Statistical Evaluation of Alpha-Powering Exponential Generalized Progressive Hybrid Censoring and Its Modeling for Medical and Engineering Sciences with Optimization Plans

Abstract

This study explores advanced methods for analyzing the two-parameter alpha-power exponential (APE) distribution using data from a novel generalized progressive hybrid censoring scheme. The APE model is inherently asymmetric, exhibiting positive skewness across all valid parameter values due to its right-skewed exponential base, with the alpha-power transformation amplifying or dampening this skewness depending on the power parameter. The proposed censoring design offers new insights into modeling lifetime data that exhibit non-monotonic hazard behaviors. It enhances testing efficiency by simultaneously imposing fixed-time constraints and ensuring a minimum number of failures, thereby improving inference quality over traditional censoring methods. We derive maximum likelihood and Bayesian estimates for the APE distribution parameters and key reliability measures, such as the reliability and hazard rate functions. Bayesian analysis is performed using independent gamma priors under a symmetric squared error loss, implemented via the Metropolis–Hastings algorithm. Interval estimation is addressed using two normality-based asymptotic confidence intervals and two credible intervals obtained through a simulated Markov Chain Monte Carlo procedure. Monte Carlo simulations across various censoring scenarios demonstrate the stable and superior precision of the proposed methods. Optimal censoring patterns are identified based on the observed Fisher information and its inverse. Two real-world case studies—breast cancer remission times and global oil reserve data—illustrate the practical utility of the APE model within the proposed censoring framework. These applications underscore the model’s capability to effectively analyze diverse reliability phenomena, bridging theoretical innovation with empirical relevance in lifetime data analysis.

1. Introduction

In life-testing experiments, a primary objective for reliability practitioners is to terminate the test before all test units have failed, often due to limitations in time or resources. Among censoring schemes, progressive Type-II (PT2) censoring has gained significant traction in reliability and survival analysis, owing to its flexibility and efficiency compared to conventional Type-II (or hybrid) censoring. This censoring is particularly suitable in contexts such as industrial testing and clinical trials, where ongoing removal of functioning units is both practical and desirable. Assume that n identical units are placed on test, and the experiment continues until exactly m failures are observed. Let $\mathbf{S}=(S_1,S_2,\dots ,S_m)$ denote a pre-specified sequence of non-negative integers such that $S_i$ units are removed at the time of the ith failure. Specifically, at the time of the first failure (denoted $Y_{1:m:n}$), $S_1$ surviving units are randomly withdrawn from the remaining $n-1$. Upon the second failure ($Y_{2:m:n}$), $S_2$ units are removed from the $n-2-S_1$ still under observation, and so on. Following the mth failure, the remaining $S_m$ units are withdrawn, and the test concludes. For further details, see Balakrishnan and Cramer [1].

One limitation of the PT2 censoring is that, when testing highly reliable items, the total testing time may become prohibitively long. To address this, Kundu and Joarder [2] introduced Type-I progressive hybrid (PHT1) censoring, in which the test is terminated at $T^{•}=min\{T,Y_{m:m:n}\}$, where T is a pre-specified time limit. While this approach reduces the risk of excessively long tests, it suffers from the drawback that when few failures are observed before time T, the resulting data may be insufficient for reliable parameter estimation. To address this issue, Cho et al. [3] proposed the generalized PH (G-PH) censoring as an improvement of the PHT1 design. This method ensures the observation of a preassigned number of failures by explicitly controlling both the number of failures and the corresponding failure times. The G-PH censoring seeks to balance test duration and cost efficiency while improving the quality of statistical inference through a reduction in the proportion of censored data.

Let $\{Y_1,Y_2,\dots ,Y_n\}$ denote the lifetimes of n identical units, assumed to follow a continuous distribution with cumulative distribution function (CDF) $G(y;\mathit{\lambda })$ and probability density function (PDF) $g(y;\mathit{\lambda })$, where $\mathit{\lambda }$ is an unknown parameter vector. Consider a pre-specified fixed time $T>0$ (with $T\in (0,\infty )$), and let $m_1<m_2\le n$ be predetermined integers. Additionally, let $\mathbf{S}=(S_1,S_2,\dots ,S_{m_2})$ be a fixed progressive censoring scheme such that $m_2+{\sum }_{i=1}^{m_2}{S}_i=n.$ Under this plan, the test proceeds as follows: when the first failure is observed at $Y_{1:m_2:n}$, $S_1$ units are randomly removed from the remaining $n-1$ surviving units. Upon the second failure at $Y_{2:m_2:n}$, $S_2$ units are withdrawn from the remaining $n-2-S_1$ units, and this process continues accordingly. The experiment is terminated at the generalized stopping time $T^{•}=max\left\{Y_{m_1:m_2:n},min\{Y_{m_2:m_2:n},T\}\right\}.$ As depicted in Figure 1, this setup gives rise to the following three mutually exclusive censoring scenarios:

• Case 1: If $T<Y_{m_1:m_2:n}<Y_{m_2:m_2:n}$, end the test at $Y_{m_1:m_2:n}$;
• Case 2: If $Y_{m_1:m_2:n}<T<Y_{m_2:m_2:n}$, end the test at T. Let d denote the number of failures observed up to time T;
• Case 3: If $Y_{m_1:m_2:n}<Y_{m_2:m_2:n}<T$, end the test at $Y_{m_2:m_2:n}$.

Now, let $\mathbf{y}=\{y_{1:m_2:n},y_{2:m_2:n},\dots ,y_{m_2:m_2:n}\}$ denote a G-PH censored sample from a continuous population with PDF $g(·)$ and CDF $G(·)$, then the joint likelihood function (say, ${\mathcal{L}}_{\varsigma }(·)$) is

$${\mathcal{L}}_{\varsigma }\left(\mathit{\lambda }\right|\mathbf{y})=C_{\varsigma }{[1-G(T;\mathit{\lambda })]}^{S^{★}}\prod _{i=1}^{D_{\varsigma }}g(y_{i:m_2:n};\mathit{\lambda }){\left[1-G(y_{i:m_2:n};\mathit{\lambda })\right]}^{S_i},\varsigma =1,2,3,$$

(1)

where $S_{D_1}=n-m_1-{\sum }_{i=1}^{m_1-1}{S}_i$ and $S_{D_3}=n-m_2-{\sum }_{i=1}^{m_2-1}{S}_i$.

We now summarize the notation associated with the proposed censoring, such as

• $\varsigma =1$, $C_1={\prod }_{x=1}^{m_1}{\sum }_{i=x}^{m_2}(S_i+1)$, $D_1=m_1$, and $S^{★}=0$ (Case 1);
• $\varsigma =2$, $C_2={\prod }_{x=1}^d{\sum }_{i=x}^{m_2}(S_i+1)$, $D_2=d$,   and $S^{★}=n-d-{\sum }_{i=1}^d{S}_i$ (Case 2);
• $\varsigma =3$, $C_3={\prod }_{x=1}^{m_2}{\sum }_{i=x}^{m_2}(S_i+1)$, $D_3=m_2$, and $S^{★}=0$ (Case 3).

This censoring offers a key advantage: it ensures the observation of at least $r_1$ failures, even when the goal is to record up to $m_2$ failures. This feature enhances experimental reliability and makes the scheme particularly useful in situations where early termination is possible or likely. Several recent studies have applied G-PH censoring in diverse statistical models. For example, Koley and Kundu [4] (for the exponential competing risks); Wang [5] (for the Weibull latent failure times); Lee and Kang [6] (for the half-logistic distribution); and Lee [7] (for the inverse Weibull entropy model); Zhu [8] (for the standard Weibull distribution); Singh et al. [9] (for the Rayleigh model with competing hazards); Elshahhat and Abu El Azm [10] (for the Nadarajah–Haghighi distribution); Maswadah [11] (for the shape-scale parameter inference); and Alotaibi et al. [12] (for the Muth distribution).

A new extended conventional exponential distribution, called the two-parameter alpha-power-exponential (APE) distribution by Mahdavi and Kundu [13], has emerged as a flexible and powerful model for analyzing lifetime data, particularly within the contexts of reliability and survival analysis. It’s conceptually similar in motivation to other extended-exponential distributions like the exponentiated-exponential and Weibull distributions. It incorporates an additional shape parameter that enhances its ability to accommodate a wider range of hazard rate behaviors, including increasing, decreasing, and bathtub-shaped forms, which the standard exponential model is unable to represent. This flexibility makes the APE distribution especially valuable for modeling complex real-world failure patterns, where simplistic assumptions of constant hazard rates may not hold. Due to its tractable mathematical form and adaptability, the APE distribution has attracted growing interest for both theoretical development and applied statistical modeling in various disciplines. Now, suppose that a random variable representing lifetime, say Y, follows the APE($\mathit{\lambda }$) distribution, denoted by $Y\sim \mathrm{A}\mathrm{P}\mathrm{E}\left(\mathit{\lambda }\right)$, where $\mathit{\lambda }={(\alpha ,\xi )}^{\top }$, with parameters $\xi >0$ (scale) and $\alpha >0$ (shape). Its PDF and CDF are then defined, respectively, as 

$$g(y;\mathit{\lambda })={\displaystyle \frac{log\left(\alpha \right)\xi e^{-\xi y}{\alpha }^{1-e^{-\xi y}}}{\alpha -1}},y>0,\alpha \ne 1$$

(2)

and

$$G(y;\mathit{\lambda })={\displaystyle \frac{1}{\alpha -1}}\left({\alpha }^{1-e^{-\xi y}}-1\right),$$

(3)

with associated reliability function (RF) and hazard rate function (HRF) (at a distinct time $t>0$), denoted by $R\left(t\right)$ and $h\left(t\right)$, are

$$R(t;\mathit{\lambda })={\displaystyle \frac{\alpha }{\alpha -1}}\left(1-{\alpha }^{-e^{-\xi t}}\right),$$

(4)

and

$$h(t;\mathit{\lambda })={\displaystyle \frac{\xi log\left(\alpha \right)e^{-\xi t}}{{\alpha }^{e^{-\xi t}}-1}},$$

(5)

respectively. The APE distribution is adopted because it offers a balance between analytical tractability and flexibility in lifetime modeling. Unlike the generalized gamma distribution, the APE admits closed-form expressions that facilitate both likelihood construction and Bayesian computation. It can accommodate diverse hazard-rate shapes, similar to the exponentiated Weibull, but with fewer computational challenges. By fixing $\xi =1$, different shapes of the PDF (2) and HRF (5) are plotted and shown in Figure 2.

Because of its importance and flexibility mentioned later, in recent years, several researchers have explored inference procedures for APE-distributed using various types of data, for example, Salah [14] based on Type-II progressive; Salah et al. [15] based on Type-II hybrid; Elsherpieny and Abdel-Hakim [16] based on unified hybrid, among others.

We now clearly articulate the unique contributions of this study. Specifically:

(i)  

Unlike existing works that treat the APE distribution and the G-PH censoring scheme separately, this study provides the first comprehensive statistical evaluation of the APE distribution under the G-PH framework;

(ii) 

We develop the complete likelihood-based and Bayesian inferential framework, deriving asymptotic properties and assessing bootstrap performance under severe censoring conditions;

(iii)

We design optimized censoring and test plans that minimize experimental cost subject to precision constraints, a novel contribution for this class of models; and

(iv)

We validate the proposed methodology through applications in both medical and engineering sciences, offering concrete guidelines for practical test design.

Although these contributions, among others to the literature, highlight the increasing applicability of APE distribution in censored data environments, especially where modeling flexibility is required to address non-monotonic failure behavior, to the best of our knowledge, the challenge of estimating APE parameters and/or their reliability features (RF or HRF) in the presence of data collected via a G-PH censored sampling strategy has not yet been studied. To close this gap, the main goals of this work are represented in fivefold, such as:

• Driving maximum likelihood and Bayesian estimations for the APE parameters $\alpha$ and $\xi$, and associated reliability measures $R\left(t\right)$ and $h\left(t\right)$.
• Employing independent gamma priors, the Bayesian estimation is implemented using the Metropolis-Hastings algorithm under a squared-error loss (SEL) function.
• Two asymptotic confidence intervals alongside two Bayesian credible intervals for all model parameters are created.
• Extensive Monte Carlo simulations are conducted to assess the accuracy and precision of the developed estimators across various thresholds, sample sizes, censoring levels, and prior choices. The proposed methodology is validated on two real datasets (breast cancer and oil reserve data) to illustrate the practical utility and flexibility of the examined model.
• The study performs a comparative evaluation of different progressive censoring patterns, revealing distinct strengths of each plan depending on the parameter of interest.

The rest of the parts are organized as follows: Section 2 and Section 3 provide the likelihood and Bayes’ estimations, respectively. Simulation discussions are highlighted in Section 4. Two real applications are illustrated in Section 5. In Section 6, optimum censoring plans are recommended. Lastly, in Section 7, different conclusions and recommendations are presented.

2. Likelihood Inference

In this part, the likelihood method of estimation is used to derive both point and asymptotic interval estimations of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ of the APE distribution based on the G-PH censored sample.

2.1. Point Estimators

Using (2), (3), and (1), we can re-express the likelihood Function (1), where $y_i=y_{i:m:n}$ for simplicity, after ignoring the parameter-free term, as follows:

$$\begin{array}{c}\hfill {\mathcal{L}}_{\varsigma }\left(\mathit{\lambda }\right|\mathbf{y})\propto {\left({\displaystyle \frac{\alpha }{\alpha -1}}\right)}^n{\left(log\left(\alpha \right)\xi \right)}^{D_{\varsigma }}{e}^{-{\sum }_{i=1}^{D_{\varsigma }}\vartheta \left(y_i;\alpha ,\xi \right)}{\left(1-{\alpha }^{-e^{-\xi T^{★}}}\right)}^{S^{★}}\prod _{i=1}^{D_{\varsigma }}{\left(1-{\alpha }^{-e^{-\xi y_i}}\right)}^{S_i},\end{array}$$

(6)

where $\vartheta \left(y_i;\alpha ,\xi \right)=\xi y_i+log\left(\alpha \right)e^{-\xi y_i}$. The natural logarithm of (6), say ${\mathcal{L}}_{\varsigma }^{★}(·)$, becomes

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\varsigma }^{★}\left(\mathit{\lambda }\right|\mathbf{y})& \propto nlog\left({\displaystyle \frac{\alpha }{\alpha -1}}\right)+D_{\varsigma }log\left(log\left(\alpha \right)\right)+D_{\varsigma }log\left(\xi \right)-\sum _{i=1}^{D_{\varsigma }}\vartheta \left(y_i;\alpha ,\xi \right)\hfill \\ \hfill & +S^{★}log\left(1-{\alpha }^{-e^{-\xi T^{★}}}\right)+\sum _{i=1}^{D_{\varsigma }}{S}_{i}log\left(1-{\alpha }^{-e^{-\xi y_i}}\right).\hfill \end{array}$$

(7)

Thus, the maximum likelihood estimators (MLEs) of $\alpha$ and $\xi$, denoted by $\widehat{\alpha }$ and $\widehat{\xi }$, respectively, can be offered by directly maximizing (7). We now acquire the normal equations by equating the first partial derivatives of (7) with respect to $\alpha$ and $\xi$, respectively, to zero, as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathcal{L}}_{\varsigma }^{★}}{\partial \alpha }}& \propto -n\left({\displaystyle \frac{1}{\alpha \left(\alpha -1\right)}}\right)+{\displaystyle \frac{D_{\varsigma }}{\alpha \left(log\left(\alpha \right)\right)}}-\sum _{i=1}^{D_{\varsigma }}{\vartheta }^{\circ }\left(y_i;\alpha ,\xi \right)\hfill \\ \hfill & {\left.+S^{★}\left[{\displaystyle \frac{{\alpha }^{-e^{-\xi T^{★}}-1}{e}^{-\xi T^{★}}}{1-{\alpha }^{-e^{-\xi T^{★}}}}}\right]+\sum _{i=1}^{D_{\varsigma }}{S}_i^{★}\left[{\displaystyle \frac{{\alpha }^{-e^{-\xi y_i}-1}{e}^{-\xi y_i}}{1-{\alpha }^{-e^{-\xi y_i}}}}\right]\right|}_{\left(\alpha =\widehat{\alpha },\xi =\widehat{\xi }\right)}=0\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathcal{L}}_{\varsigma }^{★}}{\partial \xi }}& \propto {\displaystyle \frac{D_{\varsigma }}{\xi }}-\sum _{i=1}^{D_{\varsigma }}{\vartheta }^{•}\left(y_i;\alpha ,\xi \right)\hfill \\ \hfill & {\left.-S^{★}{T}^{★}\left[{\displaystyle \frac{log\left(\alpha \right){\alpha }^{-e^{-\xi T^{★}}}{e}^{-\xi T^{★}}}{1-{\alpha }^{-e^{-\xi T^{★}}}}}\right]-\sum _{i=1}^{D_{\varsigma }}{S}_i{y}_i\left[{\displaystyle \frac{log\left(\alpha \right){\alpha }^{-e^{-\xi y_i}}{e}^{-\xi y_i}}{1-{\alpha }^{-e^{-\xi y_i}}}}\right]\right|}_{\left(\alpha =\widehat{\alpha },\xi =\widehat{\xi }\right)}=0,\hfill \end{array}$$

(9)

where ${\vartheta }^{\circ }\left(y_i;\alpha ,\xi \right)={\alpha }^{-1}{e}^{-\xi y_i}$ and ${\vartheta }^{•}\left(y_i;\alpha ,\xi \right)=y_i\left(1-log\left(\alpha \right)e^{-\xi y_i}\right)$.

Obviously, from Equations (8) and (9), the estimators $\widehat{\alpha }$ and $\widehat{\xi }$ of $\alpha$ and $\xi$, respectively, cannot be defined straightforwardly in the linear system of equations. For this purpose, the Newton-Raphson (NR) algorithm through the $\mathsf{maxNR}$ optimization function in $\mathsf{maxLik}$ package (by Henningsen and Toomet [17]) is recommended to maximize the objective functions (8) and (9) with respect to $\alpha$ and $\xi$, respectively. Then, following the MLEs’ invariance property, one can acquire the MLEs of $R\left(t\right)$ and $h\left(t\right)$ (symbolized by $\widehat{R}\left(t\right)$ and $\widehat{h}\left(t\right)$, respectively) as follows:

$$\widehat{R}\left(t\right)={\displaystyle \frac{\widehat{\alpha }}{\widehat{\alpha }-1}}\left(1-{\widehat{\alpha }}^{-e^{-\widehat{\xi }t}}\right),$$

and

$$\widehat{h}\left(t\right)=\widehat{\xi }log\left(\widehat{\alpha }\right)e^{-\widehat{\xi }t}{\left({\widehat{\alpha }}^{e^{-\widehat{\xi }t}}-1\right)}^{-1},$$

respectively.

