The XLindley Survival Model Under Generalized Progressively Censored Data: Theory, Inference, and Applications

Abstract

This paper introduces a novel extension of the classical Lindley distribution, termed the X-Lindley model, obtained by a specific mixture of exponential and Lindley distributions, thereby substantially enriching the distributional flexibility. To enhance its inferential scope, a comprehensive reliability analysis is developed under a generalized progressive hybrid censoring scheme, which unifies and extends several traditional censoring mechanisms and allows practitioners to accommodate stringent experimental and cost constraints commonly encountered in reliability and life-testing studies. Within this unified censoring framework, likelihood-based estimation procedures for the model parameters and key reliability characteristics are derived. Fisher information is obtained, enabling the establishment of asymptotic properties of the frequentist estimators, including consistency and normality. A Bayesian inferential paradigm using Markov chain Monte Carlo techniques is proposed by assigning a conjugate gamma prior to the model parameter under the squared error loss, yielding point estimates, highest posterior density credible intervals, and posterior reliability summaries with enhanced interpretability. Extensive Monte Carlo simulations, conducted under a broad range of censoring configurations and assessed using four precision-based performance criteria, demonstrate the stability and efficiency of the proposed estimators. The results reveal low bias, reduced mean squared error, and shorter interval lengths for the XLindley parameter estimates, while maintaining accurate coverage probabilities. The practical relevance of the proposed methodology is further illustrated through two real-life data applications from engineering and physical sciences, where the XLindley model provides a markedly improved fit and more realistic reliability assessment. By integrating an innovative lifetime model with a highly flexible censoring strategy and a dual frequentist–Bayesian inferential framework, this study offers a substantive contribution to modern survival theory.

1. Introduction

The Lindley distribution, originally introduced by Lindley [1] and later rediscussed by Ghitany et al. [2], has attracted considerable attention in recent decades owing to its tractable form and useful applications in lifetime and reliability analysis; see Ghitany et al. [3]. Recent research has introduced several extensions and generalizations of Lindley-type models, further enhancing their applicability across diverse engineering and physical contexts; see, for example, Qayoom et al. [4]. Meanwhile, censoring mechanisms, such as progressive and hybrid schemes, have become increasingly important in reliability studies, as they offer greater experimental control and enable the efficient use of incomplete lifetime data; see Balakrishnan et al. [5]. Collectively, these developments highlight the need for more generalized models and advanced estimation strategies capable of handling complex censoring structures while preserving statistical efficiency. However, despite its popularity, the single-parameter Lindley model often lacks the flexibility required to adequately capture diverse data behaviors observed in practice.

To address this limitation, Chouia and Zeghdoudi [6] proposed the XLindley distribution as another version of the traditional Lindley law. By mixing exponential and Lindley distributions, the XLindley distribution exhibits greater flexibility in its hazard rate structure and tail behavior, allowing it to accommodate a wider range of empirical data patterns. Its statistical properties, including explicit expressions for moments, hazard rate, and distributional characteristics, demonstrate the potential of the XLindley distribution as a competitive model for reliability studies and survival analysis.

Assume that the lifetime variable Y is continuous and distributed as $\mathrm{XLindley}\left(\gamma \right)$, where $\gamma >0$ represents the scale parameter. Following Chouia and Zeghdoudi [6], the probability density function (PDF) and cumulative distribution function (CDF) are given, respectively, by

$$g(y;\gamma )={\displaystyle \frac{{\gamma }^2}{{\left(1+\gamma \right)}^2}}\left(\gamma +y+2\right)e^{-\gamma y},y>0,$$

(1)

and

$$G(y;\gamma )=1-\left(1+{\displaystyle \frac{\gamma y}{{\left(1+\gamma \right)}^2}}\right)e^{-\gamma y},$$

(2)

with associated reliability function (RF) (at a distinct time $x>0$), denoted by $R\left(x\right)$, as

$$R(x;\gamma )=\left(1+{\displaystyle \frac{\gamma x}{{\left(1+\gamma \right)}^2}}\right)e^{-\gamma x},x>0,$$

(3)

and hazard rate function (HRF), at $x>0$, denoted by $h\left(x\right)$, as

$$h(x;\gamma )={\displaystyle \frac{{\gamma }^2\left(\gamma +x+2\right)}{{\left(1+\gamma \right)}^2+\gamma x}}.$$

(4)

The reliability characteristics of a lifetime distribution play a central role in assessing the performance of electronic systems and are widely employed by reliability practitioners. Accordingly, in this study, the reliability metrics $R\left(x\right)$ and $h\left(x\right)$ are treated as unknown lifespan parameters. Recently, numerous research studies have investigated several inferential issues for the XLindley distribution; see Alotaibi et al. [7] and Elbatal et al. [8], among others.

