)
	150	0.3098 (0.955)	0.3289 (0.931)	0.3082 (0.941)	0.1933 (0.866)	0.4159 (0.835)

display the 95% ACIs, Bootstrap CIs, and also the HPD credible intervals for $\alpha$, ${\beta }_1$, and ${\beta }_2$. In the following tables, the Bootstrap-p CI is represented as Boot-p, and the Bootstrap-t CI is represented as Boot-t. As for the HPD credible intervals, we let $info.$ denote informative priors and non-info. denote noninformative priors.

Although their ALs were substantially longer than those of the Bootstrap Cls and the HPD credible intervals, it is notable that the ACls had the largest CPs. According to the comparison of the bootstrap methods, the Bootstrap-t CIs had higher CPs and shorter ALs than the Bootstrap-p Cls, which was more suitable for interval construction. Both the ACLs and Bootstrap CIs were close to the nominal level (95%). In most cases, the ALs of the HPD credible intervals for noninformative priors tended to be longer and the CPs were less than those of informative priors. However, the CPs of the HPD credible intervals were relatively lower than the norminal level (95%), which may have a significant effect on the interval construction. Further research is anticipated on creating appropriate techniques to raise the values of the CPs.

The plots of the CPs under different methods for the three parameters are shown in Figure 7, Figure 8 and Figure 9 (taking $(m,n,r)=(40,40,70),(60,65,85),\mathrm{and}(80,80,150)$ as an example). It can be observed from the plots that the Gamma priors performed better, and the Bootstrap-t method was more suitable for interval estimates. Histograms along with the simulation values of the parameters produced by the MCMC method under the SEL and LINEX loss functions are shown in Figure 10, Figure 11 and Figure 12 (taking $(m,n,r)=(40,40,70)$ as an example).

6.2. Real Data Analysis

In this subsection, an analysis of real engineering datasets is reported to illustrate the performance of all the preceding methods. We analyzed the data provided by [25], which contained the time to break down an insulating fluid between electrodes. Here, we selected two sets when the voltage was at 32 KV and 34 KV, which are shown in

Table 6. The lifetimes of an insulating fluid at 32 KV and 34 KV.

Datasets
Set 1	0.27	0.40	0.69	0.79	2.75
(Breakdown at 32 KV)	3.91	9.88	13.95	15.93	27.80
	53.24	82.85	89.29	100.58	215.10
Set 2	0.19	0.78	0.96	1.31	2.78
(Breakdown at 34 KV)	3.16	4.15	4.67	4.85	6.50
	7.35	8.01	8.27	12.06	31.75
	32.52	33.91	36.71	72.89

.

The goodness-of-fit tests were conducted using Kolmogrov–Smirnov (K-S) statistics and other measures, such as the Anderson–Darling (A-D) test and the Cramer–Von Mises (C-M) test. The results for illustrating the fitting power of the Gamma model are shown in

Table 7. The MLEs and test statistics.

Measure	Datasets	Test Statistics	Critical Values
K-S test	Set 1	0.1267	0.2430
	Set 2	0.1847	0.2042
A-D test	Set 1	0.3241	0.8718
	Set 2	0.4890	0.7820
C-M test	Set 1	0.0468	0.1393
	Set 2	0.0914	0.1431

.

Some other lifecycle distributions, including the Lindley distribution and the exponentiated exponential (EE) distribution, were also compared with the Gamma model under the K-S test. Their PDFs were defined by:

$$Lindleydistribution:f(x;\alpha )=\left(\frac{{\alpha }^2}{1+\alpha }\right)e^{-\alpha x}(1+x),x>0,\alpha >0;$$

(47)

$$EEdistribution:f(x;\alpha ,\beta )=\alpha \beta e^{-\alpha x}{\left(1-e^{-\alpha x}\right)}^{\beta -1},x>0,\alpha ,\beta >0.$$

(48)

Table 8. The MLEs and K-S distances for different distributions.

Distribution	Datasets	$\mathit{\alpha }$	$\mathit{\beta }$	K-S Distances	p-Value
Gamma distribution	Set 1	0.4359	0.0106	0.1267	0.9446
	Set 2	0.6897	0.0480	0.1847	0.4802
Lindley distribution	Set 1	0.0475	-	0.4082	0.0090
	Set 2	0.1312	-	0.3462	0.0154
EE distribution	Set 1	0.0132	0.4288	0.1252	0.9494
	Set 2	0.0534	0.6825	0.1886	0.4535

presents that the K-S distances and p-values of the Gamma model and EE model were close in dataset 1; however, the Gamma model performed better in dataset 2 for having a significantly shorter K-S distance than the EE distribution. Both of the two models fit the data better than the Lindley distribution.