2.2. Interval Estimators

Aside from deriving the point estimators, constructing the two-sided $(1-\upsilon )100\%$ ACIs of $\alpha$, $\xi$, $R\left(t\right)$, or $h\left(t\right)$ is also an important issue. To address this objective, we utilize the asymptotic behavior of the offered MLEs $\widehat{\alpha }$ and $\widehat{\xi }$. According to large-sample theory, the joint asymptotic distribution of ${\left(\widehat{\mathit{\lambda }}\right)}^{\top }$ follows a bivariate normal distribution centered with mean $\mathit{\lambda }$ and variance-covariance (VC) matrix (say, $\mathrm{\Sigma }(·)$) given by inverting the Fisher information (FI) matrix, ${\mathrm{\Sigma }}^{-1}(·)$. However, deriving the explicit form of the Fisher information matrix is analytically challenging due to the complexity of its nonlinear components. As mentioned in Lawless [18], in this part, we get the observed FI matrix in turn to estimate the $2\times 2$ VC matrix at $(\mathit{\lambda }=\widehat{\mathit{\lambda }})$, as follows:

$$\begin{array}{ccc}\hfill \mathrm{\Sigma }\left(\mathit{\lambda }\right)& =& {\left[\begin{array}{cc}-{\ell }_{11}& -{\ell }_{12}\\ -{\ell }_{21}& -{\ell }_{22}\end{array}\right]}^{-1}\hfill \\ & =& \left[\begin{array}{cc}{\sigma }_{11}& {\sigma }_{12}\\ {\sigma }_{21}& {\sigma }_{22}\end{array}\right],\hfill \end{array}$$

(10)

where ${\ell }_{ij},i,j=1,2$, for short, are presented in Appendix A.

At a $100(1-\upsilon )\%$ confidence level, the two-sided asymptotic ACI estimators through their normality approximation (say, ACI-NA) for each APE parameter, $\alpha$ and $\xi$, are given by

$$\widehat{\alpha }\pm z_{\frac{\upsilon }{2}}\sqrt{{\sigma }_{11}},\mathrm{and}\widehat{\xi }\pm z_{\frac{\upsilon }{2}}\sqrt{{\sigma }_{22}},$$

(11)

respectively, where $z_{\frac{\upsilon }{2}}$ represents the upper 0.5$\upsilon$th quantile of the standard Gaussian distribution; see Elshahhat and Abu El Azm [10].

Additionally, to build the $100(1-\upsilon )\%$ ACI-NA estimator of $R\left(t\right)$ or $h\left(t\right)$ (at $t>0$), their approximated variances (denoted by ${\widehat{\mathrm{\Sigma }}}_{\widehat{R}}$ and ${\widehat{\mathrm{\Sigma }}}_{\widehat{h}}$, respectively) should be first acquired. To achieve this, the delta approach is one of the most often used ways for estimating these variances; for further details, see Greene [19]. However, we can get the respective variances ${\widehat{\mathrm{\Sigma }}}_{\widehat{R}}$ and ${\widehat{\mathrm{\Sigma }}}_{\widehat{h}}$ of $\widehat{R}\left(t\right)$ and $\widehat{h}\left(t\right)$, as 

$$\begin{array}{c}\hfill {\widehat{\mathrm{\Sigma }}}_{\widehat{R}}={\left.\left[R_1{R}_2\right]\mathrm{\Sigma }\left(\mathit{\lambda }\right){\left[R_1{R}_2\right]}^{\top }\right|}_{\left(\mathit{\lambda }=\widehat{\mathit{\lambda }}\right)}\end{array}$$

and

$$\begin{array}{c}\hfill {\widehat{\mathrm{\Sigma }}}_{\widehat{h}}={\left.\left[h_1{h}_2\right]\mathrm{\Sigma }\left(\mathit{\lambda }\right){\left[h_1{h}_2\right]}^{\top }\right|}_{\left(\mathit{\lambda }=\widehat{\mathit{\lambda }}\right)},\end{array}$$

where

$$\begin{array}{c}\hfill R_1={\displaystyle \frac{1}{\alpha -1}}\left[e^{-\xi t}{\alpha }^{-e^{-\xi t}}+{\displaystyle \frac{{\alpha }^{-e^{-\xi t}}-1}{\alpha -1}}\right],\end{array}$$

$$\begin{array}{c}\hfill R_2={\displaystyle \frac{\alpha log\left(\alpha \right)t{e}^{-\xi t}{\alpha }^{-e^{-\xi t}}}{\alpha -1}},\end{array}$$

$$\begin{array}{c}\hfill h_1=\xi \left[{\displaystyle \frac{\left[1-log\left(\alpha \right)e^{-\xi t}\right]{\alpha }^{-e^{-\xi t}}-1}{\alpha e^{\xi t}{({\alpha }^{-e^{-\xi t}}-1)}^2}}\right]\end{array}$$

and

$$\begin{array}{c}\hfill h_2=log\left(\alpha \right)\left[{\displaystyle \frac{\left[1+\xi t\left(log\left(\alpha \right)e^{-\xi t}-1\right)\right]{\alpha }^{-e^{-\xi t}}+\xi t-1}{e^{\xi t}{({\alpha }^{-e^{-\xi t}}-1)}^2}}\right].\end{array}$$

Subsequently, the $100(1-\upsilon )\%$ ACI-NA estimators of $R\left(t\right)$ and $h\left(t\right)$ can be represented, respectively, as 

$$\widehat{R}\left(t\right)\mp z_{\frac{\upsilon }{2}}\sqrt{{\widehat{\mathrm{\Sigma }}}_{\widehat{R}}}\mathrm{and}\widehat{h}\left(t\right)\mp z_{\frac{\upsilon }{2}}\sqrt{{\widehat{\mathrm{\Sigma }}}_{\widehat{h}}}.$$

In practice, a key limitation of the ACI-NA is that it can yield a negative lower bound when estimating a lifetime parameter, which is not meaningful in reliability contexts. To address this issue, Meeker and Escobar [20] proposed a normal log-transformed (NL) approximation for the MLE. They further demonstrated that, among various normal approximations, the ACI constructed using the NL transformation (ACI-NL) tends to exhibit the highest coverage percentage. Accordingly, the $100(1-\upsilon )\%$ ACI-NL of $\alpha$ (as an example) is given by

$$\widehat{\alpha }exp\left(\mp z_{\frac{\upsilon }{2}}{\displaystyle \frac{\sqrt{{\widehat{\sigma }}_{11}}}{\widehat{\alpha }}}\right),$$

where, in a similar pattern, the $100(1-\upsilon )\%$ ACI-NL estimator of $\xi$, $R\left(t\right)$, or $h\left(t\right)$ can be easily derived. Again, via employing the NR technique by installing the ’$\mathsf{maxLik}$’ package, the ACI-NA (or ACI-NL) bounds of $\widehat{\alpha }$, $\widehat{\xi }$, $\widehat{R}\left(t\right)$, or $\widehat{h}\left(t\right)$ can be obtained.

3. Bayesian Inference

The Bayesian paradigm enables the incorporation of prior beliefs or expert knowledge into the inferential process for unknown parameters. To implement this framework, the parameters $\alpha$ and $\xi$ of the APE lifespan distribution are treated as random variables, with their prior distributions reflecting available prior information. A particularly flexible and analytically convenient choice is the gamma conjugate prior, as discussed by Kundu [21]. The gamma distribution offers considerable versatility, making it suitable for modeling a wide spectrum of subjective or empirical prior beliefs. However, we assume that $\alpha$ and $\xi$ follow independent gamma distributions, specified as $\alpha \sim \mathrm{Gamma}(a_1,b_1)$ and $\xi \sim \mathrm{Gamma}(a_2,b_2)$, where $a_i,b_i>0$ for $i=1,2$. These hyperparameters are selected to encode prior information relevant to $\mathrm{APE}(\mathit{\lambda }$) distribution parameters. Under the assumption of independence, the joint prior PDF of $\alpha$ and $\xi$, say $p(·)$, takes the form:

$$p\left(\mathit{\lambda }\right)\propto {\alpha }^{a_1-1}{\xi }^{a_2-1}{e}^{-\left(b_1\alpha +b_2\xi \right)},\alpha ,\xi >0,$$

(12)

where $a_i$ and $b_i,i=1,2,$ are assumed to be known.

Substituting (6) and (12) into the continuous Bayes’ theorem, the joint posterior PDF (say $P_{\varsigma }(·)$) of $\alpha$ and $\xi$ can be written as

$$\begin{array}{cc}\hfill P_{\varsigma }\left(\mathit{\lambda }\right|\mathbf{y})& \propto {\displaystyle \frac{{\alpha }^{n+a_1-1}{\left(log\left(\alpha \right)\right)}^{D_{\varsigma }}}{{\left(\alpha -1\right)}^n}}{\xi }^{D_{\varsigma }+a_2-1}{e}^{-\left(b_1\alpha +b_2\xi +{\sum }_{i=1}^{D_{\varsigma }}\vartheta \left(y_i;\alpha ,\xi \right)\right)}\hfill \\ \hfill & \times {\left(1-{\alpha }^{-e^{-\xi T^{★}}}\right)}^{S^{★}}\prod _{i=1}^{D_{\varsigma }}{\left(1-{\alpha }^{-e^{-\xi y_i}}\right)}^{S_i},\hfill \end{array}$$

(13)

where its normalized term, say ${\rm Y}$, is given by ${\rm Y}={\int }_{\alpha }{\int }_{\xi }{\mathcal{L}}_{\varsigma }\left(\mathit{\lambda }\right|\mathbf{y})\times p\left(\mathit{\lambda }\right)\mathrm{d}\alpha \mathrm{d}\xi .$

It is important to highlight that the SEL function is employed in this study primarily due to its widespread use and symmetric nature, making it a natural choice for Bayesian estimation. Nevertheless, the proposed methodology is sufficiently general and can be readily extended to accommodate alternative loss functions, depending on the decision-making context or specific application requirements. It should also be noted that the Bayes estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ are each expressed as a ratio of two integrals, as given in Equation (13). However, these integrals typically lack closed-form solutions due to the nonlinear structure of  (6), rendering analytical computation intractable in most cases. Therefore, we recommend implementing MCMC algorithms to generate Markovian samples from the posterior distribution in Equation (13). This enables us to obtain the Bayes estimates and to construct the corresponding BCI/HPD intervals for each of the unknown parameters.

To proceed with the sampling procedure, we must first derive the full conditional distributions of $\alpha$ and $\xi$, denoted by $P_{\varsigma }^{\alpha }$ and $P_{\varsigma }^{\xi }$, respectively, which are given as follows:

$$\begin{array}{c}\hfill P_{\varsigma }^{\alpha }\left(\alpha \right|\xi ,\mathbf{y})\propto {\displaystyle \frac{{\alpha }^{n+a_1-1}{\left(log\left(\alpha \right)\right)}^{D_{\varsigma }}}{{\left(\alpha -1\right)}^n}}{e}^{-\left(b_1\alpha +{\sum }_{i=1}^{D_{\varsigma }}\vartheta \left(y_i;\alpha ,\xi \right)\right)}{\left(1-{\alpha }^{-e^{-\xi T^{★}}}\right)}^{S^{★}}\prod _{i=1}^{D_{\varsigma }}{\left(1-{\alpha }^{-e^{-\xi y_i}}\right)}^{S_i},\end{array}$$

(14)

and

$$\begin{array}{c}\hfill P_{\varsigma }^{\xi }\left(\xi \right|\alpha ,\mathbf{y})\propto {\xi }^{D_{\varsigma }+a_2-1}{e}^{-\left(b_2\xi +{\sum }_{i=1}^{D_{\varsigma }}\vartheta \left(y_i;\alpha ,\xi \right)\right)}{\left(1-{\alpha }^{-e^{-\xi T^{★}}}\right)}^{S^{★}}\prod _{i=1}^{D_{\varsigma }}{\left(1-{\alpha }^{-e^{-\xi y_i}}\right)}^{S_i},\end{array}$$

(15)

respectively. From (14) and (15), it is evident that the full conditional distributions of $\alpha$ and $\xi$ do not correspond to any known standard density. As a result, direct sampling from these conditionals using conventional methods is not feasible. To address this challenge, we adopt the Metropolis–Hastings (M-H) algorithm, which provides a flexible framework for sampling from non-standard distributions. To implement the M–H procedure, Figure 3 shows that the distributions (14) and (15) behave like a normal density; thus, we use the normal distribution as the proposal density to generate candidate values. The steps required to generate posterior samples from the conditionals (14) and (15), and subsequently obtain Bayes point estimates or credible intervals for $\alpha$, $\xi$, $R\left(t\right)$, or $h\left(t\right)$, are outlined in Algorithm 1.

It is important to clarify how the Bayesian formulation extends across different censoring mechanisms. The prior specification $\pi \left(\mathit{\lambda }\right)$ remains fixed, while the likelihood contribution ${\mathcal{L}}_{\varsigma }^{★}(\mathit{\lambda }\mid \mathbf{y})$ changes structurally with the censoring scheme. Thus, the posterior takes the form $\pi (\mathit{\lambda }\mid \mathbf{y})\propto {\mathcal{L}}_{\varsigma }^{★}(\mathit{\lambda }\mid \mathbf{y})\pi \left(\mathit{\lambda }\right),$ where

• Case 1: ${\mathcal{L}}_1^{★}\propto {\prod }_{i=1}^{m_1}g(y_i;\mathit{\lambda }){\left[R(y_i^{★};\mathit{\lambda })\right]}^{S_i}$.
• Case 2: ${\mathcal{L}}_2^{★}\propto {\left[R(T;\mathit{\lambda })\right]}^{n-d-{\sum }_{i=1}^d{S}_i}{\prod }_{i=1}^{D_2}g(y_i;\mathit{\lambda }){\left[R(y_d;\mathit{\lambda })\right]}^{S_i}$.
• Case 3: ${\mathcal{L}}_3^{★}\propto {\prod }_{i=1}^{m_2}g(y_i;\mathit{\lambda }){\left[R(y_i;\mathit{\lambda })\right]}^{S_i}$.

Accordingly, the posterior distribution under censoring Case k ($k=1,2,3$) is ${\pi }_k(\mathit{\lambda };\mathbf{y})\propto {\mathcal{L}}_k^{★}(\mathit{\lambda };\mathbf{y})\pi \left(\mathit{\lambda }\right),$ which ensures that the Bayesian inference procedure is valid and reproducible across all censoring scenarios considered.

Algorithm 1 The M-H Sampling:

Step 1: Start with initial guesses ${\alpha }^{\left(0\right)}=\widehat{\alpha }$ and ${\xi }^{\left(0\right)}=\widehat{\xi }$. Step 2: Set $j=1$ Step 3: Generate ${\alpha }^{★}$ and ${\xi }^{★}$ from $N(\widehat{\alpha },{\widehat{\sigma }}_{11})$ and $N(\widehat{\xi },{\widehat{\sigma }}_{22})$, respectively. Step 4: Obtain ${\chi }_{\alpha }={\displaystyle \frac{P_{\varsigma }^{\alpha }\left({\alpha }^{★}\right|{\xi }^{(j-1)},\mathbf{y})}{P_{\varsigma }^{\alpha }\left({\alpha }^{(j-1)}\right|{\xi }^{(j-1)},\mathbf{y})}}$ and ${\chi }_{\xi }={\displaystyle \frac{P_{\varsigma }^{\xi }\left({\xi }^{★}\right|{\alpha }^{\left(j\right)},\mathbf{y})}{P_{\varsigma }^{\xi }\left({\xi }^{(j-1)}\right|{\alpha }^{\left(j\right)},\mathbf{y})}}$. Step 5: Generate sample variates $u_i,i=1,2,$ from the uniform $U(0,1)$ distribution and set $$\begin{array}{c}\hfill {\alpha }^{\left(j\right)}=\left\{\begin{array}{cc}{\alpha }^{★},\hfill & \mathrm{if}{u}_1\le min\left\{1,{\chi }_{\alpha }\right\},\hfill \\ {\alpha }^{(j-1)},& \mathrm{otherwise}.\end{array}\right.\mathrm{and}{\xi }^{\left(j\right)}=\left\{\begin{array}{cc}{\xi }^{★},\hfill & \mathrm{if}{u}_2\le min\left\{1,{\chi }_{\xi }\right\},\hfill \\ {\xi }^{(j-1)},& \mathrm{otherwise}.\end{array}\right.\end{array}$$ Step 6: Obtain $R^{\left(j\right)}\left(t\right)$ and $h^{\left(j\right)}\left(t\right)$ as $R^{\left(j\right)}(t;{\alpha }^{\left(j\right)},{\xi }^{\left(j\right)})$ and $h^{\left(j\right)}(t;{\alpha }^{\left(j\right)},{\xi }^{\left(j\right)})$, respectively. Step 7: Put $j=j+1$. Step 8: Repeat Steps 3–7 $\mathbb{B}$ times and obtain ${\alpha }^{\left(j\right)}$ and ${\xi }^{\left(j\right)}$ for $j=1,2,\dots ,\mathbb{B}$. Step 9: Compute the Bayes estimates of $\alpha$, where ${\mathbb{B}}^{★}=\mathbb{B}-{\mathbb{B}}_0$ and ${\mathbb{B}}_0$ is burn-in, as  $$\tilde{\alpha }={\displaystyle \frac{1}{{\mathbb{B}}^{★}}}{\sum }_{j={\mathbb{B}}_0+1}^{\mathbb{B}}{\alpha }^{\left(j\right)}.$$ Step 10: Compute the $100(1-\upsilon )\%$ BCI of $\alpha$ by ordering ${\alpha }^{\left(j\right)},j={\mathbb{B}}_0+1,{\mathbb{B}}_0+2,\dots ,\mathbb{B}$ as ${\alpha }_{({\mathbb{B}}_0+1)},{\alpha }_{({\mathbb{B}}_0+2)},\dots ,{\alpha }_{\left(\mathbb{B}\right)}$. Thus the $100(1-\upsilon )\%$ BCI of $\alpha$ is obtained as $$\left\{{\alpha }_{\left(\frac{\upsilon }{2}\right){\mathbb{B}}^{★}},{\alpha }_{(1-\frac{\upsilon }{2}){\mathbb{B}}^{★}}\right\}.$$ Step 11: Compute the $100(1-\upsilon )\%$ HPD interval of $\alpha$ as $$\begin{array}{c}\hfill \left\{{\alpha }_{\left(j^{★}\right)},{\alpha }_{(j^{★}+(1-\upsilon ){\mathbb{B}}^{★})}\right\},\end{array}$$ where $j^{★}={\mathbb{B}}_0+1,{\mathbb{B}}_0+2,\dots ,\mathbb{B}$ is selected so that $$\begin{array}{c}\hfill {\alpha }_{\left(j^{★}+\left[\left(1-\upsilon \right){\mathbb{B}}^{★}\right]\right)}-{\alpha }_{\left(j^{★}\right)}=\underset{1⩽j⩽\upsilon {\mathbb{B}}^{★}}{min}\left[{\alpha }_{\left(j+\left[{\mathbb{B}}^{★}\left(1-\upsilon \right)\right]\right)}-{\alpha }_{\left(j\right)})\right].\end{array}$$ Step 12: Redo Steps 9 to 11 for $\xi$, $R\left(t\right)$, and $h\left(t\right)$.