The primary objective of a reliability practitioner in life-testing research is to terminate the experiment before all units fail. A hybrid censoring scheme combines features of both Type-I (time-strategy) and Type-II (failure-strategy) methods. However, a limitation of conventional Type-I, Type-II, or hybrid censoring approaches is that they only allow the withdrawal of test units at the final termination point, but not at intermediate stages of the experiment; see Balakrishnan and Kundu [9]. In practice, intermediate removals may be desirable to balance shorter experimental duration with the need to observe extreme lifetimes, or to permit early-removed units to be redesigned and reused in subsequent studies. Such considerations motivate the development of progressive censoring schemes. Among these schemes, the progressive Type-II censoring scheme (P-T2-CS) has been extensively employed in reliability investigations.

A P-T2-CS is particularly useful in industrial and biomedical settings, as it allows planned removal of surviving units at various stages of the experiment. Suppose m failures are to be observed from n identical units, where $1⩽m⩽n$. Let $\mathbf{S}=(S_1,S_2,\dots ,S_m)$ denote the predetermined removal scheme. At the first failure time $Y_1$, $S_1$ of $(n-1)$ units are randomly outside the test. At the second failure time $Y_2$, $S_2$ of $(n-2-S_1)$ units are withdrawn, and so forth. Upon the mth failure, the remaining units $S_m$ are removed, and the test terminates; see Balakrishnan and Cramer [10]. One drawback of P-T2-CS is that when inspection units are extremely trustworthy, the experiment may become excessively long. To mitigate this, Kundu and Joarder [11] introduced the progressive hybrid censoring Type-I scheme (PH-T1-CS), where the test ends at ${\mathcal{T}}^{★}=min\{\tau ,Y_m\},$ with $\tau$ a predetermined censoring time. However, if only a small number of failures occur before $\tau$, PH-T1-CS becomes ineffective, often rendering parameter estimation infeasible or unstable.

To address this restriction, Cho et al. [12] introduced the generalized progressive hybrid censoring scheme (G-PH-CS), which extends PH-T1-CS by ensuring that a minimum number of failures are always observed. This framework provides greater flexibility, enabling both reduced test duration and lower costs associated with unit failures. Let $\mathbf{Y}=\{Y_1,Y_2,\dots ,Y_n\}$ denote the ordered times of failure generated from a continuous distribution with CDF $G(·)$ and PDF $g(·)$. Suppose $\tau \in (0,\infty )$ is a predetermined censoring time, d is the failure size up to time $\tau$, and let $m_{min}<m_{max}⩽n$ be fixed integers under a specified removal scheme $\mathbf{S}$, such that $m_{max}+{\sum }_{i=1}^{m_{max}}{S}_i=n$. Finally, the experiment terminates at ${\mathcal{T}}^{★}=max\left\{Y_{m_{min}},min\{Y_{m_{max}},\tau \}\right\}.$ Thus, three censoring cases may occur:

• If $\tau <Y_{m_{max}}<Y_{m_{max}}$, the test ends at $Y_{m_{max}}$ (Case 1);
• If $Y_{m_{max}}<\tau <Y_{m_{max}}$, the test ends at $\tau$ (Case 2);
• If $Y_{m_{max}}<Y_{m_{max}}<\tau$, the test ends at $Y_{m_{max}}$ (Case 3).

Hence, the joint likelihood function (LF) of $\{\mathbf{Y},\mathbf{S}\}$, denoted by ${\mathcal{L}}_t(·)$, is given by

$${\mathcal{L}}_t\left(\gamma \right|\mathbf{Y})\propto {\left[S(\tau ;\gamma )\right]}^{S^{★}}\prod _{i=1}^{J_t}g(y_i;\gamma ){\left[S(y_i;\gamma )\right]}^{S_i},t=1,2,3,$$

(5)

where $J_t=(m_{min},d,m_{max})$ for $t=1,2,3,$ $S_{J_1}=n-m_{min}-{\sum }_{i=1}^{m_{min}-1}{S}_i$, and $S_{J_3}=n-m_{max}-{\sum }_{i=1}^{m_{max}-1}{S}_i$. The main advantage of the G-PH-CS is that it guarantees the observation of at least $m_{min}$ failures, even when the experimenter initially intends to capture $m_{max}$ failures. This flexibility makes the method particularly attractive in practice.

To illustrate the practical implications of a hybrid censoring design, consider a lifetime testing experiment for light bulbs. Suppose n identical light bulbs are placed on test, and the experimenter is interested in observing failures to assess product reliability. Due to time and cost constraints, the experiment is terminated either when a pre-specified time $\tau$ is reached (Type-I censoring) or when a fixed number m of bulbs have failed (Type-II censoring), whichever occurs first. For example, the test may stop after 500 h or once 20 bulbs fail, whichever condition is satisfied first. Such a scheme ensures that the experiment does not run excessively long while still guaranteeing the observation of a sufficient number of failures for meaningful statistical inference. This setting naturally motivates the use of hybrid censoring and reflects realistic reliability testing scenarios encountered in industrial applications. Moreover, if some surviving bulbs are deliberately removed at intermediate failure times for inspection or reuse, the experiment follows a G-PH-CS, as considered in this study.