Next, we calculated the MLEs for the four parameters. In addition, it is of importance to study the null hypothesis $H_0:{\alpha }_1={\alpha }_2$ (i.e., the shape parameters are equal) versus the alternative hypothesis $H_1:{\alpha }_1\ne {\alpha }_2$ based on the likelihood ratio test. For the given data, the test statistic was calculated as ${\rm Y}=L_1/L_2$, and the p-value was computed as 0.5263, so the assumption of ${\alpha }_1={\alpha }_2$ could not be rejected. Thus, we considered that ${\alpha }_1={\alpha }_2=\alpha$.

We ran the joint Type-II censoring scheme on these datasets with r as 27, and the censored data obtained from two samples are shown in

Table 9. The joint Type-II censoring data.

	Censored Data
	0.19	0.27	0.40	0.69	0.78	0.79	0.96	1.31	2.75
$\mathit{w}$	2.78	3.16	3.91	4.15	4.67	4.85	6.50	7.35	8.01
	8.27	9.88	12.06	13.95	15.93	27.80	31.75	32.52	33.91
	0	1	1	1	0	1	0	0	1
$\mathit{z}$	0	0	1	0	0	0	0	0	0
	0	1	0	1	1	1	0	0	0

.

Based on the observed censored data in Table 9 and the methods adopted in Section 2, we calculated the MLEs of $\alpha$, ${\beta }_1$, and ${\beta }_2$. The values are shown in

Table 10. The MLEs and Bayes estimates under the SEL and LINEX loss functions.

Parameter	MLE	Bayes Estimates
		SEL	LINEX
$\alpha$	0.5777	0.5902	0.5811
${\beta }_1$	0.0187	0.0200	0.0199
${\beta }_2$	0.0400	0.0430	0.0430

.

The three-dimensional diagram and the contour plot of the negative value of the log-likelihood function are presented in Figure 13. Both of the plots show that if the value of $\alpha$ is fixed, the best estimate values for ${\beta }_1$ and ${\beta }_2$ are close to 0.0200 and 0.0400, respectively. Furthermore, the plots demonstrate the existence and uniqueness of MLEs.

In order to obtain the Bayes estimates of $\alpha$, ${\beta }_1$, and ${\beta }_2$, 10,000 samples were generated based on the Metropolis–Hastings algorithm. These Bayes estimates were obtained after discarding the initial 2000 burn-in iterative values. Since we do not know the model parameters in advance, Bayes estimates were calculated with the noninformative priors, and the hyperparameters $(a,b,c_1,c_2,d_1,d_2)$ were set to be zeros. The MLEs of the unknown parameters were considered as their initial values, while the diagonal elements of the reciprocal of the observed Fisher information matrix were considered as the variance of the MLEs. The estimates are displayed in Table 10. The 95% Bootstrap-p CIs and Bootstrap-t CIs, together with the HPD credible intervals, are presented in

Table 11. The 95% interval estimations of three parameters under diverse methods.

Parameter	ACI	Boot-p CI	Boot-t CI	HPD Credible Interval
				Non-Info.
$\alpha$	(0.3238, 0.8317)	(0.4160, 0.9204)	(0.3313, 0.7964)	(0.4002, 0.7200)
${\beta }_1$	(0.0000, 0.0395)	(0.0046, 0.0807)	(0.0000, 0.0338)	(0.0101, 0.0167)
${\beta }_2$	(0.0075, 0.0726)	(0.0137, 0.1362)	(0.0000, 0.0654)	(0.0371, 0.0499)

. Again, the Bootstrap-t method performed better in terms of the AL.

7. Conclusions

There are situations where experimenters are used to test the reliability of various populations from separated lines of production. In such cases, the joint censoring scheme is recommended. This article takes into account the statistical inference of two Gamma distributions with the same shape parameter but distinct scale parameters under the joint Type-II censoring scheme. The Newton–Raphson method was applied to obtain the maximum likelihood estimates. Moreover, the asymptotic confidence intervals were constructed. Bootstrap methods were also utilized for interval estimations.

We considered the Bayes estimations using independent Gamma priors and noninformative priors; then, we constructed HPD credible intervals. The difference in diverse loss functions was also displayed. Due to the complexity of the posterior distribution, the Metropolis–Hastings algorithm was utilized to create samples for Bayesian inference. With these procedures, we carried out Monte Carlo simulations under various censoring schemes in which the point estimators and confidence/credible intervals were compared. Two real engineering datasets were analyzed in order to appraise the estimation methods throughout the study.

Regarding future studies, more work is expected to be conducted on the study of two Gamma populations under other censoring schemes. The case of different shape parameters is also worthy of further exploration. The concept of generalizing two populations to k$(k>2)$ populations can also be applied to the Gamma distribution.