4. Numerical Evaluations

A comprehensive series of Monte Carlo analyses is employed in this section to investigate the statistical accuracy and operational performance of the estimates of APE parameters $\alpha$ and $\xi$, as well as of APE reliability metrics $R\left(t\right)$ and $h\left(t\right)$ introduced earlier.

4.1. Simulation Design

To get the point (or interval) estimate of $\alpha$, $\xi$, $R\left(t\right)$, or $h\left(t\right)$, we repeat the G-PH censoring plan 1000 times from two distinct $\mathrm{APE}\left(\mathit{\lambda }\right)$, namely Pop–1: $(0.5,1.5)$ and Pop–2: $(1.5,2.5)$. Taking $t=0.1$, the plausible value of ($R\left(t\right)$, $h\left(t\right)$) is taken as (0.95403, 0.09184) and (0.812, 2.126) from Pop-$i,i=1,2$, respectively. Specifically, each proposed test is carried out based on various choices of T(threshold time), n(total experimental units), $(m_i,i=1,2)$ (effective sample sizes), and $\mathbf{S}$ (progressive censoring plan (PCP)). For each propped population, we assigned $T(=0.5,1.5)$ and $n(=30,50,80)$. In 

Table 1. Several progressive designs in Monte Carlo comparisons.

$\mathit{n}\downarrow$	S
$(m_1,m_2)\to$	(10, 15)	(15, 20)
30	PCP[1]:$(5^3,0^{12})$	PCP[1]:$(5^2,0^{18})$
	PCP[2]:$(0^6,5^3,0^6)$	PCP[2]:$(0^9,5^2,0^9)$
	PCP[3]:$(0^{12},5^3)$	PCP[3]:$(0^{18},5^2)$
$(m_1,m_2)\to$	(20, 30)	(40, 30)
50	PCP[1]:$(5^4,0^{26})$	PCP[1]:$(5^2,0^{38})$
	PCP[2]:$(0^{13},5^4,0^{13})$	PCP[2]:$(0^{19},5^2,0^{19})$
	PCP[3]:$(0^{26},5^4)$	PCP[3]:$(0^{38},5^2)$
$(m_1,m_2)\to$	(50, 40)	(50, 60)
80	PCP[1]:$(5^6,0^{44})$	PCP[1]:$(5^4,0^{56})$
	PCP[2]:$(0^{22},5^6,0^{22})$	PCP[2]:$(0^{28},5^4,0^{28})$
	PCP[3]:$(0^{44},5^6)$	PCP[3]:$(0^{56},5^4)$

, for each value of n, several options of $m_i,i=1,2,$ and various patterns of PCPs, where $5^2$ (for instance) implies that five survival items are removed at the first two stages, are provided. For clarity, to produce a G-PH censored dataset from APE$\left(\mathit{\lambda }\right)$, do the generation steps described in Algorithm 2.

Algorithm 2 Generation Procedure of G-PH censoring plan:

1: Input: Assign the parameter values of $\alpha$ and $\xi$   2: Input: Assign the censoring values of T, n, $\mathbf{S}$, and $m_i,i=1,2$   3: Output: Simulate ${\varrho }_i,i=1,2,\dots ,m,$ from uniform $U(0,1)$ distribution   4: Input: Set ${\rho }_i={\varrho }_i^{{\left(i+{\sum }_{l=m-i+1}^m{S}_l\right)}^{-1}},i=1,2,\dots ,m$   5: Input: Set $U_i=1-{\rho }_m{\rho }_{m-1}\dots {\rho }_{m-i+1}$ for $i=1,2,\dots ,m$   6: Input: Get $y_i=-{\displaystyle \frac{1}{\xi }}log\left[1-{\displaystyle \frac{log(u_i(\alpha -1)+1)}{log\left(\alpha \right)}}\right],i=1,2,\dots ,m$   7: Output: Find d at T   8: Output: Get the G-PH censoring data vector as:   9: Find $\mathbf{y}$ as 10: if $T<y_r<y_m$ then 11:     Data $\{y_1,\dots ,y_r\}$ 12: end if 13: if $y_r<T<y_m$ then 14:     Data $\{y_1,\dots ,y_d\}$ 15: end if 16: if $y_r<y_m<T$ then 17:     Data $\{y_1,\dots ,y_m\}$ 18: end if

Upon generation of the 1000 G-PH right-censored datasets, the MLEs of the model parameters $\alpha$ and $\xi$, as well as of the model reliability measures $R\left(t\right)$ and $h\left(t\right)$ along with their corresponding 95% ACI-NA/ACI-NL, are obtained through installing the $\mathsf{maxLik}$ package (by Henningsen and Toomet [17]) in $\mathsf{R}$ software (v4.2.2).

To establish explicitly and describe our NR convergence strategy, the convergence strategy for the log-likelihood in (7) is declared only when:

$${\mathit{\lambda }}^{(k+1)}={\mathit{\lambda }}^{\left(k\right)}-{\left[{\nabla }^2{\mathcal{L}}_{\varsigma }^{★}\left({\mathit{\lambda }}^{\left(k\right)}\mid \mathbf{y}\right)\right]}^{-1}\nabla {\mathcal{L}}_{\varsigma }^{★}\left({\mathit{\lambda }}^{\left(k\right)}\mid \mathbf{y}\right),$$

where ∇ and ${\nabla }^2$ denote the score vector and observed information matrix, respectively. Convergence is declared when the following conditions are simultaneously satisfied:

• $\parallel \nabla {\mathcal{L}}_{\varsigma }^{★}(\widehat{\mathit{\lambda }}\mid \mathbf{y}){\parallel }_{\infty }\le {10}^{-6}$,
• ${\displaystyle \frac{\parallel {\mathit{\lambda }}^{(k+1)}-{\mathit{\lambda }}^{\left(k\right)}{\parallel }_{\infty }}{1+\parallel {\mathit{\lambda }}^{\left(k\right)}{\parallel }_{\infty }}}\le {10}^{-8}$,
• ${\mathcal{L}}_{\varsigma }^{★}({\mathit{\lambda }}^{(k+1)}\mid \mathbf{y})-{\mathcal{L}}_{\varsigma }^{★}({\mathit{\lambda }}^{\left(k\right)}\mid \mathbf{y})\le {10}^{-8}$, or after a maximum of 200 iterations.

We also verify positive definiteness of the observed information at the solution and report the gradient norm. These additions address potential non-convergence and ensure that, especially under censoring where the likelihood surface can be relatively flat, the algorithm behaves robustly.

Numerically, since the likelihood equations in (8) and (9) involve highly nonlinear terms that may lead to instabilities for small or heavily censored samples, we address these issues by adopting one or more of the following strategies:

(i)  

Reparameterizing as $\alpha \equiv 1+e^{\eta }$ and $\xi \equiv e^{\gamma }$ to automatically enforce $\alpha >1$ and $\xi >0$;

(ii) 

Evaluating terms using numerically stable primitives (e.g., $log1p(·)$ and $\mathrm{expm}1(·)$) to avoid cancellation when $1-{\alpha }^{-e^{-\xi t}}$ is close to zero;

(iii)

Scaling the data by $\mathrm{median}\left(y\right)$ to keep $\xi t$ in a moderate range and clipping $e^{-\xi t}$ away from exact 0 or 1.

Figure 4 demonstrates that the MLEs of both $\alpha$ and $\xi$ are highly sensitive to sample size and censoring level under extreme conditions. When $n=20$, the estimates are unstable, particularly at moderate and heavy censoring (30% and 50%), where the mean values deviate substantially from the true parameters. However, as the sample size increases ($n\ge 80$), the estimates rapidly stabilize and become robust to different censoring levels, indicating that the proposed method yields reliable results under realistic sample sizes. This analysis highlights the importance of sample adequacy for practical applications of the model.

For Bayesian inference, the M-H sampling strategy described in this study is applied, discarding the initial ${\mathbb{B}}^{★}=2000$ iterations as burn-in from a total of $\mathbb{B}=12,000$ MCMC samples for each parameter. The subsequent posterior summaries, including point estimates and 95% BCI/HPD interval estimates, are computed using the $\mathsf{coda}$ package (by Plummer et al. [22]) in the same version of $\mathsf{R}$ software. Following the framework of specifying values of prior parameters suggested by Kundu [21], the hyperparameters $(a_i,b_i)$ for $i=1,2$, corresponding to the prior distributions for the APE parameters $\alpha$ and $\xi$ under Priors I and II, are selected to ensure that their prior means align with the respective expected values of the model parameters.

Now, to evaluate the influence of prior specification on Bayesian inference, both point and credible interval estimates are analyzed under two distinct sets of hyperparameters $(a_i,b_i)$ for each population under consideration, namely:

• For Pop-1:APE(0.5, 1.5):

  –

  Prior I: $(a_1,a_2)=(2.5,7.5)$ and $b_i=5,i=1,2$;

  –

  Prior II: $(a_1,a_2)=(5,15)$ and $b_i=10,i=1,2$,

• For Pop-2:APE(1.5, 2.5):

  –

  Prior I: $(a_1,a_2)=(7.5,12.5)$ and $b_i=5,i=1,2$;

  –

  Prior II: $(a_1,a_2)=(15,25)$ and $b_i=10,i=1,2$.

As soon as the 1000 G-PH censored datasets are collected, the offered MLEs alongside with associated ACI-NA/ACI-NL estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ are evaluated via ‘$\mathsf{maxLik}$’ package introduced by Henningsen and Toomet [17]. Implementing the M-H approach examined in this study, the first ${\mathbb{B}}^{★}=2000$ (of $\mathbb{B}$ = 12,000) MCMC iterations are used as burn-in for each unknown quantity. Following that, the Bayes-MCMC estimates and associated 95% BCI/HPD intervals of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ are produced using the ‘$\mathsf{coda}$’ package suggested by Plummer et al. [22].

Following the methodology outlined by Kundu [21], the hyperparameter values $(a_i,b_i),i=1,2,$ of APE parameters $\alpha$ and $\xi$ in priors I and II provided below are chosen such that the corresponding prior means match the expected values of the respective model parameters. For this purpose, to see the effects of prior choices on Bayes results (point and interval) are examined using two different sets of the hyperparameters $(a_i,b_i),i=1,2,$ for each given population, namely:

Next, we calculate the following precision metrics of $\alpha$ (for instance):

• Root Mean Squared Error: $\mathrm{RMSE}\left(\ddot{\alpha }\right)=\sqrt{{\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}{\left({\ddot{\alpha }}^{\left[i\right]}-\alpha \right)}^2},$
• Mean Relative Absolute Bias: $\mathrm{MRAB}\left(\ddot{\alpha }\right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}{\displaystyle \frac{1}{\alpha }}\left|{\ddot{\alpha }}^{\left[i\right]}-\alpha \right|,$
• Average Interval Length: ${\mathrm{AIL}}_{95\%}\left(\alpha \right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}\left({\mathcal{U}}_{{\ddot{\alpha }}^{\left[i\right]}}-{\mathcal{L}}_{{\ddot{\alpha }}^{\left[i\right]}}\right),$
• Coverage Percentage: ${\mathrm{CP}}_{95\%}\left(\alpha \right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}{\mathbb{D}}_{\left({\mathcal{L}}_{{\ddot{\alpha }}^{\left[i\right]}};{\mathcal{U}}_{{\ddot{\alpha }}^{\left[i\right]}}\right)}\left(\alpha \right),$
  where ${\ddot{\alpha }}^{\left[i\right]}$ is the calculated point estimate of $\alpha$ at ith sample, $\mathbb{D}(·)$ is the indicator function, $\left(\mathcal{L}\right(·),\mathcal{U}(·\left)\right)$ is two-sided of $(1-\upsilon )$ ACI-NA/ACI-NL (or BCI/HPD) of $\alpha$.

4.2. Simulation Results and Discussions

Table 2. The RMSE (1st Col.) and MRAB (2nd Col.) results of $\alpha$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	MLE		MCMC					MLE		MCMC
		$\mathit{T}=\mathbf{0.5}$							$\mathit{T}=\mathbf{1.5}$
Prior→				I		II					I		II
Pop-1
(30,15,10)	PCP[1]	4.1354	1.2642	0.2444	0.2063	0.1980	0.1698		3.4061	1.1304	0.2283	0.2012	0.1794	0.1692
	PCP[2]	4.2151	1.3430	0.2906	0.2388	0.2039	0.1707		3.9360	1.2081	0.2414	0.2088	0.1805	0.1692
	PCP[3]	3.5341	1.2394	0.2301	0.2058	0.1879	0.1691		3.3023	1.0223	0.2258	0.1983	0.1794	0.1689
(30,20,15)	PCP[1]	2.9043	0.8034	0.2216	0.1909	0.1789	0.1636		2.0837	0.7586	0.2160	0.1877	0.1769	0.1630
	PCP[2]	3.2644	1.1335	0.2256	0.1974	0.1833	0.1670		3.0338	0.9289	0.2214	0.1928	0.1794	0.1645
	PCP[3]	3.1648	0.8997	0.2219	0.1917	0.1793	0.1636		2.4519	0.8977	0.2168	0.1889	0.1791	0.1636
(50,30,20)	PCP[1]	2.5234	0.5811	0.2187	0.1877	0.1721	0.1489		1.3066	0.5681	0.2121	0.1800	0.1602	0.1373
	PCP[2]	2.6603	0.5932	0.2201	0.1903	0.1752	0.1600		1.5488	0.5922	0.2144	0.1854	0.1614	0.1385
	PCP[3]	2.4101	0.5214	0.2178	0.1835	0.1691	0.1382		1.2482	0.4363	0.2102	0.1780	0.1560	0.1372
(50,40,30)	PCP[1]	2.2283	0.4199	0.2112	0.1827	0.1660	0.1372		1.1561	0.3559	0.2078	0.1771	0.1433	0.1332
	PCP[2]	2.1762	0.3889	0.2107	0.1806	0.1625	0.1332		1.0838	0.3184	0.2063	0.1769	0.1433	0.1315
	PCP[3]	2.0457	0.3740	0.2037	0.1763	0.1554	0.1315		1.0486	0.3094	0.2026	0.1741	0.1421	0.1302
(80,50,40)	PCP[1]	1.5479	0.3563	0.1910	0.1630	0.1474	0.1283		0.8735	0.2868	0.1907	0.1623	0.1407	0.1281
	PCP[2]	1.6581	0.3613	0.2008	0.1739	0.1514	0.1299		0.9731	0.2979	0.1964	0.1676	0.1409	0.1284
	PCP[3]	1.5431	0.3487	0.1896	0.1614	0.1428	0.1278		0.8599	0.2779	0.1896	0.1614	0.1401	0.1274
(80,60,50)	PCP[1]	1.4439	0.3147	0.1840	0.1556	0.1393	0.1184		0.8181	0.2381	0.1805	0.1555	0.1392	0.1217
	PCP[2]	1.4667	0.3418	0.1894	0.1609	0.1403	0.1195		0.8544	0.2742	0.1861	0.1580	0.1397	0.1171
	PCP[3]	1.3399	0.3090	0.1828	0.1531	0.1379	0.1160		0.6979	0.2267	0.1678	0.1384	0.1379	0.1150
Pop-2
(30,15,10)	PCP[1]	6.2369	1.3162	0.5490	0.5147	0.2084	0.1956		3.7562	1.2906	0.5208	0.4864	0.2012	0.1880
	PCP[2]	6.5361	1.4431	0.5606	0.5258	0.2106	0.2019		3.8960	1.3227	0.5344	0.4979	0.2020	0.1898
	PCP[3]	5.8117	1.2879	0.5447	0.5102	0.2020	0.1896		3.5386	1.2243	0.5108	0.4758	0.2002	0.1856
(30,20,15)	PCP[1]	5.3495	1.0840	0.4746	0.4129	0.1976	0.1799		3.1078	1.0510	0.4537	0.3886	0.1906	0.1720
	PCP[2]	5.6283	1.2259	0.4857	0.4509	0.1995	0.1842		3.3892	1.1610	0.4726	0.4419	0.1976	0.1816
	PCP[3]	5.4979	1.1732	0.4848	0.4153	0.1991	0.1803		3.2430	1.1101	0.4560	0.4008	0.1936	0.1799
(50,30,20)	PCP[1]	4.9242	0.8761	0.4439	0.3889	0.1827	0.1517		2.7762	0.6965	0.4346	0.3806	0.1748	0.1492
	PCP[2]	5.1598	0.9775	0.4477	0.4071	0.1848	0.1541		2.9746	0.9560	0.4439	0.3877	0.1765	0.1507
	PCP[3]	4.4894	0.7993	0.4262	0.3806	0.1807	0.1478		2.5543	0.6451	0.4173	0.3805	0.1736	0.1478
(50,40,30)	PCP[1]	4.1980	0.6539	0.4145	0.3794	0.1776	0.1458		2.1952	0.5341	0.4139	0.3777	0.1573	0.1445
	PCP[2]	3.9823	0.6482	0.4081	0.3759	0.1706	0.1451		2.1079	0.5279	0.4040	0.3704	0.1565	0.1445
	PCP[3]	3.8817	0.6162	0.3988	0.3668	0.1655	0.1441		2.0042	0.4883	0.3988	0.3598	0.1553	0.1434
(80,50,40)	PCP[1]	3.0194	0.5558	0.3843	0.3521	0.1584	0.1383		1.7741	0.4422	0.3740	0.3391	0.1540	0.1382
	PCP[2]	3.8142	0.5706	0.3935	0.3598	0.1615	0.1392		2.0022	0.4551	0.3773	0.3404	0.1541	0.1390
	PCP[3]	2.7186	0.5529	0.3773	0.3404	0.1541	0.1374		1.5914	0.3974	0.3637	0.3306	0.1533	0.1338
(80,60,50)	PCP[1]	2.6114	0.5272	0.3186	0.2748	0.1520	0.1294		1.5606	0.3795	0.3124	0.2690	0.1512	0.1293
	PCP[2]	2.7064	0.5420	0.3345	0.2895	0.1530	0.1302		1.5673	0.3945	0.3136	0.2706	0.1525	0.1294
	PCP[3]	2.5171	0.4764	0.3120	0.2702	0.1509	0.1283		1.5254	0.3219	0.2982	0.2576	0.1452	0.1219