An important feature of the G-PH-CS is that it guarantees at least $m_{min}$ failures, even when the censoring plan permits up to $m_{max}$ failures to be recorded. This safeguard enhances the reliability of the collected data and makes the scheme particularly well suited for studies where premature termination may otherwise occur. Owing to these practical advantages, the G-PH-CS framework has received considerable attention and has been applied to a variety of lifetime distributions. Notable applications include the inverse Weibull entropy model by Lee [13], the Weibull distribution by Zhu [14], the Rayleigh model with competing risks by Singh et al. [15], inference on shape and scale parameters by Maswadah [16], and, most recently, the Chris–Jerry model by Alotaibi et al. [17].

Although Lindley-type distributions have been extensively studied in reliability modeling, most existing works focus on uncensored data or simple censoring schemes, leaving limited methodological development for progressive hybrid censoring. This represents a critical gap, as hybrid censoring designs frequently arise in engineering and physical sciences experiments, where tests are often terminated by a combination of failures and time constraints. Furthermore, while classical inference methods for Lindley variants have been proposed, few effort has been made to systematically compare frequentist and Bayesian approaches within such complex censoring frameworks.

At the same time, the recently introduced XLindley distribution has attracted attention for its ability to accommodate diverse hazard rate shapes and for its suitability in real-world reliability applications. Despite this promise, its inferential properties under advanced censoring schemes remain underdeveloped.

This study is therefore motivated by the dual need to (i) incorporate realistic censoring mechanisms into reliability analysis and (ii) leverage the modeling flexibility of the XLindley distribution. By developing and analyzing generalized progressive hybrid XLindley censored data, we construct a framework that not only mirrors practical testing environments more closely but also extends the methodological toolkit available to reliability practitioners. The main contributions of this work are fivefold:

• Develop maximum likelihood estimation procedures for the model parameter and associated reliability indices, and establish their asymptotic properties for rigorous inference.
• Construct Bayesian estimators under a gamma prior distribution, implemented via the Metropolis–Hastings algorithm with a symmetric squared-error loss function.
• Design efficient numerical algorithms for parameter estimation and reliability evaluation, ensuring computational feasibility under complex censoring schemes.
• Conduct a simulation study to investigate the proposed estimators’ superiority in terms of bias, efficiency, and coverage.
• Validate the proposed inferential frameworks through two real-life data applications from physical and engineering contexts, demonstrating superior model fit and enhanced applicability.

The structure of the paper is as follows. Section 2 and Section 3 describe the LF-based and Bayesian estimation approaches. Section 4 compares the proposed methodologies by simulations. Section 5 analyzes two datasets representing rainfall records in New South Wales and vehicle fatalities in South Carolina. Concluding remarks are offered in Section 6.

2. Likelihood Inference

This section uses a LF-based estimation approach to produce point and asymptotic interval estimates of the XLindley distribution parameters $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ using the G-PH-CS dataset.

2.1. Point Estimators

Substituting (1) and (3) into (5), the LF (5) and its $log{\mathcal{L}}_t\left(\gamma \right|\mathbf{Y})\equiv {\mathcal{L}}_t(·)$ can be updated as follows:

$$\begin{array}{c}\hfill {\mathcal{L}}_t\left(\gamma \right|\mathbf{y})\propto {\left(\gamma {\gamma }^{★}\right)}^{J_t}{e}^{{\sum }_{i=1}^{J_t}log\left(\xi \left(\gamma ;y_i\right)\right)-\gamma \zeta }\prod _{i=1}^{J_t}{\left(1+{\gamma }^{★}{y}_i\right)}^{S_i}{\left(1+{\gamma }^{★}\tau \right)}^{S^{★}},\end{array}$$

(6)

and

$$\begin{array}{cc}\hfill log{\mathcal{L}}_t\left(\gamma \right|\mathbf{y})& \propto J_{t}log\left(\gamma {\gamma }^{★}\right)-\gamma \zeta \hfill \\ \hfill & +{\sum }_{i=1}^{J_t}log\left(\xi \left(\gamma ;y_i\right)\right)+{\sum }_{i=1}^{J_t}{S}_{i}log\left(1+{\gamma }^{★}{y}_i\right)+S^{★}log\left(1+{\gamma }^{★}\tau \right).\hfill \end{array}$$

(7)

respectively, where

$$\zeta =\tau S^{★}+{\sum }_{i=1}^{J_t}{y}_i\left(S_i+1\right),{\gamma }^{★}=\gamma {\left(1+\gamma \right)}^{-2},\mathrm{and}\xi \left(\gamma ;y_i\right)=\gamma +y_i+2.$$

From (7), the corresponding MLE $\widehat{\gamma }$ cannot be expressed in closed form:

$$\begin{array}{c}\hfill {\left.{\displaystyle \frac{2J_t}{\gamma \left(1+\gamma \right)}}-\zeta +{\sum }_{i=1}^{J_t}{\xi }^{-1}\left(\gamma ;y_i\right)+{\sum }_{i=1}^{J_t}{\displaystyle \frac{\dot{\gamma }{y}_i{S}_i}{1+{\gamma }^{★}{y}_i}}+{\displaystyle \frac{\dot{\gamma }\tau S^{★}}{1+{\gamma }^{★}\tau }}\right|}_{\gamma =\widehat{\gamma }}=0,\end{array}$$

(8)

where $\dot{\gamma }=\left(1-\gamma \right){\left(1+\gamma \right)}^{-3}$.

Remark 1.

Owing to the nonlinear structure of the score equation under the G-PH-CS, deriving global analytical conditions for the existence and uniqueness of the MLE is analytically challenging. In Appendix A, additional proof showing that the MLE $\widehat{\gamma }$ covers both its existence and uniqueness properties is presented.

Nevertheless, for $\gamma >0$, a fixed censoring scheme, and at least $m_{min}\ge 2$ observed failures, the log-LF is smooth and locally concave in a neighborhood of the true parameter value. Under these regularity conditions, the model is locally identifiable and the solution to the normal equation is unique in that neighborhood. The numerical illustrations presented in Figure 1 are therefore intended to support local existence and uniqueness under representative censoring configurations, rather than to claim a general analytical result.

In particular, owing to the structure of the LF (5), the solution to the LF equation lacks an analytical form. Consequently, iterative numerical methods must be employed to evaluate $\widehat{\gamma }$ when a closed-form solution is not available. It is significant to show the existence and uniqueness of $\widehat{\gamma }$. Due to the complicated structure of (8), it is analytically challenging to establish these properties in a theoretical way. To address this challenge, two different G-PH-CS samples are created from XLindley($\gamma$) when $\gamma \in \{0.5,1.5\}$ with $(\tau ,m_{min},m_{max},n)=(1,10,20,40)$ and $S_i=1,i=1,\dots ,m_{max}$. To fulfill this purpose, the Newton-Raphson (NR) method is recommended by employing the statement maxNR(·) beyond installing maxLik software package (by Henningsen and Toomet [18]. Subsequently, the fitted values of $\widehat{\gamma }$ from XLindley(0.5) and (1.5) are obtained as 0.5001 and 1.6302, respectively. Figure 1 illustrates the log-LF and normal-equation functions corresponding to (7) and (8), respectively. It demonstrates that the vertical line representing $\widehat{\gamma }$ intersects the log-LF curve at its maximum point and the normal-equation curve at its root. This graphical evidence confirms that the MLE $\widehat{\gamma }$ of $\gamma$ exists and is unique.

Corollary 1.

Once the MLE $\widehat{\gamma }$ is obtained, the respective MLEs of $R\left(x\right)$ and $h\left(x\right)$, denoted by $\widehat{R}\left(x\right)$ and $\widehat{h}\left(x\right)$ for $x>0$, are given as follows:

$$\widehat{R}\left(x\right)=\left(1+{\displaystyle \frac{\widehat{\gamma }x}{{\left(1+\widehat{\gamma }\right)}^2}}\right)e^{-\widehat{\gamma }x}and\widehat{h}\left(x\right)={\displaystyle \frac{{\widehat{\gamma }}^2\left(\widehat{\gamma }+x+2\right)}{{\left(1+\widehat{\gamma }\right)}^2+\widehat{\gamma }x}}.$$

2.2. Interval Estimators

This part employs both the standard asymptotic normal approximation and its log-transformed variant, based on the MLE $\widehat{\gamma }$. The distribution of $\widehat{\gamma }$ is approximately normal $N(\gamma ,{\sigma }_{\gamma })$. However, the nonlinear formula in (6) renders the exact analytical derivation of the Fisher information (FI) matrix intractable. Therefore, following Lawless [19], the variance estimator ${\widehat{\sigma }}_{\widehat{\gamma }}$ is computed by evaluating the observed FI item at $\gamma \equiv \widehat{\gamma }$, as given below:

$${\widehat{\sigma }}_{\widehat{\gamma }}={\mathbf{I}}^{-1}{\left(\gamma \right)|}_{\gamma =\widehat{\gamma }},$$

with

$$\mathbf{I}\left(\gamma \right)={\displaystyle \frac{2J_t\left(1+2\gamma \right)}{{\left(\gamma \left(1+\gamma \right)\right)}^2}}+\sum _{i=1}^{J_t}{\xi }^{-2}\left(\gamma ;y_i\right)-\sum _{i=1}^{J_t}{\displaystyle \frac{\ddot{\gamma }\left(y_i\right)y_i{S}_i}{1+{\gamma }^{★}{y}_i}}-{\displaystyle \frac{\ddot{\gamma }\left(\tau \right)\tau S^{★}}{1+{\gamma }^{★}\tau }},$$

(9)

where $\ddot{\gamma }=-2\left(2-\gamma \right){\left(1+\gamma \right)}^{-4}$.

At the $100(1-\pi )\%$ confidence level, the two-sided ACI bounds for $\gamma$ based on the normal approximation (ACI-NA) are

$$\widehat{\gamma }\pm z_{\pi /2}\sqrt{{\widehat{\sigma }}_{\widehat{\gamma }}},$$

where $\widehat{\gamma }$ is the estimator of $\gamma$, $\widehat{Var}\left(\widehat{\gamma }\right)$ its estimated variance, and $z_{\pi /2}$ denotes the $(\pi /2)$-quantile of the standard normal distribution.