,

Table 3. The RMSE (1st Col.) and MRAB (2nd Col.) results of $\xi$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	MLE		MCMC					MLE		MCMC
		$\mathit{T}=\mathbf{0.5}$							$\mathit{T}=\mathbf{1.5}$
Prior→				I		II					I		II
Pop-1
(30,15,10)	PCP[1]	2.1662	1.5862	0.5873	0.3845	0.2078	0.1955		2.1632	1.5716	0.5572	0.3436	0.2032	0.1915
	PCP[2]	1.5848	1.2360	0.3693	0.3220	0.2004	0.1888		1.4199	1.1365	0.3608	0.3108	0.1928	0.1817
	PCP[3]	1.9363	1.4477	0.4637	0.3667	0.2034	0.1917		1.5359	1.2505	0.4162	0.3165	0.1983	0.1868
(30,20,15)	PCP[1]	1.4278	1.1526	0.3610	0.3174	0.1921	0.1688		1.3751	1.0943	0.3433	0.3038	0.1890	0.1659
	PCP[2]	1.2913	1.0173	0.3586	0.3117	0.1879	0.1650		1.2407	0.9667	0.3315	0.3034	0.1785	0.1626
	PCP[3]	1.4071	1.0759	0.3601	0.3155	0.1890	0.1659		1.3271	1.0546	0.3350	0.3036	0.1890	0.1659
(50,30,20)	PCP[1]	1.2637	0.9878	0.3556	0.3084	0.1685	0.1593		1.2021	0.9383	0.3239	0.2776	0.1685	0.1593
	PCP[2]	1.1368	0.8846	0.3439	0.3012	0.1623	0.1538		1.0077	0.7982	0.3171	0.2718	0.1587	0.1299
	PCP[3]	1.2258	0.9142	0.3497	0.3022	0.1685	0.1570		1.0435	0.8370	0.3179	0.2754	0.1659	0.1564
(50,40,30)	PCP[1]	1.0632	0.8722	0.3362	0.2849	0.1587	0.1299		0.9068	0.7247	0.2965	0.2413	0.1553	0.1295
	PCP[2]	0.9991	0.8148	0.3120	0.2589	0.1571	0.1283		0.8150	0.6612	0.2743	0.2276	0.1370	0.1193
	PCP[3]	1.0257	0.8219	0.3230	0.2769	0.1575	0.1287		0.8393	0.6741	0.2897	0.2352	0.1471	0.1268
(80,50,40)	PCP[1]	0.9716	0.7807	0.3010	0.2496	0.1327	0.1033		0.8018	0.6465	0.2594	0.2124	0.1277	0.0993
	PCP[2]	0.9504	0.7654	0.2648	0.2162	0.1254	0.0989		0.7679	0.6292	0.2553	0.2076	0.1243	0.0986
	PCP[3]	0.9540	0.7711	0.2711	0.2241	0.1290	0.1007		0.7812	0.6453	0.2592	0.2084	0.1263	0.0990
(80,60,50)	PCP[1]	0.9345	0.7432	0.2541	0.2076	0.1162	0.0983		0.7629	0.6287	0.2541	0.2035	0.1155	0.0979
	PCP[2]	0.8864	0.6833	0.2534	0.2016	0.1146	0.0970		0.7285	0.5751	0.2533	0.2009	0.1138	0.0964
	PCP[3]	0.8892	0.7014	0.2540	0.2030	0.1160	0.0979		0.7486	0.5910	0.2537	0.2021	0.1152	0.0971
Pop-2
(30,15,10)	PCP[1]	2.2653	1.6760	0.5963	0.5020	0.2214	0.2008		2.1998	1.6499	0.5628	0.4373	0.2132	0.2004
	PCP[2]	1.7048	1.3757	0.4723	0.4155	0.2132	0.2004		1.4601	1.1621	0.4495	0.3923	0.2064	0.1942
	PCP[3]	1.9937	1.6234	0.5262	0.4560	0.2132	0.2004		1.7551	1.3828	0.4715	0.4102	0.2065	0.1942
(30,20,15)	PCP[1]	1.5423	1.2700	0.4652	0.4083	0.1971	0.1728		1.4497	1.1360	0.4451	0.3761	0.1971	0.1728
	PCP[2]	1.4278	1.1817	0.4521	0.3890	0.1964	0.1721		1.3404	1.0665	0.3942	0.3487	0.1851	0.1618
	PCP[3]	1.5326	1.2360	0.4625	0.4070	0.1971	0.1728		1.3556	1.0787	0.4376	0.3660	0.1971	0.1728
(50,30,20)	PCP[1]	1.3738	1.1209	0.4116	0.3741	0.1680	0.1590		1.2091	0.9612	0.3758	0.3469	0.1664	0.1565
	PCP[2]	1.2915	1.0264	0.3711	0.3281	0.1664	0.1574		1.0320	0.8096	0.3655	0.3025	0.1655	0.1552
	PCP[3]	1.3055	1.0928	0.3824	0.3387	0.1678	0.1589		1.1552	0.9093	0.3718	0.3152	0.1664	0.1558
(50,40,30)	PCP[1]	1.1751	0.9877	0.3655	0.3185	0.1664	0.1359		1.0020	0.8054	0.3561	0.2952	0.1648	0.1359
	PCP[2]	1.1442	0.9015	0.3625	0.3025	0.1632	0.1330		0.9425	0.7094	0.3331	0.2833	0.1511	0.1222
	PCP[3]	1.1464	0.9362	0.3629	0.3125	0.1640	0.1359		0.9600	0.7356	0.3342	0.2882	0.1618	0.1320
(80,50,40)	PCP[1]	1.1304	0.9011	0.3342	0.2805	0.1333	0.1035		0.9420	0.6965	0.3323	0.2787	0.1306	0.1017
	PCP[2]	1.0616	0.8262	0.3228	0.2657	0.1311	0.1021		0.8799	0.6556	0.2992	0.2448	0.1302	0.1014
	PCP[3]	1.1280	0.8927	0.3266	0.2787	0.1314	0.1022		0.9418	0.6927	0.3034	0.2490	0.1306	0.1017
(80,60,50)	PCP[1]	1.0571	0.8165	0.3143	0.2577	0.1175	0.0998		0.8610	0.6405	0.2599	0.2133	0.1173	0.0997
	PCP[2]	0.9820	0.7688	0.2505	0.2088	0.1170	0.0995		0.8052	0.5962	0.2480	0.2058	0.1159	0.0987
	PCP[3]	1.0436	0.8132	0.2982	0.2460	0.1170	0.0995		0.8473	0.6315	0.2505	0.2088	0.1167	0.0993

,

Table 4. The RMSE (1st Col.) and MRAB (2nd Col.) results of $R\left(t\right)$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	MLE		MCMC					MLE		MCMC
		$\mathit{T}=\mathbf{0.5}$							$\mathit{T}=\mathbf{1.5}$
Prior→				I		II					I		II
Pop-1
(30,15,10)	PCP[1]	0.7525	0.5831	0.4565	0.3838	0.0651	0.0550		0.7602	0.4782	0.4016	0.2783	0.0586	0.0476
	PCP[2]	0.8790	0.6101	0.5254	0.3877	0.0660	0.0570		0.7995	0.5609	0.5004	0.3219	0.0613	0.0525
	PCP[3]	0.9817	0.6453	0.5626	0.4094	0.0700	0.0605		0.8346	0.6232	0.5305	0.3467	0.0637	0.0555
(30,20,15)	PCP[1]	0.5561	0.4539	0.3619	0.2783	0.0557	0.0480		0.5527	0.4435	0.3618	0.2572	0.0495	0.0390
	PCP[2]	0.6088	0.5089	0.4068	0.2965	0.0621	0.0524		0.5782	0.4660	0.3782	0.2738	0.0540	0.0445
	PCP[3]	0.6054	0.4867	0.3684	0.2867	0.0616	0.0499		0.5635	0.4536	0.3648	0.2663	0.0534	0.0410
(50,30,20)	PCP[1]	0.4525	0.3694	0.3373	0.2424	0.0506	0.0420		0.4354	0.3488	0.2940	0.2336	0.0461	0.0370
	PCP[2]	0.5495	0.4406	0.3437	0.2572	0.0516	0.0437		0.4704	0.3806	0.3140	0.2392	0.0475	0.0377
	PCP[3]	0.5551	0.4466	0.3499	0.2614	0.0548	0.0449		0.4841	0.3939	0.3499	0.2565	0.0486	0.0379
(50,40,30)	PCP[1]	0.4111	0.3262	0.2667	0.2157	0.0437	0.0350		0.3816	0.3026	0.2649	0.2090	0.0420	0.0337
	PCP[2]	0.4479	0.3693	0.3234	0.2352	0.0504	0.0397		0.4140	0.3299	0.2887	0.2254	0.0450	0.0361
	PCP[3]	0.4151	0.3285	0.2958	0.2290	0.0475	0.0370		0.4001	0.3163	0.2810	0.2097	0.0437	0.0350
(80,50,40)	PCP[1]	0.3822	0.3087	0.2323	0.1961	0.0352	0.0281		0.3307	0.2634	0.2017	0.1701	0.0352	0.0280
	PCP[2]	0.3832	0.3129	0.2383	0.1967	0.0368	0.0289		0.3431	0.2711	0.2073	0.1735	0.0365	0.0288
	PCP[3]	0.3984	0.3136	0.2403	0.2024	0.0437	0.0350		0.3592	0.2866	0.2342	0.1969	0.0368	0.0289
(80,60,50)	PCP[1]	0.3305	0.2626	0.2152	0.1612	0.0331	0.0269		0.3131	0.2478	0.1795	0.1321	0.0331	0.0236
	PCP[2]	0.3527	0.2789	0.2294	0.1936	0.0342	0.0278		0.3292	0.2622	0.1974	0.1653	0.0340	0.0269
	PCP[3]	0.3401	0.2656	0.2167	0.1811	0.0337	0.0269		0.3150	0.2504	0.1874	0.1561	0.0331	0.0264
Pop-2
(30,15,10)	PCP[1]	0.6090	0.4887	0.4131	0.3048	0.0485	0.0428		0.5858	0.4718	0.3884	0.2887	0.0447	0.0390
	PCP[2]	0.6350	0.5068	0.4161	0.3064	0.0485	0.0436		0.6198	0.5005	0.3976	0.2973	0.0468	0.0398
	PCP[3]	0.6747	0.5518	0.4420	0.3915	0.0494	0.0455		0.6368	0.5249	0.4131	0.3048	0.0485	0.0447
(30,20,15)	PCP[1]	0.5716	0.4570	0.3623	0.2678	0.0394	0.0346		0.5699	0.4552	0.3217	0.2607	0.0387	0.0334
	PCP[2]	0.5908	0.4756	0.4115	0.2978	0.0396	0.0414		0.5804	0.4689	0.3731	0.2678	0.0424	0.0381
	PCP[3]	0.5787	0.4700	0.3871	0.2783	0.0396	0.0380		0.5725	0.4648	0.3470	0.2637	0.0391	0.0337
(50,30,20)	PCP[1]	0.4583	0.3689	0.3112	0.2584	0.0364	0.0322		0.4479	0.3569	0.3088	0.2473	0.0329	0.0273
	PCP[2]	0.4651	0.3722	0.3217	0.2589	0.0390	0.0341		0.4555	0.3622	0.3102	0.2512	0.0377	0.0303
	PCP[3]	0.5341	0.4401	0.3391	0.2652	0.0393	0.0341		0.4971	0.3978	0.3160	0.2599	0.0377	0.0313
(50,40,30)	PCP[1]	0.4350	0.3465	0.2875	0.2263	0.0306	0.0253		0.4268	0.3400	0.2795	0.2236	0.0256	0.0207
	PCP[2]	0.4503	0.3635	0.3085	0.2493	0.0329	0.0273		0.4394	0.3482	0.3067	0.2374	0.0311	0.0264
	PCP[3]	0.4473	0.3571	0.2886	0.2333	0.0326	0.0270		0.4271	0.3419	0.2875	0.2263	0.0299	0.0245
(80,50,40)	PCP[1]	0.3805	0.3049	0.2033	0.1659	0.0222	0.0181		0.3647	0.2940	0.1882	0.1507	0.0217	0.0172
	PCP[2]	0.4246	0.3447	0.2387	0.1933	0.0223	0.0182		0.3847	0.3073	0.2115	0.1694	0.0220	0.0180
	PCP[3]	0.4345	0.3458	0.2415	0.1934	0.0234	0.0189		0.4097	0.3311	0.2139	0.1697	0.0220	0.0180
(80,60,50)	PCP[1]	0.3546	0.2810	0.1662	0.1404	0.0204	0.0153		0.3499	0.2781	0.1555	0.1325	0.0204	0.0143
	PCP[2]	0.3674	0.2946	0.1948	0.1565	0.0210	0.0164		0.3606	0.2890	0.1630	0.1342	0.0205	0.0164
	PCP[3]	0.3578	0.2877	0.1932	0.1555	0.0205	0.0154		0.3526	0.2819	0.1562	0.1333	0.0204	0.0154

and

Table 5. The RMSE (1st Col.) and MRAB (2nd Col.) results of $h\left(t\right)$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	MLE		MCMC					MLE		MCMC
		$\mathit{T}=\mathbf{0.5}$							$\mathit{T}=\mathbf{1.5}$
Prior→				I		II					I		II
Pop-1
(30,15,10)	PCP[1]	0.7630	0.6900	0.6028	0.4539	0.0675	0.0574		0.6922	0.6416	0.5538	0.3666	0.0654	0.0552
	PCP[2]	0.7421	0.7095	0.6337	0.4651	0.0708	0.0605		0.7188	0.7013	0.5732	0.3938	0.0660	0.0558
	PCP[3]	0.6992	0.5547	0.4958	0.4470	0.0673	0.0548		0.6288	0.5167	0.4543	0.3230	0.0650	0.0542
(30,20,15)	PCP[1]	0.6753	0.5295	0.4818	0.3408	0.0642	0.0539		0.6227	0.4958	0.4415	0.3196	0.0541	0.0443
	PCP[2]	0.6706	0.5231	0.4347	0.3381	0.0638	0.0512		0.6204	0.4848	0.4291	0.3141	0.0535	0.0405
	PCP[3]	0.6232	0.4934	0.4233	0.3196	0.0572	0.0479		0.6081	0.4839	0.4232	0.3005	0.0507	0.0401
(50,30,20)	PCP[1]	0.6165	0.4837	0.3942	0.2984	0.0529	0.0432		0.5183	0.4188	0.3658	0.2785	0.0489	0.0384
	PCP[2]	0.6127	0.4855	0.4015	0.3075	0.0548	0.0441		0.5213	0.4242	0.4015	0.2979	0.0500	0.0393
	PCP[3]	0.4838	0.3953	0.3788	0.2807	0.0518	0.0417		0.4736	0.3936	0.3422	0.2720	0.0474	0.0381
(50,40,30)	PCP[1]	0.4824	0.3938	0.3573	0.2734	0.0518	0.0405		0.4497	0.3782	0.3355	0.2611	0.0468	0.0373
	PCP[2]	0.4359	0.3462	0.3424	0.2663	0.0489	0.0384		0.4319	0.3444	0.3264	0.2430	0.0460	0.0371
	PCP[3]	0.4342	0.3438	0.3096	0.2507	0.0460	0.0373		0.4055	0.3239	0.3071	0.2419	0.0450	0.0365
(80,50,40)	PCP[1]	0.4045	0.3282	0.2771	0.2305	0.0420	0.0357		0.3650	0.2888	0.2416	0.2029	0.0430	0.0335
	PCP[2]	0.4154	0.3299	0.2806	0.2353	0.0460	0.0373		0.3910	0.3120	0.2734	0.2308	0.0420	0.0342
	PCP[3]	0.3964	0.3205	0.2714	0.2297	0.0406	0.0342		0.3565	0.2847	0.2354	0.1990	0.0410	0.0322
(80,60,50)	PCP[1]	0.3768	0.2981	0.2681	0.2270	0.0405	0.0342		0.3421	0.2715	0.2300	0.1924	0.0399	0.0300
	PCP[2]	0.3477	0.2731	0.2527	0.2110	0.0399	0.0338		0.3344	0.2658	0.2182	0.1816	0.0390	0.0272
	PCP[3]	0.3428	0.2716	0.2508	0.1894	0.0380	0.0333		0.3249	0.2572	0.2092	0.1563	0.0383	0.0244
Pop-2
(30,15,10)	PCP[1]	0.7088	0.5571	0.5011	0.3736	0.0804	0.0738		0.7046	0.5535	0.4779	0.3602	0.0750	0.0688
	PCP[2]	0.7707	0.6268	0.5356	0.4760	0.0814	0.0760		0.7108	0.5657	0.5011	0.3736	0.0803	0.0750
	PCP[3]	0.6760	0.5356	0.4958	0.3725	0.0803	0.0728		0.6695	0.5187	0.4666	0.3500	0.0725	0.0678
(30,20,15)	PCP[1]	0.6673	0.5350	0.4892	0.3478	0.0676	0.0685		0.6653	0.5123	0.4351	0.3140	0.0676	0.0582
	PCP[2]	0.6647	0.5121	0.4560	0.3271	0.0676	0.0658		0.6472	0.5121	0.4053	0.3080	0.0669	0.0577
	PCP[3]	0.6379	0.5008	0.4241	0.3202	0.0665	0.0621		0.6239	0.4915	0.3787	0.3060	0.0641	0.0573
(50,30,20)	PCP[1]	0.5156	0.4091	0.3787	0.3104	0.0634	0.0578		0.4853	0.3856	0.3673	0.3028	0.0602	0.0561
	PCP[2]	0.5965	0.4923	0.4006	0.3140	0.0640	0.0601		0.5355	0.4262	0.3677	0.3048	0.0618	0.0561
	PCP[3]	0.4915	0.3905	0.3688	0.3075	0.0601	0.0545		0.4852	0.3820	0.3657	0.2905	0.0557	0.0480
(50,40,30)	PCP[1]	0.4870	0.3859	0.3620	0.3016	0.0557	0.0461		0.4625	0.3625	0.3611	0.2902	0.0552	0.0461
	PCP[2]	0.4734	0.3806	0.3455	0.2775	0.0552	0.0458		0.4541	0.3612	0.3402	0.2731	0.0509	0.0423
	PCP[3]	0.4700	0.3747	0.3402	0.2703	0.0522	0.0428		0.4462	0.3564	0.3319	0.2632	0.0435	0.0357
(80,50,40)	PCP[1]	0.4537	0.3607	0.2990	0.2439	0.0376	0.0305		0.4008	0.3167	0.2661	0.2139	0.0370	0.0300
	PCP[2]	0.4603	0.3653	0.3071	0.2545	0.0396	0.0319		0.4361	0.3490	0.2728	0.2237	0.0370	0.0300
	PCP[3]	0.3972	0.3120	0.2697	0.2246	0.0374	0.0303		0.3868	0.3081	0.2414	0.1979	0.0364	0.0295
(80,60,50)	PCP[1]	0.3890	0.3090	0.2598	0.2095	0.0304	0.0269		0.3661	0.2909	0.2213	0.1803	0.0340	0.0282
	PCP[2]	0.3654	0.2910	0.2482	0.2044	0.0304	0.0249		0.3641	0.2881	0.2177	0.1776	0.0330	0.0268
	PCP[3]	0.3578	0.2828	0.2231	0.1906	0.0303	0.0222		0.3562	0.2823	0.2106	0.1760	0.0302	0.0248

provide the simulated values of RMSE and MRAB for the estimators of $\alpha$, $\xi$, and the reliability metrics $R\left(t\right)$ and $h\left(t\right)$, respectively. Additionally,