Since both $R\left(x\right)$ and $h\left(x\right)$ are smooth and continuously differentiable functions of $\gamma$ at $\gamma >0$, their estimators $\widehat{R}\left(x\right)$ and $\widehat{h}\left(x\right)$ inherit their asymptotic behavior from the MLE $\widehat{\gamma }$. In particular, under the regularity conditions ensuring the asymptotic normality of $\widehat{\gamma }$, the delta method yields

$$\sqrt{m_{min}}\left(\widehat{R}\left(x\right)-R\left(x\right)\right)\stackrel{d}{\to }N\left(0,{\left({\displaystyle \frac{\mathrm{d}R\left(x\right)}{\mathrm{d}\gamma }}\right)}^2{\mathbf{I}}^{-1}\left(\gamma \right)\right),$$

with an analogous result holding for $\widehat{h}\left(x\right)$. Consequently, the variance approximations and ACIs constructed for $R\left(x\right)$ and $h\left(x\right)$ are asymptotically valid under the proposed censoring. Although the exact FI does not admit a closed-form expression due to the nonlinear structure induced by the proposed censoring technique, the observed FI provides a consistent estimator of the asymptotic variance. These results justify the use of normal-based confidence intervals for $\gamma$ and, via the delta method, for the associated reliability measures.

Subsequently, to construct the $100(1-\pi )\%$ ACI-NA estimators of $R\left(x\right)$ and $h\left(x\right)$ (for $x>0$), it is first necessary to obtain their approximate variances, denoted by ${\widehat{\sigma }}_{\widehat{R}}$ and ${\widehat{\sigma }}_{\widehat{h}}$, respectively. A common approach for deriving these variances is the delta method, which provides a convenient large-sample approximation; see Greene [20] for further details. Accordingly, $\xi \left(\gamma ;\tau \right)=\gamma +\tau +2$, the variances associated with $\widehat{R}\left(x\right)$ and $\widehat{h}\left(x\right)$ can be derived as

$${\widehat{\sigma }}_{\widehat{R}}={\left.{\mathcal{Z}}_R^2{\sigma }_{\gamma }\right|}_{\sigma =\widehat{\sigma }}\mathrm{and}{\widehat{\sigma }}_{\widehat{h}}={\left.{\mathcal{Z}}_h^2{\sigma }_{\gamma }\right|}_{\sigma =\widehat{\sigma }},$$

where

$${\widehat{\mathcal{Z}}}_{\widehat{R}}=-{\displaystyle \frac{\widehat{{\gamma }^{★}}x{e}^{-\widehat{\gamma }x}[4+x+\widehat{\gamma }(x+\widehat{\gamma }+3)]}{1+\widehat{\gamma }}}$$

and

$${\widehat{\mathcal{Z}}}_{\widehat{h}}={\displaystyle \frac{\widehat{\gamma }[3\widehat{\gamma }+2(x+2)]}{{(1+\widehat{\gamma })}^2+\widehat{\gamma }x}}-{\displaystyle \frac{{\widehat{\gamma }}^2\xi \left(\gamma ;\tau \right)[2(\widehat{\gamma }+1)+x]}{{[{(1+\widehat{\gamma })}^2+\widehat{\gamma }x]}^2}}.$$

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

$$(\widehat{R}\left(x\right)\mp z_{\frac{\pi }{2}}\sqrt{{\widehat{\sigma }}_{\widehat{R}}})\mathrm{and}(\widehat{h}\left(x\right)\mp z_{\frac{\pi }{2}}\sqrt{{\widehat{\sigma }}_{\widehat{h}}}).$$

A practical drawback of the ACI-NA is that its lower bound may turn negative when estimating lifetime parameters, which is unrealistic in reliability applications. To mitigate this issue, Meeker and Escobar [21] advocated the use of a log-normal transformation of the MLE $\widehat{\gamma }$. They also demonstrated that, among competing normal-based methods, the log-transformed interval (ACI-NL) generally achieves superior coverage performance.

Corollary 2.

The $100(1-\pi )\%$ ACI-NL for γ (for instance) becomes

$$\left(\widehat{\gamma }exp\left(\mp z_{\frac{\pi }{2}}{\widehat{\gamma }}^{-1}\sqrt{{\widehat{\sigma }}_{\widehat{\gamma }}}\right),\right)$$

where the ACI-NL estimators of $R\left(x\right)$ and $h\left(x\right)$ can be directly established.

3. Bayesian Inference

The gamma distribution can accommodate a wide range of subjective or empirically motivated prior specifications; see Kundu [22]. Despite its appeal, the selection of an appropriate prior often remains a challenging task. As emphasized by Arnold and Press [23], no universally accepted criterion exists for prior choice in Bayesian inference. Since the XLindley parameter $\gamma$ is constrained to the positive real line $(0,\infty )$, the gamma prior arises as a natural and mathematically coherent option. However, the prior PDF of $\gamma$, say $\Lambda (·)$, becomes

$$\Lambda \left(\gamma \right)\propto {\gamma }^{\alpha -1}{e}^{-\theta \gamma },$$

(10)

where $\alpha ,\theta >0$.