Table 6. The AIL (1st Col.) and CP (2nd Col.) results of $\alpha$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	ACI-NA		ACI-NL		BCI				HPD
Prior→						I		II		I		II
Pop-1 ($T=0.5$)
(30,15,10)	PCP[1]	0.981	0.960	1.226	0.956	0.809	0.962	0.781	0.963	0.270	0.967	0.267	0.968
	PCP[2]	1.200	0.957	1.290	0.955	0.859	0.962	0.832	0.962	0.281	0.967	0.271	0.967
	PCP[3]	0.920	0.961	1.150	0.957	0.762	0.963	0.728	0.964	0.269	0.968	0.266	0.968
(30,20,15)	PCP[1]	0.793	0.963	0.991	0.960	0.710	0.964	0.689	0.964	0.239	0.969	0.238	0.969
	PCP[2]	0.865	0.962	1.081	0.958	0.748	0.963	0.723	0.964	0.250	0.968	0.246	0.969
	PCP[3]	0.817	0.962	1.022	0.959	0.715	0.964	0.702	0.964	0.239	0.969	0.238	0.969
(50,30,20)	PCP[1]	0.774	0.963	0.967	0.960	0.670	0.965	0.650	0.965	0.229	0.970	0.226	0.970
	PCP[2]	0.780	0.963	0.975	0.960	0.675	0.964	0.668	0.965	0.231	0.969	0.228	0.970
	PCP[3]	0.761	0.963	0.952	0.960	0.645	0.965	0.625	0.965	0.228	0.970	0.221	0.970
(50,40,30)	PCP[1]	0.715	0.964	0.893	0.961	0.560	0.966	0.545	0.966	0.224	0.971	0.210	0.971
	PCP[2]	0.708	0.964	0.885	0.961	0.554	0.966	0.534	0.967	0.206	0.971	0.207	0.972
	PCP[3]	0.703	0.964	0.879	0.961	0.521	0.967	0.504	0.967	0.203	0.972	0.204	0.972
(80,50,40)	PCP[1]	0.580	0.966	0.844	0.962	0.458	0.968	0.445	0.968	0.198	0.973	0.192	0.973
	PCP[2]	0.621	0.965	0.856	0.962	0.495	0.967	0.461	0.968	0.199	0.972	0.197	0.973
	PCP[3]	0.562	0.966	0.780	0.963	0.450	0.968	0.442	0.968	0.195	0.973	0.186	0.973
(80,60,50)	PCP[1]	0.531	0.967	0.664	0.965	0.445	0.968	0.436	0.968	0.194	0.973	0.179	0.973
	PCP[2]	0.558	0.966	0.763	0.963	0.448	0.968	0.441	0.968	0.194	0.973	0.183	0.973
	PCP[3]	0.448	0.968	0.657	0.965	0.439	0.968	0.259	0.971	0.192	0.973	0.172	0.974
Pop-1 ($T=1.5$)
(30,15,10)	PCP[1]	0.937	0.961	1.171	0.957	0.799	0.966	0.767	0.967	0.270	0.971	0.262	0.972
	PCP[2]	0.976	0.960	1.220	0.956	0.852	0.965	0.813	0.966	0.277	0.970	0.268	0.971
	PCP[3]	0.887	0.961	1.108	0.958	0.754	0.966	0.715	0.968	0.269	0.971	0.260	0.972
(30,20,15)	PCP[1]	0.759	0.963	0.949	0.960	0.699	0.967	0.668	0.969	0.239	0.972	0.232	0.973
	PCP[2]	0.824	0.962	1.030	0.959	0.745	0.966	0.710	0.968	0.249	0.971	0.240	0.973
	PCP[3]	0.813	0.962	1.017	0.959	0.711	0.967	0.701	0.968	0.239	0.972	0.238	0.973
(50,30,20)	PCP[1]	0.681	0.964	0.851	0.962	0.662	0.968	0.642	0.969	0.228	0.973	0.221	0.974
	PCP[2]	0.720	0.964	0.899	0.961	0.675	0.967	0.667	0.969	0.229	0.972	0.223	0.974
	PCP[3]	0.645	0.965	0.807	0.962	0.640	0.968	0.621	0.970	0.226	0.973	0.212	0.975
(50,40,30)	PCP[1]	0.606	0.965	0.757	0.963	0.557	0.969	0.544	0.970	0.219	0.974	0.208	0.975
	PCP[2]	0.596	0.966	0.746	0.963	0.547	0.969	0.531	0.971	0.205	0.974	0.202	0.976
	PCP[3]	0.590	0.966	0.737	0.964	0.507	0.970	0.492	0.971	0.202	0.975	0.201	0.976
(80,50,40)	PCP[1]	0.464	0.968	0.676	0.964	0.454	0.971	0.437	0.972	0.195	0.975	0.190	0.977
	PCP[2]	0.497	0.967	0.685	0.964	0.479	0.970	0.446	0.972	0.198	0.975	0.193	0.977
	PCP[3]	0.455	0.968	0.624	0.965	0.449	0.971	0.437	0.972	0.195	0.976	0.182	0.977
(80,60,50)	PCP[1]	0.449	0.968	0.545	0.966	0.440	0.971	0.436	0.972	0.191	0.976	0.174	0.977
	PCP[2]	0.449	0.968	0.610	0.965	0.447	0.971	0.436	0.972	0.192	0.976	0.181	0.977
	PCP[3]	0.442	0.968	0.526	0.967	0.432	0.971	0.207	0.973	0.189	0.976	0.167	0.978
Pop-2 ($T=0.5$)
(30,15,10)	PCP[1]	1.886	0.946	2.811	0.933	1.195	0.957	1.181	0.957	0.298	0.962	0.295	0.962
	PCP[2]	2.954	0.931	3.192	0.927	1.201	0.957	1.194	0.957	0.299	0.962	0.296	0.962
	PCP[3]	1.791	0.948	2.521	0.937	1.176	0.957	1.168	0.957	0.294	0.963	0.291	0.962
(30,20,15)	PCP[1]	1.397	0.954	1.926	0.946	1.158	0.957	1.125	0.958	0.271	0.964	0.270	0.963
	PCP[2]	1.549	0.951	2.257	0.941	1.168	0.957	1.160	0.957	0.279	0.963	0.274	0.962
	PCP[3]	1.543	0.952	2.144	0.943	1.166	0.957	1.147	0.957	0.271	0.964	0.270	0.962
(50,30,20)	PCP[1]	1.376	0.954	1.751	0.948	0.960	0.960	0.922	0.961	0.250	0.965	0.247	0.966
	PCP[2]	1.392	0.954	1.908	0.946	0.966	0.960	0.948	0.960	0.252	0.965	0.249	0.965
	PCP[3]	1.139	0.958	1.490	0.952	0.931	0.961	0.911	0.961	0.250	0.966	0.247	0.966
(50,40,30)	PCP[1]	1.108	0.958	1.489	0.952	0.917	0.961	0.885	0.961	0.227	0.966	0.226	0.966
	PCP[2]	1.028	0.959	1.468	0.953	0.916	0.961	0.865	0.962	0.224	0.966	0.223	0.967
	PCP[3]	1.027	0.959	1.466	0.953	0.883	0.961	0.863	0.962	0.224	0.966	0.222	0.967
(80,50,40)	PCP[1]	0.987	0.960	1.233	0.956	0.792	0.963	0.785	0.963	0.217	0.968	0.215	0.968
	PCP[2]	1.022	0.959	1.390	0.954	0.818	0.962	0.810	0.962	0.219	0.967	0.218	0.967
	PCP[3]	0.885	0.961	1.106	0.958	0.785	0.963	0.780	0.963	0.216	0.968	0.212	0.968
(80,60,50)	PCP[1]	0.684	0.964	0.908	0.961	0.667	0.965	0.659	0.965	0.211	0.970	0.209	0.970
	PCP[2]	0.777	0.963	1.082	0.958	0.702	0.964	0.695	0.964	0.212	0.969	0.210	0.969
	PCP[3]	0.665	0.965	0.681	0.964	0.559	0.966	0.447	0.968	0.210	0.971	0.208	0.970
Pop-2 ($T=1.5$)
(30,15,10)	PCP[1]	1.509	0.952	2.617	0.936	1.184	0.960	1.180	0.961	0.295	0.965	0.292	0.966
	PCP[2]	2.925	0.931	3.085	0.929	1.194	0.960	1.189	0.961	0.297	0.965	0.294	0.965
	PCP[3]	1.433	0.953	2.257	0.941	1.172	0.960	1.160	0.961	0.294	0.965	0.291	0.966
(30,20,15)	PCP[1]	1.160	0.957	1.541	0.952	1.126	0.961	1.117	0.962	0.271	0.966	0.270	0.967
	PCP[2]	1.239	0.956	2.059	0.944	1.164	0.960	1.158	0.961	0.278	0.965	0.273	0.966
	PCP[3]	1.235	0.956	1.948	0.946	1.160	0.960	1.145	0.961	0.271	0.965	0.270	0.966
(50,30,20)	PCP[1]	1.101	0.958	1.401	0.954	0.951	0.963	0.914	0.965	0.250	0.968	0.247	0.970
	PCP[2]	1.114	0.958	1.527	0.952	0.958	0.963	0.937	0.965	0.250	0.968	0.247	0.970
	PCP[3]	0.940	0.961	1.192	0.957	0.912	0.964	0.905	0.965	0.247	0.969	0.244	0.970
(50,40,30)	PCP[1]	0.927	0.961	1.191	0.957	0.887	0.964	0.883	0.965	0.226	0.969	0.224	0.970
	PCP[2]	0.925	0.961	1.174	0.957	0.876	0.964	0.822	0.966	0.224	0.969	0.222	0.971
	PCP[3]	0.885	0.961	1.173	0.957	0.873	0.965	0.822	0.966	0.223	0.970	0.220	0.971
(80,50,40)	PCP[1]	0.805	0.963	1.006	0.960	0.786	0.966	0.777	0.967	0.215	0.971	0.211	0.972
	PCP[2]	0.824	0.962	1.112	0.958	0.816	0.965	0.804	0.967	0.216	0.970	0.212	0.972
	PCP[3]	0.787	0.963	0.977	0.960	0.782	0.966	0.776	0.967	0.213	0.971	0.210	0.972
(80,60,50)	PCP[1]	0.682	0.964	0.726	0.964	0.666	0.968	0.534	0.971	0.210	0.973	0.205	0.975
	PCP[2]	0.714	0.964	0.865	0.962	0.701	0.967	0.622	0.969	0.210	0.972	0.208	0.974
	PCP[3]	0.658	0.965	0.678	0.964	0.552	0.969	0.442	0.972	0.209	0.973	0.205	0.975

,

Table 7. The AIL (1st Col.) and CP (2nd Col.) results of $\xi$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	ACI-NA		ACI-NL		BCI				HPD
Prior→						I		II		I		II
Pop-1 ($T=0.5$)
(30,15,10)	PCP[1]	1.060	0.944	1.458	0.926	1.031	0.945	0.822	0.955	0.275	0.981	0.268	0.981
	PCP[2]	0.987	0.947	1.357	0.930	0.962	0.949	0.695	0.961	0.267	0.981	0.262	0.981
	PCP[3]	0.994	0.947	1.367	0.930	0.979	0.948	0.798	0.956	0.268	0.981	0.263	0.981
(30,20,15)	PCP[1]	0.977	0.948	1.343	0.931	0.956	0.949	0.645	0.963	0.256	0.981	0.254	0.982
	PCP[2]	0.927	0.950	1.274	0.934	0.903	0.951	0.616	0.965	0.255	0.981	0.252	0.982
	PCP[3]	0.956	0.949	1.314	0.932	0.937	0.950	0.616	0.965	0.255	0.981	0.252	0.982
(50,30,20)	PCP[1]	0.895	0.952	1.231	0.936	0.893	0.952	0.570	0.967	0.255	0.982	0.252	0.982
	PCP[2]	0.866	0.953	1.191	0.938	0.857	0.954	0.542	0.968	0.252	0.982	0.244	0.982
	PCP[3]	0.874	0.953	1.202	0.937	0.866	0.953	0.549	0.968	0.252	0.982	0.249	0.982
(50,40,30)	PCP[1]	0.844	0.954	1.161	0.939	0.826	0.955	0.538	0.968	0.243	0.982	0.238	0.982
	PCP[2]	0.809	0.956	1.112	0.942	0.787	0.957	0.497	0.970	0.241	0.982	0.233	0.982
	PCP[3]	0.841	0.954	1.156	0.940	0.813	0.956	0.500	0.970	0.242	0.982	0.236	0.982
(80,50,40)	PCP[1]	0.773	0.957	1.062	0.944	0.753	0.958	0.478	0.971	0.215	0.983	0.211	0.984
	PCP[2]	0.738	0.959	1.015	0.946	0.730	0.959	0.471	0.971	0.205	0.984	0.194	0.984
	PCP[3]	0.762	0.958	1.048	0.945	0.738	0.959	0.475	0.971	0.215	0.983	0.210	0.984
(80,60,50)	PCP[1]	0.711	0.960	0.978	0.948	0.698	0.961	0.440	0.973	0.194	0.984	0.190	0.984
	PCP[2]	0.639	0.964	0.878	0.953	0.624	0.964	0.418	0.974	0.187	0.985	0.173	0.985
	PCP[3]	0.688	0.961	0.945	0.949	0.656	0.963	0.422	0.974	0.192	0.984	0.188	0.985
Pop-1 ($T=1.5$)
(30,15,10)	PCP[1]	1.054	0.944	1.246	0.935	1.018	0.946	0.821	0.955	0.270	0.981	0.263	0.981
	PCP[2]	0.981	0.948	1.159	0.939	0.950	0.949	0.533	0.969	0.264	0.981	0.259	0.981
	PCP[3]	0.982	0.948	1.168	0.939	0.968	0.948	0.580	0.966	0.267	0.981	0.262	0.981
(30,20,15)	PCP[1]	0.961	0.949	1.147	0.940	0.945	0.949	0.506	0.970	0.256	0.981	0.254	0.982
	PCP[2]	0.923	0.950	1.089	0.943	0.897	0.952	0.439	0.973	0.253	0.982	0.249	0.982
	PCP[3]	0.944	0.949	1.123	0.941	0.933	0.950	0.477	0.971	0.253	0.982	0.250	0.982
(50,30,20)	PCP[1]	0.895	0.952	1.052	0.944	0.893	0.952	0.429	0.973	0.252	0.982	0.249	0.982
	PCP[2]	0.863	0.953	1.018	0.946	0.852	0.954	0.375	0.976	0.248	0.982	0.242	0.982
	PCP[3]	0.869	0.953	1.027	0.946	0.866	0.953	0.418	0.974	0.251	0.982	0.244	0.982
(50,40,30)	PCP[1]	0.840	0.954	0.992	0.947	0.819	0.955	0.356	0.977	0.242	0.982	0.236	0.982
	PCP[2]	0.802	0.956	0.950	0.949	0.783	0.957	0.348	0.977	0.239	0.982	0.232	0.983
	PCP[3]	0.837	0.954	0.988	0.947	0.807	0.956	0.355	0.977	0.239	0.982	0.232	0.983
(80,50,40)	PCP[1]	0.763	0.958	0.908	0.951	0.737	0.959	0.327	0.978	0.215	0.983	0.211	0.984
	PCP[2]	0.736	0.959	0.868	0.953	0.717	0.960	0.320	0.978	0.202	0.984	0.191	0.984
	PCP[3]	0.744	0.959	0.895	0.952	0.729	0.959	0.324	0.978	0.211	0.984	0.207	0.984
(80,60,50)	PCP[1]	0.703	0.961	0.836	0.954	0.690	0.961	0.304	0.979	0.193	0.984	0.189	0.985
	PCP[2]	0.632	0.964	0.750	0.958	0.624	0.964	0.190	0.985	0.177	0.985	0.115	0.988
	PCP[3]	0.678	0.962	0.808	0.956	0.642	0.964	0.253	0.982	0.191	0.984	0.187	0.985
Pop-2 ($T=0.5$)
(30,15,10)	PCP[1]	1.132	0.941	1.556	0.921	1.085	0.943	0.880	0.952	0.277	0.980	0.271	0.981
	PCP[2]	1.077	0.943	1.481	0.924	1.061	0.944	0.747	0.959	0.272	0.981	0.267	0.981
	PCP[3]	1.084	0.943	1.491	0.924	1.077	0.943	0.861	0.953	0.273	0.981	0.268	0.981
(30,20,15)	PCP[1]	1.073	0.943	1.475	0.925	1.015	0.946	0.697	0.961	0.265	0.981	0.264	0.981
	PCP[2]	1.069	0.944	1.470	0.925	1.004	0.947	0.647	0.963	0.261	0.981	0.260	0.981
	PCP[3]	1.069	0.944	1.470	0.925	1.009	0.946	0.651	0.963	0.265	0.981	0.260	0.981
(50,30,20)	PCP[1]	1.039	0.945	1.429	0.927	1.002	0.947	0.615	0.965	0.260	0.981	0.259	0.981
	PCP[2]	1.011	0.946	1.389	0.929	0.982	0.948	0.578	0.966	0.260	0.981	0.258	0.981
	PCP[3]	1.038	0.945	1.428	0.927	0.986	0.947	0.589	0.966	0.260	0.981	0.258	0.981
(50,40,30)	PCP[1]	1.003	0.947	1.379	0.929	0.972	0.948	0.532	0.969	0.258	0.981	0.251	0.982
	PCP[2]	0.985	0.948	1.354	0.930	0.963	0.949	0.500	0.970	0.250	0.982	0.247	0.982
	PCP[3]	0.986	0.947	1.356	0.930	0.971	0.948	0.512	0.970	0.251	0.982	0.247	0.982
(80,50,40)	PCP[1]	0.982	0.948	1.350	0.931	0.959	0.949	0.486	0.971	0.213	0.983	0.209	0.984
	PCP[2]	0.935	0.950	1.285	0.934	0.918	0.951	0.460	0.972	0.213	0.983	0.209	0.984
	PCP[3]	0.975	0.948	1.340	0.931	0.935	0.950	0.467	0.972	0.213	0.983	0.209	0.984
(80,60,50)	PCP[1]	0.908	0.951	1.249	0.935	0.888	0.952	0.454	0.972	0.198	0.984	0.195	0.984
	PCP[2]	0.867	0.953	1.193	0.938	0.846	0.954	0.414	0.974	0.197	0.984	0.193	0.984
	PCP[3]	0.888	0.952	1.221	0.937	0.876	0.953	0.436	0.973	0.197	0.984	0.193	0.984
Pop-2 ($T=1.5$)
(30,15,10)	PCP[1]	1.126	0.941	1.330	0.932	1.079	0.943	0.880	0.952	0.271	0.981	0.266	0.981
	PCP[2]	1.064	0.944	1.266	0.935	1.055	0.944	0.606	0.965	0.270	0.981	0.265	0.981
	PCP[3]	1.082	0.943	1.274	0.934	1.069	0.944	0.621	0.964	0.271	0.981	0.266	0.981
(30,20,15)	PCP[1]	1.059	0.944	1.260	0.935	1.008	0.946	0.563	0.967	0.265	0.981	0.264	0.981
	PCP[2]	1.054	0.944	1.256	0.935	0.993	0.947	0.518	0.969	0.260	0.981	0.259	0.981
	PCP[3]	1.056	0.944	1.256	0.935	1.006	0.947	0.547	0.968	0.264	0.981	0.259	0.981
(50,30,20)	PCP[1]	1.014	0.946	1.221	0.937	0.991	0.947	0.471	0.971	0.260	0.981	0.259	0.981
	PCP[2]	0.997	0.947	1.187	0.938	0.963	0.949	0.441	0.973	0.259	0.981	0.251	0.982
	PCP[3]	1.010	0.946	1.220	0.937	0.978	0.948	0.442	0.973	0.259	0.981	0.251	0.982
(50,40,30)	PCP[1]	0.982	0.948	1.179	0.939	0.950	0.949	0.395	0.975	0.258	0.981	0.251	0.982
	PCP[2]	0.967	0.948	1.157	0.940	0.945	0.949	0.367	0.976	0.248	0.982	0.243	0.982
	PCP[3]	0.980	0.948	1.159	0.939	0.947	0.949	0.370	0.976	0.248	0.982	0.243	0.982
(80,50,40)	PCP[1]	0.960	0.949	1.153	0.940	0.938	0.950	0.347	0.977	0.212	0.983	0.208	0.984
	PCP[2]	0.934	0.950	1.098	0.942	0.918	0.951	0.326	0.978	0.210	0.984	0.206	0.984
	PCP[3]	0.955	0.949	1.145	0.940	0.918	0.951	0.344	0.977	0.212	0.984	0.208	0.984
(80,60,50)	PCP[1]	0.888	0.952	1.067	0.944	0.876	0.953	0.309	0.979	0.197	0.984	0.194	0.984
	PCP[2]	0.852	0.954	1.019	0.946	0.829	0.955	0.279	0.980	0.195	0.984	0.191	0.984
	PCP[3]	0.882	0.952	1.043	0.945	0.871	0.953	0.300	0.979	0.196	0.984	0.193	0.984

,

Table 8. The AIL (1st Col.) and CP (2nd Col.) results of $R\left(t\right)$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	ACI-NA		ACI-NL		BCI				HPD
Prior→						I		II		I		II
Pop-1 ($T=0.5$)
(30,15,10)	PCP[1]	0.217	0.958	0.213	0.958	0.139	0.977	0.138	0.977	0.137	0.977	0.137	0.977
	PCP[2]	0.233	0.954	0.227	0.955	0.143	0.976	0.141	0.976	0.140	0.977	0.139	0.977
	PCP[3]	0.243	0.951	0.236	0.953	0.146	0.975	0.143	0.976	0.142	0.976	0.141	0.976
(30,20,15)	PCP[1]	0.196	0.963	0.191	0.964	0.133	0.978	0.129	0.979	0.118	0.982	0.114	0.983
	PCP[2]	0.211	0.959	0.210	0.959	0.138	0.977	0.134	0.978	0.132	0.978	0.129	0.979
	PCP[3]	0.201	0.961	0.199	0.962	0.134	0.978	0.131	0.979	0.121	0.981	0.115	0.983
(50,30,20)	PCP[1]	0.162	0.971	0.150	0.974	0.113	0.983	0.111	0.984	0.110	0.984	0.105	0.985
	PCP[2]	0.168	0.969	0.157	0.972	0.114	0.983	0.111	0.984	0.110	0.984	0.105	0.985
	PCP[3]	0.183	0.966	0.161	0.971	0.117	0.982	0.115	0.983	0.114	0.983	0.111	0.984
(50,40,30)	PCP[1]	0.149	0.974	0.133	0.978	0.102	0.986	0.097	0.987	0.091	0.989	0.087	0.989
	PCP[2]	0.159	0.972	0.149	0.974	0.112	0.983	0.109	0.984	0.107	0.985	0.101	0.986
	PCP[3]	0.154	0.973	0.142	0.976	0.108	0.984	0.103	0.986	0.101	0.986	0.096	0.987
(80,50,40)	PCP[1]	0.132	0.978	0.122	0.981	0.071	0.994	0.070	0.994	0.069	0.994	0.069	0.994
	PCP[2]	0.135	0.978	0.125	0.980	0.076	0.992	0.075	0.993	0.074	0.993	0.073	0.993
	PCP[3]	0.142	0.976	0.127	0.980	0.084	0.990	0.082	0.991	0.082	0.991	0.081	0.991
(80,60,50)	PCP[1]	0.124	0.980	0.066	0.995	0.063	0.996	0.060	0.996	0.055	0.997	0.054	0.998
	PCP[2]	0.126	0.980	0.121	0.981	0.069	0.994	0.068	0.994	0.066	0.995	0.065	0.995
	PCP[3]	0.125	0.980	0.077	0.992	0.068	0.994	0.067	0.994	0.062	0.996	0.062	0.996
Pop-1 ($T=1.5$)
(30,15,10)	PCP[1]	0.215	0.958	0.210	0.959	0.138	0.977	0.138	0.977	0.137	0.977	0.136	0.977
	PCP[2]	0.229	0.954	0.215	0.958	0.142	0.976	0.141	0.976	0.140	0.977	0.138	0.977
	PCP[3]	0.239	0.952	0.230	0.954	0.145	0.975	0.142	0.976	0.141	0.976	0.140	0.976
(30,20,15)	PCP[1]	0.194	0.963	0.189	0.964	0.132	0.978	0.128	0.979	0.117	0.982	0.113	0.983
	PCP[2]	0.211	0.959	0.208	0.960	0.137	0.977	0.133	0.978	0.131	0.979	0.128	0.980
	PCP[3]	0.199	0.962	0.197	0.962	0.133	0.978	0.130	0.979	0.120	0.981	0.115	0.983
(50,30,20)	PCP[1]	0.161	0.971	0.148	0.974	0.112	0.983	0.110	0.984	0.109	0.984	0.104	0.985
	PCP[2]	0.164	0.970	0.156	0.973	0.113	0.983	0.111	0.984	0.110	0.984	0.104	0.985
	PCP[3]	0.178	0.967	0.159	0.972	0.116	0.982	0.115	0.983	0.114	0.983	0.110	0.984
(50,40,30)	PCP[1]	0.148	0.975	0.132	0.978	0.101	0.986	0.096	0.987	0.090	0.989	0.087	0.990
	PCP[2]	0.158	0.972	0.148	0.975	0.111	0.984	0.108	0.984	0.106	0.985	0.100	0.986
	PCP[3]	0.153	0.973	0.140	0.976	0.105	0.985	0.101	0.986	0.100	0.986	0.096	0.987
(80,50,40)	PCP[1]	0.131	0.979	0.121	0.981	0.070	0.994	0.069	0.994	0.069	0.994	0.068	0.994
	PCP[2]	0.134	0.978	0.124	0.980	0.076	0.992	0.075	0.993	0.074	0.993	0.072	0.993
	PCP[3]	0.141	0.976	0.126	0.980	0.083	0.990	0.082	0.991	0.081	0.991	0.080	0.991
(80,60,50)	PCP[1]	0.123	0.981	0.064	0.995	0.060	0.996	0.059	0.996	0.055	0.998	0.054	0.998
	PCP[2]	0.125	0.980	0.120	0.981	0.069	0.994	0.068	0.994	0.065	0.995	0.064	0.995
	PCP[3]	0.124	0.980	0.076	0.992	0.068	0.994	0.067	0.995	0.062	0.996	0.061	0.996
Pop-2 ($T=0.5$)
(30,15,10)	PCP[1]	0.230	0.954	0.229	0.954	0.119	0.982	0.117	0.982	0.116	0.982	0.115	0.983
	PCP[2]	0.242	0.951	0.237	0.952	0.121	0.981	0.119	0.982	0.118	0.982	0.117	0.982
	PCP[3]	0.263	0.946	0.253	0.949	0.122	0.981	0.121	0.981	0.120	0.981	0.119	0.982
(30,20,15)	PCP[1]	0.216	0.958	0.213	0.959	0.112	0.983	0.111	0.984	0.110	0.984	0.108	0.984
	PCP[2]	0.223	0.956	0.221	0.956	0.117	0.982	0.116	0.982	0.116	0.982	0.114	0.983
	PCP[3]	0.219	0.957	0.215	0.958	0.114	0.983	0.113	0.983	0.112	0.983	0.110	0.984
(50,30,20)	PCP[1]	0.176	0.968	0.175	0.968	0.110	0.984	0.107	0.985	0.105	0.985	0.103	0.985
	PCP[2]	0.182	0.966	0.181	0.966	0.110	0.984	0.107	0.984	0.107	0.985	0.105	0.985
	PCP[3]	0.199	0.962	0.192	0.964	0.111	0.984	0.110	0.984	0.108	0.984	0.105	0.985
(50,40,30)	PCP[1]	0.169	0.969	0.166	0.970	0.101	0.986	0.098	0.987	0.091	0.989	0.086	0.990
	PCP[2]	0.172	0.969	0.171	0.969	0.103	0.986	0.101	0.986	0.100	0.986	0.098	0.987
	PCP[3]	0.169	0.969	0.167	0.970	0.103	0.986	0.100	0.986	0.099	0.987	0.097	0.987
(80,50,40)	PCP[1]	0.141	0.976	0.139	0.977	0.073	0.993	0.071	0.993	0.069	0.994	0.068	0.994
	PCP[2]	0.149	0.974	0.147	0.975	0.085	0.990	0.081	0.991	0.078	0.992	0.074	0.993
	PCP[3]	0.159	0.972	0.158	0.972	0.085	0.990	0.084	0.990	0.082	0.991	0.076	0.992
(80,60,50)	PCP[1]	0.137	0.977	0.134	0.978	0.056	0.997	0.055	0.997	0.055	0.998	0.053	0.998
	PCP[2]	0.139	0.977	0.138	0.977	0.064	0.995	0.063	0.996	0.055	0.997	0.054	0.998
	PCP[3]	0.137	0.977	0.136	0.977	0.060	0.996	0.059	0.996	0.055	0.997	0.054	0.998
Pop-2 ($T=1.5$)
(30,15,10)	PCP[1]	0.230	0.954	0.227	0.955	0.119	0.982	0.116	0.982	0.115	0.983	0.115	0.983
	PCP[2]	0.239	0.952	0.234	0.953	0.120	0.981	0.119	0.982	0.118	0.982	0.117	0.982
	PCP[3]	0.261	0.947	0.250	0.949	0.121	0.981	0.120	0.981	0.119	0.982	0.118	0.982
(30,20,15)	PCP[1]	0.215	0.958	0.210	0.959	0.111	0.984	0.110	0.984	0.109	0.984	0.107	0.985
	PCP[2]	0.222	0.956	0.220	0.957	0.117	0.982	0.116	0.982	0.115	0.983	0.113	0.983
	PCP[3]	0.217	0.957	0.212	0.959	0.113	0.983	0.112	0.983	0.111	0.984	0.110	0.984
(50,30,20)	PCP[1]	0.175	0.968	0.173	0.968	0.109	0.984	0.106	0.985	0.104	0.985	0.103	0.986
	PCP[2]	0.181	0.966	0.179	0.967	0.109	0.984	0.107	0.985	0.106	0.985	0.104	0.985
	PCP[3]	0.197	0.962	0.190	0.964	0.110	0.984	0.109	0.984	0.107	0.985	0.104	0.985
(50,40,30)	PCP[1]	0.167	0.970	0.165	0.970	0.100	0.986	0.097	0.987	0.090	0.989	0.085	0.990
	PCP[2]	0.172	0.969	0.170	0.969	0.102	0.986	0.100	0.986	0.099	0.987	0.097	0.987
	PCP[3]	0.167	0.970	0.165	0.970	0.102	0.986	0.100	0.986	0.099	0.987	0.096	0.987
(80,50,40)	PCP[1]	0.141	0.976	0.138	0.977	0.072	0.993	0.071	0.994	0.069	0.994	0.067	0.994
	PCP[2]	0.148	0.974	0.146	0.975	0.084	0.990	0.081	0.991	0.078	0.992	0.073	0.993
	PCP[3]	0.158	0.972	0.157	0.972	0.085	0.990	0.083	0.990	0.081	0.991	0.076	0.992
(80,60,50)	PCP[1]	0.136	0.978	0.134	0.978	0.056	0.997	0.055	0.998	0.054	0.998	0.053	0.998
	PCP[2]	0.138	0.977	0.137	0.977	0.064	0.995	0.062	0.996	0.055	0.998	0.054	0.998
	PCP[3]	0.136	0.977	0.135	0.978	0.060	0.996	0.058	0.997	0.055	0.998	0.053	0.998

and

Table 9. The AIL (1st Col.) and CP (2nd Col.) results of $h\left(t\right)$.

$(\mathit{n},{\mathit{m}}_2,{\mathit{m}}_1)$	PCP	ACI-NA		ACI-NL		BCI				HPD
Prior→						I		II		I		II
Pop-1 ($T=0.5$)
(30,15,10)	PCP[1]	1.634	0.929	2.553	0.900	0.166	0.970	2.401	0.905	1.595	0.931	0.159	0.971
	PCP[2]	1.721	0.927	2.842	0.891	0.169	0.969	2.629	0.898	1.635	0.929	0.165	0.970
	PCP[3]	1.618	0.930	2.514	0.902	0.157	0.970	2.347	0.907	1.588	0.931	0.153	0.971
(30,20,15)	PCP[1]	1.594	0.931	2.373	0.906	0.152	0.971	2.249	0.910	1.560	0.932	0.149	0.972
	PCP[2]	1.579	0.931	2.261	0.910	0.150	0.971	2.145	0.913	1.425	0.936	0.148	0.972
	PCP[3]	1.564	0.932	2.142	0.913	0.149	0.971	2.034	0.917	1.398	0.937	0.148	0.972
(50,30,20)	PCP[1]	1.336	0.939	1.791	0.924	0.147	0.972	1.728	0.926	1.304	0.940	0.146	0.973
	PCP[2]	1.372	0.938	1.962	0.919	0.148	0.972	1.791	0.924	1.325	0.939	0.147	0.973
	PCP[3]	1.335	0.939	1.779	0.925	0.146	0.972	1.649	0.929	1.273	0.941	0.145	0.973
(50,40,30)	PCP[1]	1.241	0.942	1.681	0.928	0.145	0.973	1.618	0.930	1.235	0.942	0.143	0.974
	PCP[2]	1.222	0.942	1.644	0.929	0.144	0.973	1.560	0.932	1.175	0.944	0.141	0.974
	PCP[3]	1.188	0.943	1.580	0.931	0.141	0.973	1.448	0.935	1.053	0.948	0.141	0.974
(80,50,40)	PCP[1]	0.870	0.953	1.426	0.936	0.139	0.974	1.322	0.939	0.852	0.954	0.137	0.975
	PCP[2]	0.971	0.950	1.529	0.933	0.140	0.974	1.385	0.937	0.946	0.951	0.138	0.975
	PCP[3]	0.814	0.955	1.408	0.937	0.136	0.974	1.308	0.940	0.803	0.956	0.137	0.975
(80,60,50)	PCP[1]	0.801	0.956	1.311	0.940	0.127	0.975	1.283	0.940	0.761	0.957	0.124	0.976
	PCP[2]	0.788	0.956	1.302	0.940	0.125	0.975	0.809	0.955	0.723	0.958	0.122	0.976
	PCP[3]	0.703	0.959	1.293	0.940	0.122	0.975	0.706	0.959	0.619	0.961	0.121	0.976
Pop-1 ($T=1.5$)
(30,15,10)	PCP[1]	1.623	0.930	2.544	0.901	0.161	0.971	2.397	0.905	1.592	0.931	0.153	0.973
	PCP[2]	1.681	0.928	2.764	0.894	0.168	0.970	2.549	0.901	1.629	0.930	0.154	0.972
	PCP[3]	1.594	0.931	2.453	0.904	0.153	0.971	2.295	0.909	1.583	0.931	0.152	0.973
(30,20,15)	PCP[1]	1.583	0.931	2.373	0.906	0.151	0.972	2.249	0.910	1.522	0.933	0.148	0.974
	PCP[2]	1.556	0.932	2.233	0.911	0.149	0.972	2.114	0.914	1.361	0.938	0.148	0.974
	PCP[3]	1.534	0.933	2.089	0.915	0.148	0.972	1.985	0.918	1.359	0.938	0.147	0.974
(50,30,20)	PCP[1]	1.305	0.940	1.742	0.926	0.147	0.973	1.683	0.928	1.246	0.942	0.145	0.975
	PCP[2]	1.347	0.938	1.878	0.922	0.147	0.973	1.733	0.926	1.308	0.940	0.146	0.975
	PCP[3]	1.294	0.940	1.721	0.927	0.145	0.973	1.601	0.930	1.230	0.942	0.144	0.975
(50,40,30)	PCP[1]	1.237	0.942	1.630	0.929	0.144	0.973	1.572	0.931	1.180	0.944	0.141	0.976
	PCP[2]	1.190	0.943	1.595	0.931	0.144	0.974	1.521	0.933	1.120	0.946	0.140	0.976
	PCP[3]	1.132	0.945	1.534	0.933	0.141	0.974	1.415	0.936	1.024	0.949	0.138	0.976
(80,50,40)	PCP[1]	0.854	0.954	1.392	0.937	0.139	0.976	1.297	0.940	0.844	0.954	0.135	0.977
	PCP[2]	0.962	0.951	1.488	0.934	0.139	0.974	1.358	0.938	0.944	0.951	0.137	0.977
	PCP[3]	0.813	0.955	1.379	0.937	0.139	0.976	1.284	0.940	0.800	0.956	0.133	0.977
(80,60,50)	PCP[1]	0.796	0.956	1.287	0.940	0.125	0.977	1.260	0.941	0.754	0.957	0.123	0.978
	PCP[2]	0.785	0.956	1.277	0.941	0.123	0.977	0.791	0.956	0.719	0.958	0.121	0.978
	PCP[3]	0.700	0.959	1.269	0.941	0.121	0.977	0.704	0.959	0.593	0.962	0.120	0.978
Pop-2 ($T=0.5$)
(30,15,10)	PCP[1]	1.457	0.935	2.775	0.893	0.136	0.973	2.597	0.899	1.437	0.936	0.133	0.974
	PCP[2]	1.472	0.934	2.916	0.889	0.140	0.972	2.888	0.890	1.455	0.935	0.136	0.973
	PCP[3]	1.435	0.936	2.734	0.895	0.135	0.973	2.581	0.900	1.415	0.936	0.138	0.974
(30,20,15)	PCP[1]	1.388	0.937	2.560	0.900	0.133	0.974	2.430	0.904	1.384	0.937	0.134	0.975
	PCP[2]	1.383	0.937	2.478	0.903	0.131	0.974	2.353	0.907	1.357	0.938	0.132	0.975
	PCP[3]	1.360	0.938	2.361	0.906	0.131	0.974	2.247	0.910	1.336	0.939	0.130	0.975
(50,30,20)	PCP[1]	1.322	0.939	1.976	0.919	0.130	0.975	1.908	0.921	1.282	0.940	0.126	0.976
	PCP[2]	1.339	0.939	2.210	0.911	0.130	0.975	2.103	0.915	1.314	0.939	0.129	0.976
	PCP[3]	1.298	0.940	1.928	0.920	0.129	0.975	1.867	0.922	1.232	0.942	0.122	0.976
(50,40,30)	PCP[1]	1.251	0.941	1.817	0.924	0.128	0.976	1.764	0.925	1.205	0.943	0.120	0.977
	PCP[2]	1.219	0.942	1.806	0.924	0.126	0.976	1.752	0.926	1.174	0.944	0.118	0.977
	PCP[3]	1.205	0.943	1.782	0.925	0.124	0.976	1.731	0.926	1.107	0.946	0.123	0.977
(80,50,40)	PCP[1]	0.978	0.950	1.588	0.931	0.113	0.977	1.533	0.933	0.966	0.950	0.111	0.978
	PCP[2]	1.050	0.948	1.716	0.927	0.116	0.977	1.672	0.928	1.004	0.949	0.119	0.978
	PCP[3]	0.905	0.952	1.492	0.934	0.110	0.977	1.463	0.935	0.864	0.954	0.109	0.978
(80,60,50)	PCP[1]	0.815	0.955	1.434	0.936	0.109	0.978	1.407	0.937	0.723	0.957	0.105	0.979
	PCP[2]	0.763	0.957	1.413	0.936	0.106	0.978	1.387	0.937	0.713	0.958	0.102	0.979
	PCP[3]	0.727	0.958	1.403	0.937	0.103	0.978	1.378	0.937	0.703	0.958	0.100	0.980
Pop-2 ($T=1.5$)
(30,15,10)	PCP[1]	1.445	0.935	2.748	0.894	0.136	0.974	2.589	0.899	1.425	0.936	0.136	0.976
	PCP[2]	1.457	0.935	2.801	0.893	0.139	0.973	2.758	0.894	1.437	0.936	0.140	0.975
	PCP[3]	1.432	0.936	2.678	0.896	0.134	0.974	2.503	0.902	1.404	0.937	0.131	0.976
(30,20,15)	PCP[1]	1.388	0.937	2.559	0.900	0.132	0.975	2.429	0.904	1.384	0.937	0.129	0.977
	PCP[2]	1.362	0.938	2.375	0.906	0.131	0.975	2.253	0.910	1.338	0.939	0.128	0.977
	PCP[3]	1.339	0.939	2.343	0.907	0.131	0.975	2.230	0.911	1.335	0.939	0.127	0.977
(50,30,20)	PCP[1]	1.316	0.939	1.928	0.920	0.128	0.976	1.861	0.922	1.262	0.941	0.125	0.978
	PCP[2]	1.318	0.939	2.119	0.914	0.129	0.976	2.016	0.917	1.293	0.940	0.126	0.978
	PCP[3]	1.276	0.941	1.922	0.920	0.127	0.976	1.861	0.922	1.226	0.942	0.124	0.978
(50,40,30)	PCP[1]	1.230	0.942	1.808	0.924	0.125	0.976	1.755	0.926	1.181	0.944	0.122	0.979
	PCP[2]	1.194	0.943	1.754	0.926	0.125	0.977	1.701	0.927	1.164	0.944	0.120	0.979
	PCP[3]	1.181	0.944	1.739	0.926	0.121	0.977	1.689	0.928	1.069	0.947	0.119	0.979
(80,50,40)	PCP[1]	0.975	0.950	1.552	0.932	0.110	0.979	1.498	0.934	0.919	0.952	0.109	0.980
	PCP[2]	1.006	0.949	1.675	0.928	0.115	0.977	1.632	0.929	0.947	0.951	0.113	0.980
	PCP[3]	0.891	0.953	1.490	0.934	0.110	0.979	1.461	0.935	0.854	0.954	0.109	0.980
(80,60,50)	PCP[1]	0.808	0.955	1.415	0.936	0.108	0.980	1.388	0.937	0.712	0.958	0.099	0.981
	PCP[2]	0.755	0.957	1.410	0.936	0.104	0.980	1.384	0.937	0.705	0.959	0.093	0.981
	PCP[3]	0.717	0.958	1.399	0.937	0.100	0.980	1.373	0.938	0.705	0.959	0.091	0.981