The posterior PDF (say, ${\Lambda }^{★}(·)$) of $\gamma$ using (6) and (10) is

$$\begin{array}{c}\hfill {\Lambda }^{★}\left(\gamma \right|\mathbf{y})={\complement }^{-1}{\gamma }^{J_t+\alpha -1}{{\gamma }^{★}}^{J_t}{e}^{{\sum }_{i=1}^{J_t}log\left(\xi \left(\gamma ;y_i\right)\right)-\gamma \left(\zeta +\theta \right)}\prod _{i=1}^{J_t}{\left(1+{\gamma }^{★}{y}_i\right)}^{S_i}{\left(1+{\gamma }^{★}\tau \right)}^{S^{★}},\end{array}$$

(11)

where ∁ is given by

$$\begin{array}{c}\hfill \complement ={\int }_{\gamma }{\mathcal{L}}_t\left(\gamma \right|\mathbf{y})\times \Lambda \left(\gamma \right)d\gamma .\end{array}$$

The symmetric squared error loss (SEL) function is employed because it penalizes estimation errors equally, yielding the posterior mean as the Bayes estimator and ensuring balanced inference. This property makes SEL particularly suitable for reliability analysis, where underestimation and overestimation are of comparable importance. Nevertheless, in situations where the costs of these errors differ, alternative asymmetric loss functions, such as the linear-exponential loss or general entropy loss, may be adopted, depending on the practical requirements of the analysis. Owing to the structure in (6), the integrals in ((11) rarely admit closed-form solutions, rendering analytical evaluation infeasible in most practical cases. To solve this issue, we utilize the MCMC methodology to generate Markovian draws from (11)). From these samples, we compute the Bayes estimates and construct the associated BCI and HPD intervals.

Using the same censoring setting used in Figure 1, Figure 2 further illustrates that the posterior distribution in (11) closely resembles a normal density. Consequently, in Algorithm 1, the normal proposal density within the Metropolis–Hastings (M-H) algorithm for updating $\gamma$ is utilized. This choice ensures efficient sampling and facilitates reliable computation of Bayes point (or credible) estimates for all unknown parameters.

Algorithm 1 MCMC Sampling for $\gamma$

1: Input: Initial estimate $\widehat{\gamma }$, variance estimate ${\widehat{\sigma }}_{\widehat{\gamma }}$, total iterations ${\Im }^{•}$, burn-in period ${\Im }^{\circ }$, confidence level $(1-\pi )$ 2: Initialize ${\gamma }^{\left(0\right)}\leftarrow \widehat{\gamma }$ 3: Set iteration counter $a\leftarrow 1$ 4: while  $a\le {\Im }^{•}$do 5:    Generate candidate ${\gamma }^{*}\sim N({\gamma }^{(a-1)},{\widehat{\sigma }}_{\widehat{\gamma }})$ 6:    Compute $O\leftarrow min\left(1,{\displaystyle \frac{{\Lambda }^{★}\left({\gamma }^{*}\right|\mathbf{y})}{{\Lambda }^{★}\left({\gamma }^{(a-1)}\right|\mathbf{y})}}\right)$ 7:    Generate $u\sim U(0,1)$ 8:    if $u\le O$ then 9:        ${\gamma }^{\left(a\right)}\leftarrow {\gamma }^{*}$ 10:    else 11:       ${\gamma }^{\left(a\right)}\leftarrow {\gamma }^{(a-1)}$ 12:    end if 13:    Update $R\left(x\right)$ and $h\left(x\right)$ by replacing $\gamma$ with ${\gamma }^{\left(a\right)}$ 14:     $a\leftarrow a+1$ 15: end while 16: Discard first ${\Im }^{\circ }$ samples (burn-in) and retain $\{{\gamma }^{({\Im }^{\circ }+1)},\dots ,{\gamma }^{\left({\Im }^{•}\right)}\}$ 17: Compute posterior mean estimate $$\tilde{\gamma }\leftarrow {\displaystyle \frac{1}{{\Im }^{★}}}\sum _{a={\Im }^{\circ }+1}^{{\Im }^{•}}{\gamma }^{\left(a\right)},{\Im }^{★}={\Im }^{•}-{\Im }^{\circ }$$ 18: Compute $(1-\pi )100\%$ HPD interval: 1. Sort retained ${\gamma }^{\left(a\right)}$ values as ${\gamma }_{\left(a\right)}$ 2. For each $a=1,\dots ,\pi {\Im }^{★}$ compute $$\Delta \left(a\right)={\gamma }_{(a+\left[(1-\pi ){\Im }^{★}\right])}-{\gamma }_{\left(a\right)}$$ 3. Find $a^{★}=arg{min}_a\Delta \left(a\right)$ 4. HPD interval $\leftarrow \left({\gamma }_{\left(a^{★}\right)},{\gamma }_{(a^{★}+(1-\pi ){\Im }^{★})}\right)$ 19: Redo Steps 17–18 for $R\left(x\right)$ and $h\left(x\right)$

Posterior inference for the parameter $\gamma$ is carried out using a random-walk M–H algorithm with a normal proposal distribution of the form ${\gamma }^{*}\sim N\left({\gamma }^{(a-1)},b{\widehat{\sigma }}_{\gamma }\right),$ where ${\widehat{\sigma }}_{\gamma }$ denotes the estimated standard deviation of the MLE of $\gamma$, obtained from the inverse of the observed FI matrix. The scaling factor b is selected through an adaptive tuning procedure during an initial burn-in phase to achieve a target acceptance rate between $0.25$ and $0.45$, which is commonly recommended for one-dimensional random-walk algorithms.

Figure 3 reports standard MCMC diagnostics for $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ under two representative settings, $\gamma =0.5$ and $1.5$. For each parameter, the autocorrelation functions (ACFs), trace plots, and ergodic averages are shown to assess mixing and convergence. The ACFs decay rapidly toward zero, indicating limited serial dependence and satisfactory mixing of the chains across all parameters. Trace plots exhibit stable fluctuations around constant levels after a short transient period, with no evidence of multimodality or nonstationary behavior. The posterior means remain well centered within the sampled trajectories. Moreover, the ergodic averages stabilize quickly, suggesting that the chains have reached their stationary distributions and that Monte Carlo error is well controlled. Overall, these diagnostics provide strong evidence of reliable convergence and robustness of the MCMC algorithm across different values of $\gamma$.

Figure 3. The ACF, Trace, and Ergodic diagnostics of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$.

Figure 3. The ACF, Trace, and Ergodic diagnostics of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$.

4. Monte Carlo Evaluations

In this section, extensive simulations are conducted to assess the statistical efficiency and practical performance of the XLindley parameter $\gamma$, along with the corresponding reliability measures $R\left(x\right)$ and $h\left(x\right)$ previously developed.

4.1. Simulation Setups

To obtain point or interval estimates of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$, the G-PH-CS is replicated 1000 times using two distinct $\mathrm{XLindley}\left(\gamma \right)$ settings: Pop–1 $\left(0.8\right)$ and Pop–2 $\left(1.5\right)$. For $t=0.1$, the corresponding values of $\left(R\right(x),h(x\left)\right)$ are $(0.9459,0.5590)$ and $(0.8814,1.2656)$ for Pop–i, $i=1,2$, respectively. All considered tests are performed under varying specifications of $\tau$ (threshold time), n (number of experimental units), $(m_{min},m_{max})$ (effective sample sizes), and $\mathbf{S}$ (progressive censoring). For each population, $\tau$ is set to $(0.5,1.5)$ and n takes values $(30,50,80)$.

Table 1. Several censoring setups in simulation comparisons.

$\mathit{n}\downarrow$	S
$(m_{min},m_{max})\to$	(10,15)	(15,20)
30	[A]: $(5^3,0^{12})$	[A]: $(5^2,0^{18})$
	[B]: $(0^6,5^3,0^6)$	[B]: $(0^9,5^2,0^9)$
	[C]: $(0^{12},5^3)$	[C]: $(0^{18},5^2)$
$(m_{min},m_{max})\to$	(20,30)	(40,30)
50	[A]: $(5^4,0^{26})$	[A]: $(5^2,0^{38})$
	[B]: $(0^{13},5^4,0^{13})$	[B]: $(0^{19},5^2,0^{19})$
	[C]: $(0^{26},5^4)$	[C]: $(0^{38},5^2)$
$(m_{min},m_{max})\to$	(30,40)	(50,60)
80	[A]: $(5^8,0^{32})$	[A]: $(5^4,0^{56})$
	[B]: $(0^{16},5^8,0^{16})$	[B]: $(0^{28},5^4,0^{28})$
	[C]: $(0^{32},5^8)$	[C]: $(0^{56},5^4)$

reports, for each n, the corresponding configurations of $m_i,i=1,2$, and several PCP patterns, where, for example, $2^5$ indicates that two units are censored at each of the first five stages. For reproducibility, the procedure for generating a G-PH-CS dataset from $\mathrm{XLindley}\left(\gamma \right)$ is outlined in Algorithm 2.

For the 1000 generated G-PH-CS datasets, the MLEs of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ and their associated 95% confidence intervals (ACI-NA/ACI-NL), were computed using the maxLik package [18] in R (v4.2.2). According to Algorithm 1, the posterior inference for the parameter $\gamma$ is carried out with a normal proposal distribution of the form ${\gamma }^{*}\sim N({\gamma }^{(a-1)},{\widehat{\sigma }}_{\widehat{\gamma }}),$ where ${\widehat{\sigma }}_{\gamma }$ denotes the estimated standard deviation of the MLE of $\gamma$, obtained from the inverse of the observed FI matrix. The scaling factor b is selected through an adaptive tuning procedure during an initial burn-in phase to achieve a target acceptance rate between $0.25$ and $0.45$, which is commonly recommended for one-dimensional random-walk Metropolis algorithms.