present the AIL and CP results of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$, respectively. A detailed evaluation of the numerical results in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 leads to the following methodological insights regarding estimator efficiency and interval reliability:

• The overall estimation procedures yield satisfactory outcomes for the parameters $\alpha$ and $\xi$, as well as for $R\left(t\right)$ and $h\left(t\right)$, across all simulation settings.
• Estimation accuracy, in both point and interval terms, improves with increasing sample size n, and likewise when the total number of censoring units ${\sum }_{i=1}^m{S}_i$ is reduced, highlighting the sensitivity of the inferential results to test size and censoring intensity.
• As $m_i,i=1,2$ grow, all estimation findings of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ perform well.
• As T grows, the RMSE, MRAB, and AIL results of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ decreased while their CPs increased.
• When APE($\mathit{\lambda }$) grows, we noticed that:

  –

  The RMSEs and MRABs of APE model parameters $\alpha$ and $\xi$ increased except for APE reliability parameters $R\left(t\right)$ and $h\left(t\right)$ decreased;

  –

  The AILs of $\alpha$, $\xi$, and $R\left(t\right)$ increased while those of $h\left(t\right)$ decreased;

  –

  The CPs of $\alpha$, $\xi$, and $R\left(t\right)$ decreased while those of $h\left(t\right)$ increased.

• The incorporation of prior knowledge through informative gamma priors substantially enhances the Bayesian estimates compared to the MLEs for all parameters under consideration, yielding narrower and more reliable HPD intervals.
• Bayesian inference, facilitated by prior information, consistently outperforms classical counterparts in estimating $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$, particularly under informative prior settings.
• Given that Prior-II exhibits lower variance compared to Prior-I, the MCMC-based Bayesian estimation under Prior-II yields more precise posterior estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ for both populations (Pop-i for $i=1,2$), outperforming the results obtained under alternative prior specifications.
• Comparing the proposed interval estimations of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$, it is observed that:

  –

  All credible interval estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ produced by BCI/HPD methods outperformed the asymptotic interval estimates produced by ACI-NA/ACI-NL methods;

  –

  The ACI-NA method is the best choice compared to the ACI-NL method for estimating the interval estimates of $\alpha$, $\xi$, and $h\left(t\right)$. The opposite comment is also noted for $R\left(t\right)$.

  –

  The HPD interval method is the best choice compared to the BCI method for estimating the interval estimates of all unknown subjects.

• Comparing the proposed censoring schemes PCP[i] (for $i=1,2,3$) reveals that the PCP[3] is most effective for estimating $\alpha$ and $h\left(t\right)$, the PCP[2] excels for $\xi$, and the PCP[1] is preferable for $R\left(t\right)$.
• To ensure high-quality estimation of lifetime characteristics under complex censoring, experimental designs should aim to maximize test duration within the constraints of cost and logistical feasibility.
• Overall, for data arising from the generalized progressive hybrid censoring framework, the Bayesian methodology, specifically employing the Metropolis-Hastings algorithm, proves to be a stable and reliable approach for inference on both model parameters and associated reliability functions.

All computations were performed on an $\mathsf{HP}$ laptop equipped with an Intel Core(TM) i5-5200U processor and 8 GB of RAM, HP Inc., Palo Alto, CA, USA. Using the $\mathsf{R}$ (v4.2.2) programming environment on a standard desktop machine, we evaluated the computational costs of both procedures for a G-PH censored dataset of moderate size (e.g., $n=50$, without loss of generality):

• Algorithm 1: Generating an MCMC sample via the M–H sampler with 12,000 iterations and a burn-in of 2000 required approximately 3 s. The computational cost scales linearly with the number of iterations and sample size but remains highly manageable across all simulation settings.
• Algorithm 2: Generating censored samples required less than 0.1 s, which is negligible compared to the iterative M–H sampling.

5. Real Data Applications

This section investigates two authentic datasets drawn from clinical and petroleum engineering to illustrate the practical applicability of the proposed estimation methodologies and their alignment with real-world analytical needs. These applications confirm that the recommended inferential procedures exhibit sufficient performance when applied to real data under the proposed censoring schemes. To highlight the practical significance of RF $R\left(t\right)$ and HRF $h\left(t\right)$, we emphasize their role in both medical and engineering contexts. For example, in medical device applications, the HRF $h\left(t\right)$ helps assess whether the risk of failure increases sharply after a certain period of use, thereby guiding preventive replacement or monitoring policies. In engineering systems, the HRF provides the expected additional operating time given survival up to t, which is directly relevant for warranty planning and maintenance scheduling. These measures, therefore, translate statistical inference into actionable strategies for enhancing safety, reliability, and cost-effectiveness in real-world applications.

5.1. Breast Data

Breast cancer is a malignant disease characterized by abnormal and uncontrolled growth of cells within breast tissue, usually arising from the milk ducts or lobular units. It primarily affects women and may clinically manifest through palpable lumps, morphological changes in the breast, or atypical nipple discharge. Early detection via screening, along with multimodal therapeutic approaches such as surgery, radiotherapy, and chemotherapy, substantially enhances the prognosis. In this study, we analyze a real-world dataset consisting of recurrence-free survival times (in days) for 43 breast cancer (BC) patients presenting with tumors larger than 50 mm, all of whom underwent hormonal treatment. The dataset, originally compiled by Royston and Altman [23], is rescaled by dividing survival times by one hundred for computational tractability; see

Table 10. Survival times of 43 BC patients.

0.45	0.64	0.75	1.01	1.17	1.37	1.77	2.42	2.60	3.01	3.38	3.49
3.85	3.89	4.49	4.55	4.68	4.91	5.46	5.75	6.08	6.11	6.38	6.56
6.96	7.84	8.81	8.82	9.17	9.24	9.40	9.77	10.57	11.43	11.49	13.19
13.98	15.15	17.43	17.72	18.06	22.48	24.05

.

Before to evaluating the theoretical estimation procedures, the suitability of the APE distribution was initially examined using the BC dataset summarized in Table 10. This preliminary assessment employed the Kolmogorov–Smirnov ($\mathsf{KS}$) test, alongside its associated p-value, to validate model adequacy. Additionally, MLEs for the parameters $\alpha$ and $\xi$, along with their Std.Ers and 95% ACI-NA/ACI-NL estimates including corresponding interval lengths (ILs), are computed and are presented in

Table 11. Fit results of the APE model from BC data.

Par.	MLE		95% ACI-NA			95% ACI-NL			$\mathsf{KS}$
	Est.	Std.Er	Low.	Upp.	IL	Low.	Upp.	IL	Statistic	p-Value
$\alpha$	6.4938	5.8447	0.0000	17.949	17.949	1.1127	37.899	36.786	0.0555	0.9984
$\xi$	0.1939	0.0364	0.1226	0.2652	0.1426	0.1342	0.2801	0.1459

. The outcomes clearly indicate that the APE model offers an excellent fit to the BC data. Table 11 also indicates that the estimated ILs created by ACI-NA/ACI-NL approaches are quite close to each other.

Displaying fitting results through visual tools enhances interpretability, facilitates model diagnostics, and allows for intuitive assessment of how well a model captures the underlying data structure. For this purpose, in Figure 5, we provide a set of diagnostic plots, including (a) empirical/fitted survival functions $R\left(y\right)$, (b) probability–probability (PP) plots, (c) quantile–quantile (QQ) plots, (d) empirical/fitted scaled-TTT lines, (e) likelihood contour map, and (f) boxplot within violin diagram. The subplots in Figure 5a–c corroborate the statistical evidence, affirming that the APE distribution effectively captures the underlying distributional structure of the observed data. The model diagnostic plots in Figure 5 for the APE distribution show that:

• Figure 5a demonstrates close agreement between the empirical estimates and the fitted APE reliability function.
• Figure 5b shows alignment of fitted and empirical probabilities along the 45° line.
• Figure 5c confirms that the model adequately captures the distributional shape of the data, as most points follow the diagonal.
• Figure 5d reveals that the BC data exhibit an increasing failure rate, consistent with one of the theoretical forms predicted by the APE model.
• Figure 5e highlights the parameter region of highest support, with the red dot marking the global maximum, and further confirms the existence and uniqueness of the MLEs $\widehat{\alpha }$ and $\widehat{\xi }$.
• Figure 5f summarizes the posterior sample distribution, including density shape and central tendency, indicating that the data exhibit moderate right skewness with the central tendency clustered around the median.

To evaluate the theoretical estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ from the full BC dataset, three G-PH censored samples with $(m_1,m_2)=(15,20)$ and various specifications of $\mathbf{S}$ and T are generated; see

Table 12. Artificial G-PH censored samples from BC data.

Sample	PCP	$\mathit{T}\left(\mathit{d}\right)$	${\mathit{S}}^{\mathbf{★}}$	${\mathit{T}}^{\mathbf{★}}$	Data
$\mathbb{S}\mathbb{A}$	$(2^{10},3,0^9)$	2.5 (3)	13	7.84	0.45, 1.01, 1.37, 2.60, 3.01, 3.89, 4.68, 5.75, 6.38, 7.84
$\mathbb{S}\mathbb{B}$	$(0^5,2^{10},3,0^4)$	5.5 (11)	8	10	0.45, 0.64, 0.75, 1.01, 1.17, 1.37, 2.42, 3.01, 3.89, 4.55
					4.91, 5.75, 6.56, 8.81, 9.77
$\mathbb{S}\mathbb{C}$	$(0^9,3,2^{10})$	11.5 (20)	0	11.43	0.45, 0.64, 0.75, 1.01, 1.17, 1.37, 1.77, 2.42, 2.60, 3.01,
					3.89, 4.55, 4.68, 5.75, 6.96, 7.84, 8.82, 9.24, 10.57, 11.43

. Due to the absence of prior information regarding the APE$\left(\mathit{\lambda }\right)$ parameters within the BC dataset, Bayesian estimation is conducted by employing ${\mathbb{B}}^{★}=10,000$ burn-in iterations and $\mathbb{B}=40,000$ total MCMC samples. For each dataset listed in Table 12, both point estimates (with associated Std.Ers) and interval estimates (with corresponding ILs) for $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ (for $t=5$) are obtained using maximum likelihood and Bayesian methods; see

Table 13. Estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ from BC data.

Sample	Par.	MLE		ACI-NA			BCI
		MCMC		ACI-NL			HPD
		Est.	Std.Er	Lower	Upper	IL	Lower	Upper	IL
$\mathbb{S}\mathbb{A}$	$\alpha$	7.4882	15.426	0.0000	37.722	37.722	7.4688	7.5078	0.0390
		7.4882	0.0099	0.1321	14.496	14.364	7.4689	7.5079	0.0390
	$\xi$	0.1058	0.0754	0.0021	0.2537	0.2916	0.0866	0.1237	0.0371
		0.1050	0.0094	0.0262	0.4280	0.4018	0.0871	0.1241	0.0370
	$R\left(5\right)$	0.8017	0.0598	0.6845	0.9189	0.2344	0.7639	0.8413	0.0774
		0.8034	0.0197	0.6927	0.9279	0.2352	0.7632	0.8404	0.0772
	$h\left(5\right)$	0.0552	0.0215	0.0130	0.0973	0.0843	0.0420	0.0685	0.0265
		0.0547	0.0067	0.0257	0.1185	0.0928	0.0412	0.0675	0.0263
$\mathbb{S}\mathbb{B}$	$\alpha$	0.4666	3.1792	0.0000	6.6977	6.6977	0.4469	0.4860	0.0391
		0.4663	0.0100	0.0018	16.235	16.233	0.4474	0.4864	0.0390
	$\xi$	0.0466	0.1269	0.0011	0.2953	0.2942	0.0314	0.0618	0.0305
		0.0460	0.0078	0.0002	9.6464	9.6461	0.0311	0.0614	0.0304
	$R\left(5\right)$	0.7252	0.0639	0.6000	0.8503	0.2504	0.6559	0.8036	0.1477
		0.7293	0.0380	0.6102	0.8618	0.2516	0.6552	0.8026	0.1475
	$h\left(5\right)$	0.0621	0.0171	0.0286	0.0956	0.0670	0.0427	0.0807	0.0380
		0.0613	0.0098	0.0362	0.1065	0.0703	0.0425	0.0805	0.0380
$\mathbb{S}\mathbb{C}$	$\alpha$	3.1291	4.3336	0.0000	11.623	11.623	3.1097	3.1488	0.0390
		3.1291	0.0099	0.2073	17.235	17.028	3.1100	3.1490	0.0390
	$\xi$	0.1205	0.0548	0.0131	0.2279	0.2148	0.1020	0.1383	0.0364
		0.1199	0.0092	0.0494	0.2939	0.2445	0.1017	0.1379	0.0363
	$R\left(5\right)$	0.6826	0.0646	0.5560	0.8093	0.2532	0.6395	0.7290	0.0894
		0.6844	0.0227	0.5671	0.8218	0.2547	0.6402	0.7294	0.0892
	$h\left(5\right)$	0.0868	0.0213	0.0450	0.1285	0.0834	0.0710	0.1026	0.0316
		0.0863	0.0080	0.0536	0.1403	0.0867	0.0708	0.1023	0.0315

. The results demonstrate that the Bayesian estimates, particularly those derived via MCMC, exhibit superior performance relative to MLEs, consistently yielding lower standard errors and narrower interval lengths, thereby indicating enhanced inferential efficiency and precision.

To show the existence and uniqueness characteristics of the offered MLEs $\widehat{\alpha }$ and $\widehat{\xi }$ of $\alpha$ and $\xi$, respectively, Figure 6 depicts the contour diagrams for the log-likelihood of $\alpha$ and $\xi$. At varying options of $\alpha$ and $\xi$, for all G-PH censored datasets reported in Table 12, Figure 6 stands that the acquired estimates of $\widehat{\alpha }$ and $\widehat{\xi }$ existed and are unique. It supports the same results presented in Table 13 and suggests using the values of $\widehat{\alpha }$ and $\widehat{\xi }$ in each sample as initial guesses for running the forthcoming Bayesian iterations.

To assess the convergence behavior of the Markov chain samples for the APE parameters $\mathit{\lambda }$, RF $R\left(t\right)$, and HRF $h\left(t\right)$, both trace plots and posterior density estimates are depicted in Figure 7. For visual clarity, the posterior mean and the associated 95% BCI bounds are indicated by solid and dashed cyan lines, respectively. The plots in Figure 7 suggest that the 30,000 retained MCMC samples for each parameter exhibit satisfactory convergence and display approximately symmetric posterior distributions. In addition, a suite of descriptive statistics, including the mean, mode, quartiles ($Q_1$, $Q_2$, $Q_3$), standard deviation (Std.D), and skewness (Sk.), for $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ is reported in

Table 14. Statistics of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ from BC data.

Sample	Par.	Mean	Mode	${\mathit{Q}}_1$	${\mathit{Q}}_2$	${\mathit{Q}}_3$	Std.D	Sk.
$\mathbb{S}\mathbb{A}$	$\alpha$	7.48816	7.47713	7.48140	7.48814	7.49483	0.00994	0.01162
	$\xi$	0.10495	0.09175	0.09855	0.10493	0.11128	0.00939	0.03441
	$R\left(5\right)$	0.80337	0.78946	0.79019	0.80351	0.81678	0.01959	−0.06067
	$h\left(5\right)$	0.05471	0.04546	0.05007	0.05455	0.05916	0.00670	0.16136
$\mathbb{S}\mathbb{B}$	$\alpha$	0.46634	0.45892	0.45950	0.46628	0.47311	0.01002	−0.00075
	$\xi$	0.04601	0.04545	0.04064	0.04573	0.05116	0.00779	0.16086
	$R\left(5\right)$	0.72925	0.72938	0.70356	0.72945	0.75481	0.03778	0.01932
	$h\left(5\right)$	0.06127	0.06100	0.05458	0.06101	0.06775	0.00973	0.10517
$\mathbb{S}\mathbb{C}$	$\alpha$	3.12912	3.11684	3.12236	3.12911	3.13578	0.00994	0.01114
	$\xi$	0.11985	0.10643	0.11355	0.11980	0.12608	0.00922	0.04243
	$R\left(5\right)$	0.68440	0.66837	0.66899	0.68429	0.69977	0.02268	0.00864
	$h\left(5\right)$	0.08629	0.07482	0.08078	0.08617	0.09163	0.00801	0.10262

. The numerical summaries are consistent with and further reinforce the convergence patterns observed in Figure 7. Table 14 also confirms that the MCMC distributions of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ are almost symmetric.

5.2. Oil Data

Oil remains the dominant global energy source, accounting for a substantial share of primary energy consumption. In 2020, worldwide oil consumption averaged approximately 88.6 million barrels per day, representing 30.1% of total primary energy use. Venezuela holds the largest proven oil reserves globally, exceeding 300 billion barrels, followed by Saudi Arabia with an estimated 297.5 billion barrels. Although the United States leads the world in both oil production and consumption, it continues to rely on imports from numerous oil-exporting nations. Notably, despite its leading production status, the United States ranks only ninth globally in terms of proven oil reserves. In this subsection, to examine the applicability of the theoretical results of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ to a real engineering situation, we analyze a realistic data set consisting of oil reserves (in billion barrels) in 2020 for the top 20 countries; see

Table 15. Oil reserve production in the top 20 countries.

6.1	7.0	7.8	7.9	11.9	12.2	25.2	26	30	36.9
48.4	68.8	97.8	101.5	107.8	145	157.8	168.1	297.5	303.8

.

Before proceeding with the evaluation of theoretical estimation techniques, the appropriateness of the APE distribution was preliminarily assessed using the oil dataset detailed in Table 15. To examine the model’s goodness of fit, the $\mathsf{KS}$ test (along with its $\mathsf{P}$-value), the MLEs of $\alpha$ and $\xi$ alongside their Std.Ers, and 95% ACI-NA/ACI-NL estimates (with associated ILs) are obtained; see

Table 16. Fit results of the APE model from oil data.

Par.	MLE		95% ACI-NA			95% ACI-NL			$\mathsf{KS}$
	Est.	Std.Er	Low.	Upp.	IL	Low.	Upp.	IL	Statistic	p -Value
$\alpha$	0.3337	0.5481	0.0000	1.4078	1.4078	0.0133	8.3450	8.3317	0.1361	0.8052
$\xi$	0.0091	0.0047	0.0000	0.0183	0.0183	0.0033	0.0250	0.0217

. As a result, Table 16 reveals that the APE distribution is a suitable model for analyzing the oil data.

The subplots in Figure 8a–c illustrate the model-fitting results corresponding to the outcomes summarized in Table 16. Specifically, Figure 8a compares the empirical and fitted survival functions, showing close agreement and thereby confirming the adequacy of the APE model for the oil dataset. Figure 8b presents the PP plot, which lies close to the 45-degree line, further supporting the goodness of fit. Similarly, Figure 8c displays the QQ plot, indicating that the APE distribution appropriately captures the distributional behavior of the observed oil data. In addition, Figure 8d shows the likelihood contour plot, which confirms both the existence and uniqueness of the MLEs $\widehat{\alpha }$ and $\widehat{\xi }$ corresponding to $\alpha$ and $\xi$, respectively. These MLEs are therefore recommended as initial values in subsequent Bayesian computations to improve numerical stability and convergence efficiency. Finally, Figure 8f provides a violin plot with an embedded boxplot, offering further insight into the distributional properties of the oil dataset. It reveals pronounced right skewness and heavy-tailed behavior, with observations spread across a wide range and extreme values concentrated in the upper tail. This empirical pattern underscores the flexibility of the APE distribution in accommodating skewed and heavy-tailed lifetime data frequently encountered in reliability and engineering applications.

To assess the theoretical estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ using the complete BC dataset, three G-PH censored samples are constructed under the configuration $(m_1,m_2)=(10,15)$ and varying choices of censoring schemes $\mathbf{S}$ and termination times T, as outlined in

Table 17. Artificial G-PH censored samples from oil data.

Sample	PCP	$\mathit{T}\left(\mathit{d}\right)$	${\mathit{S}}^{\mathbf{★}}$	${\mathit{T}}^{\mathbf{★}}$	Data
$\mathbb{S}1$	$(1^5,0^{10})$	12(3)	5	101.5	6.1, 7.8, 11.9, 12.2, 26, 30, 36.9, 48.4, 68.8, 101.5
$\mathbb{S}2$	$(0^5,1^5,0^5)$	150(12)	3	150	6.1, 7.0, 7.8, 7.9, 11.9, 12.2, 26, 30, 36.9, 68.8, 107.8, 145
$\mathbb{S}3$	$(0^{10},1^5)$	170(15)	0	168.1	6.1, 7.0, 7.8, 7.9, 11.9, 12.2, 25.2, 26, 30, 36.9, 48.4, 97.8, 101.5, 145, 168.1

. Given the lack of prior information on the APE$\left(\mathit{\lambda }\right)$ parameters, Bayesian inference is implemented using ${\mathbb{B}}^{★}=10,000$ burn-in iterations and a total of $\mathbb{B}=40,000$ MCMC samples. For each censoring scenario, point estimates (accompanied by Std.Ers) and interval estimates (with associated ILs) for the parameters $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ (for $t=10$) are computed using both maximum likelihood and Bayesian methodologies, as detailed in

Table 18. Estimates of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ from oil data.

Sample	Par.	MLE		ACI-NA			BCI
		MCMC		ACI-NL			HPD
		Est.	Std.Er	Lower	Upper	IL	Lower	Upper	IL
$\mathbb{S}\mathbb{A}$	$\alpha$	0.1103	0.9196	0.0000	1.9126	1.9126	0.1003	0.1199	0.0196
		0.1100	0.0050	0.0008	5.2424	5.2416	0.1006	0.1201	0.0196
	$\xi$	0.0050	0.0140	0.0000	0.0324	0.0324	0.0033	0.0067	0.0034
		0.0049	0.0009	0.0002	1.1891	1.1891	0.0032	0.0067	0.0034
	$R\left(5\right)$	0.8852	0.0376	0.8114	0.9589	0.1475	0.8505	0.9231	0.0727
		0.8876	0.0187	0.8144	0.9621	0.1476	0.8506	0.9232	0.0726
	$h\left(5\right)$	0.0120	0.0041	0.0039	0.0200	0.0161	0.0079	0.0158	0.0079
		0.0117	0.0020	0.0061	0.0234	0.0173	0.0079	0.0158	0.0079
$\mathbb{S}\mathbb{B}$	$\alpha$	0.1120	0.3872	0.0000	0.8709	0.8709	0.1021	0.1217	0.0196
		0.1118	0.0050	0.0003	98.050	98.050	0.1023	0.1219	0.0195
	$\xi$	0.0055	0.0064	0.0000	0.0181	0.0181	0.0037	0.0071	0.0034
		0.0054	0.0009	0.0005	0.0549	0.0543	0.0037	0.0071	0.0034
	$R\left(5\right)$	0.8760	0.0380	0.8015	0.9505	0.1490	0.8417	0.9136	0.0719
		0.8783	0.0184	0.8046	0.9538	0.1492	0.8415	0.9132	0.0717
	$h\left(5\right)$	0.0130	0.0042	0.0048	0.0212	0.0164	0.0089	0.0168	0.0079
		0.0127	0.0020	0.0069	0.0244	0.0175	0.0090	0.0168	0.0079
$\mathbb{S}\mathbb{C}$	$\alpha$	0.8223	1.4073	0.0000	3.5807	3.5807	0.8126	0.8321	0.0195
		0.8223	0.0050	0.0287	23.541	23.512	0.8128	0.8323	0.0195
	$\xi$	0.0165	0.0090	0.0000	0.0342	0.0342	0.0145	0.0184	0.0039
		0.0164	0.0010	0.0056	0.0483	0.0426	0.0145	0.0183	0.0039
	$R\left(5\right)$	0.8351	0.0587	0.7201	0.9501	0.2300	0.8181	0.8533	0.0351
		0.8357	0.0089	0.7277	0.9584	0.2307	0.8188	0.8538	0.0351
	$h\left(5\right)$	0.0179	0.0062	0.0057	0.0301	0.0245	0.0158	0.0199	0.0041
		0.0178	0.0011	0.0090	0.0354	0.0264	0.0157	0.0198	0.0041

. The findings indicate that the Bayesian MCMC-based estimators consistently outperform their maximum likelihood counterparts by offering reduced standard errors and more concise interval estimates, thereby supporting the efficiency and precision of the Bayesian framework.

The contour plots in Figure 9 demonstrate that the maximum likelihood estimators $\widehat{\alpha }$ and $\widehat{\xi }$ exist and are unique for all G-PH censored samples, confirming the existence of global optima. Figure 10 verifies that the MCMC chains for $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ converge satisfactorily and exhibit nearly symmetric posterior distributions. Descriptive summaries in

Table 19. Statistics of $\alpha$, $\xi$, $R\left(t\right)$, and $h\left(t\right)$ from oil data.

Sample	Par.	Mean	Mode	${\mathit{Q}}_1$	${\mathit{Q}}_2$	${\mathit{Q}}_3$	Std.D	Sk.
$\mathbb{S}\mathbb{A}$	$\alpha$	0.11004	0.11043	0.10664	0.11004	0.11339	0.00499	0.01525
	$\xi$	0.00490	0.00435	0.00430	0.00489	0.00549	0.00087	0.12879
	$R\left(5\right)$	0.88760	0.89931	0.87505	0.88768	0.90031	0.01851	−0.05361
	$h\left(5\right)$	0.01173	0.01045	0.01034	0.01171	0.01309	0.00201	0.09596
$\mathbb{S}\mathbb{B}$	$\alpha$	0.11180	0.09655	0.10840	0.11180	0.11515	0.00498	0.01223
	$\xi$	0.00537	0.00465	0.00476	0.00536	0.00597	0.00088	0.10814
	$R\left(5\right)$	0.87828	0.87104	0.86583	0.87831	0.89084	0.01831	−0.03389
	$h\left(5\right)$	0.01275	0.01131	0.01136	0.01273	0.01410	0.00201	0.07644
S3$\mathbb{S}\mathbb{C}$	$\alpha$	0.82228	0.83506	0.81890	0.82228	0.82562	0.00498	0.01779
	$\xi$	0.01643	0.01548	0.01576	0.01643	0.01709	0.00098	0.03125
	$R\left(5\right)$	0.83566	0.84524	0.82965	0.83563	0.84170	0.00892	0.00265
	$h\left(5\right)$	0.01783	0.01671	0.01712	0.01783	0.01854	0.00105	0.02685

further support the numerical findings listed in Table 18 and the graphical facts shown in Figure 10, validating the precision and stability of both point and credible interval estimations acquired through the Bayesian inference procedure.

6. Optimal PCP

In reliability tests, one of the central challenges is identifying the optimal censoring scheme from among several available alternatives. To maximize the information content regarding unknown model parameters, Balakrishnan and Aggarwala [24] first explored the issue of specifying the optimal censoring strategies among all possible under various experimental and inferential setups. Subsequently, the selection of an optimal PCP scenario across different inferential contexts has attracted growing interest in recent statistical research; see, for example, Wang and Yu [25] and Pradhan and Kundu [26]. For practitioners, determining the most informative censoring strategy from a set of competing designs is crucial for efficient estimation of unknown $\mathit{\lambda }$ parameters.

Numerous studies in the literature have also addressed the problem of comparing two (or more) progressive censoring designs, aiming to evaluate their relative efficiency and informativeness under various statistical models; see, for example, Elshahhat and Abu El Azm [10]. Next, after assigning various combinations of $(S_1,\dots ,S_m)$ given fixed values of n, $m_i(i=1,2)$, and a specified threshold time T, we outline four objective criteria for selecting an optimal PCP design, namely:

• Criterion $\mathbb{O}\left[1\right]$: Maximizing the trace of ${\mathrm{\Sigma }}^{-1}\left(\mathit{\lambda }\right)$;
• Criterion $\mathbb{O}\left[2\right]$: Minimizing the trace of $\mathrm{\Sigma }\left(\mathit{\lambda }\right)$;
• Criterion $\mathbb{O}\left[3\right]$: Minimizing the determinant of $\mathrm{\Sigma }\left(\mathit{\lambda }\right)$;
• Criterion $\mathbb{O}\left[4\right]$: Minimizing the variance of the MLE of the logarithm of $\varrho$th APE quantile (for $0<\varrho <1$), where $\kappa =1+\varrho (\alpha -1)$, as

  $$\mathbb{O}\left[4\right]=min\left({\sigma }_{11}^2{R}_1^2+2{\sigma }_{12}{R}_1{R}_2+{\sigma }_{22}^2{R}_2^2\right),$$

  with

  $$R_1={\displaystyle \frac{\alpha \varrho log\left(\alpha \right)-\kappa log\left(\kappa \right)}{\kappa \alpha log\left(\alpha \right)\left[1-log\left(\kappa \right){log}^{-1}\left(\alpha \right)\right][log\left(\kappa \right)-log\left(\alpha \right)]}}$$

  and

  $$R_2=-{\xi }^{-1}.$$

Clearly, an optimal PCP should maximize the value of $\mathbb{O}\left[1\right]$ while simultaneously minimizing the values of $\mathbb{O}\left[i\right]$ for $i=2,3,4$. To demonstrate the effectiveness of the proposed PCP mechanisms provided in the two real-world applications, the optimality criteria $\mathbb{O}\left[i\right],i=1,2,3,4$, are evaluated using the G-PH censored samples derived from the breast cancer and oil reserves datasets, as presented in Table 12, Table 13, Table 14, Table 15, Table 16 and Table 17. From

Table 20. Optimum PCP mechanisms from BC data.

Sample	$\mathbb{O}\left[1\right]$	$\mathbb{O}\left[2\right]$	$\mathbb{O}\left[3\right]$	$\mathbb{O}\left[4\right]$
$\mathbf{\varrho }\to$				0.3	0.6	0.9
$\mathbb{S}\mathbb{A}$	6297.789	237.9618	0.160199	3.204534	28.21442	222.9919
$\mathbb{S}\mathbb{B}$	1485.411	10.12321	0.001607	2.396828	35.74682	2187.111
$\mathbb{S}\mathbb{C}$	1865.942	18.78300	0.010066	1.114359	4.582958	45.16951

and

Table 21. Optimum PCP mechanisms from oil data.

Sample	$\mathbb{O}\left[1\right]$	$\mathbb{O}\left[2\right]$	$\mathbb{O}\left[3\right]$	$\mathbb{O}\left[4\right]$
$\mathbf{\varrho }\to$				0.3	0.6	0.9
$\mathbb{S}\mathbb{A}$	316,665.1	0.845848	0.000003	115.4766	905.6239	1548.539
$\mathbb{S}\mathbb{B}$	299,664.8	0.149952	0.000001	87.04930	682.4630	6981.514
$\mathbb{S}\mathbb{C}$	52,748.25	1.980712	0.000038	50.21496	214.3330	13,502.72

, it is noted that:

• For the BC data:

  –

  Using $\mathbb{O}\left[1\right]$, the optimal PCP design corresponds to Sample ‘$\mathbb{S}\mathbb{A}$’ (i.e., left-censoring);

  –

  Using $\mathbb{O}\left[i\right],i=2,3$, the optimal PCP design corresponds to Sample ‘$\mathbb{S}\mathbb{B}$’ (i.e., middle-censoring);

  –

  Using $\mathbb{O}\left[4\right]$, the optimal PCP design corresponds to Sample ‘$\mathbb{S}\mathbb{C}$’ (i.e., right-censoring),

• For the oil data:

  –

  Using $\mathbb{O}\left[i\right],i=1,4$, the optimal PCP design corresponds to Sample ‘$\mathbb{S}\mathbb{C}$’ (i.e., right-censoring);

  –

  Using $\mathbb{O}\left[i\right],i=2,3$, the optimal PCP design corresponds to Sample ‘$\mathbb{S}\mathbb{B}$’ (i.e., middle-censoring).

Notably, the optimal PCP mechanisms identified in Table 20 and Table 21 are in strong agreement with the findings previously established in Section 4, thereby reinforcing the consistency and stability of the proposed approach. As a summary, the examination of the G-PH censored mechanism in the breast cancer and oil reserves datasets provides a comprehensive evaluation of the alpha-power-exponential lifetime model, confirms the simulation findings, and illustrates the practical utility of the proposed methodology in both clinical and engineering reliability contexts.

7. Conclusions and Remarks

In this paper, we developed a comprehensive inferential framework for analyzing lifetime data from the alpha-power-exponential (APE) distribution under G-PH censoring. This censoring design offers improved control over test duration and censoring levels, making it highly relevant in practical reliability and survival analysis contexts. The MLEs for the APE parameters ($\alpha$ and $\xi$) and associated reliability functions ($R\left(t\right)$ and $h\left(t\right)$), and constructed asymptotic confidence intervals using both normal approximation and log-transformed methods. A Bayesian estimation approach was established by the SEL function with independent gamma priors. Because closed-form solutions were intractable, we implemented the Metropolis–Hastings algorithm to approximate posterior summaries and construct BCI and HPD intervals. Extensive Monte Carlo simulations were conducted to assess the performance of both MLE and Bayes estimators. Key findings showed that Bayesian estimates consistently outperformed classical counterparts, particularly under informative priors, in terms of higher coverage percentages and lower root mean squared error, mean relative absolute bias, and shorter average interval length values. Different G-PHs were compared, revealing that specific patterns (such as right-censoring for $\alpha$ and $h\left(t\right)$, middle-censoring for $\xi$, and left-censoring for $R\left(t\right)$) provide superior inferential outcomes, depending on the target parameter. The proposed methodologies were successfully applied to two genuine datasets, breast cancer survival and oil reserves, to demonstrate the model’s practical applicability and superior fit for several real-world scenarios. The findings underscore the utility of the APE distribution as a stable tool for analyzing such censored lifetime data. The proposed estimation methods, particularly the Bayesian MCMC approach, offer significant practical advantages in precision and reliability. Future work could extend this approach to multivariate models, accelerated life testing, or competing risks frameworks using the same censoring strategy explored here.