Algorithm 2 Generation of the G-PH-CS Plan

1: Input: Specify the parameter value of the $\mathrm{XLindley}\left(\gamma \right)$ model. 2: Input: Specify the censoring quantities: $\tau$, n, $\mathbf{S}$, $m_{min}$, and $m_{max}$. 3: Step 1: Generate $v_i\sim U(0,1)$ for $i=1,2,\dots ,m_{max}$. 4: Step 2: Compute $${\nu }_i=v_i^{{\left(i+{\sum }_{l=m_{max}-i+1}^{m_{max}}{S}_l\right)}^{-1}},i=1,2,\dots ,m_{max}.$$ 5: Step 3: Evaluate $$U_i=1-{\nu }_{m_{max}}{\nu }_{m_{max}-1}\dots {\nu }_{m_{max}-i+1},i=1,2,\dots ,m_{max}.$$ 6: Step 4: Obtain $$y_i=G^{-1}(U_i;\gamma ),i=1,2,\dots ,m_{max},$$ where $G^{-1}(·;\gamma )$ is the quantile function of the XLindley distribution. 7: Step 5: Determine d such that $y_d\le \tau <y_{d+1}$. 8: Output: Construct the G-PH-CS sample $\mathbf{y}$ according to the following cases: 9: if  $\tau <y_{m_{min}}<y_{m_{max}}$  then 10:    $\mathbf{y}=\{y_1,\dots ,y_{m_{min}}\}$. 11: end if 12: if  $y_{m_{min}}<\tau <y_{m_{max}}$  then 13:      $\mathbf{y}=\{y_1,\dots ,y_d\}$. 14: end if 15: if  $y_{m_{min}}<y_{m_{max}}<\tau$  then 16:      $\mathbf{y}=\{y_1,\dots ,y_{m_{max}}\}$. 17: end if

Bayesian inference was carried out via omitting the first ${\Im }^{\circ }=2000$ iterations discarded as burn-in from a total of ${\Im }^{•}=12,000$ MCMC samples for each parameter. Posterior summaries, including point estimates and 95% credible intervals (BCI/HPD), were obtained using the statement run_metropolis_MCMC$(·)$ which is available in coda package by Plummer et al. [24]. To investigate the effect of prior selection on Bayesian results, both point and interval estimates were analyzed under two alternative hyperparameter settings $(\alpha ,\theta )$ for each population. Specifically, for Pop–1 the choices are Prior–1: $(4,5)$ and Prior–2: $(8,10)$, while for Pop–2 the corresponding settings are Prior–1: $(7.5,5)$ and Prior–2: $(15,10)$. Following the prior specification strategy of Kundu [22], the hyperparameters $(\alpha ,\theta )$, corresponding to priors 1 and 2 for each XLindley population, are chosen such that the prior means coincide with the expected values of the respective model parameter.

Using 1000 G-PH-CS samples (from Pop–i for $i=1,2$) when $\tau =0.5$, $(n,m_{min},m_{max})=(10,15,30)$, and $\mathbf{S}=(5^3,0^{12})$, we examine the sensitivity analysis of the simulated posterior estimates of $\gamma$ using four different prior choices, including improper (when $\alpha =\theta =0.001$), weakly informative (say, Uniform(0.1, 10)), informative (suggested), and overdispersed gamma (say, gamma(5, 2)) priors. Figure 4 shows that posterior means and HPD interval bounds remain stable across different prior specifications, with only moderate variation in interval widths under informative priors. This indicates that the likelihood under the G-PH-CS is sufficiently informative, and the choice of prior does not unduly influence that posterior inference.

Next, we compute the following precision measures for the parameter $\gamma$ (as an illustrative example):

• Root Mean Squared Error: $\mathrm{RMSE}\left(\check{\gamma }\right)=\sqrt{{\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}{\left({\check{\gamma }}^{\left[i\right]}-\gamma \right)}^2}.$
• Mean Absolute Bias: $\mathrm{MAB}\left(\check{\gamma }\right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}\left|{\check{\gamma }}^{\left[i\right]}-\gamma \right|.$
• Average Interval Length: ${\mathrm{AIL}}_{95\%}\left(\gamma \right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}\left({\mathcal{U}}_{{\check{\gamma }}^{\left[i\right]}}-{\mathcal{L}}_{{\check{\gamma }}^{\left[i\right]}}\right).$
• Coverage Percentage: ${\mathrm{CP}}_{95\%}\left(\gamma \right)={\displaystyle \frac{1}{1000}}{\displaystyle \sum _{i=1}^{1000}}{\Xi }_{\left({\mathcal{L}}_{{\check{\gamma }}^{\left[i\right]}};{\mathcal{U}}_{{\check{\gamma }}^{\left[i\right]}}\right)}\left(\gamma \right).$

where ${\check{\gamma }}^{\left[i\right]}$ is the fitted estimate at ith simulated dataset.

4.2. Simulation Results and Interpretation

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

$(\mathit{n},{\mathit{m}}_{\mathit{max}},{\mathit{m}}_{\mathit{min}})$	S	MLE		MCMC				MLE		MCMC
		$\tau =\mathbf{0}.\mathbf{5}$						$\tau =\mathbf{1}.\mathbf{5}$
Prior→				1		2				1		2
Pop-1
(30,15,10)	[A]	0.260	0.243	0.199	0.184	0.191	0.153	0.260	0.243	0.195	0.157	0.182	0.152
	[B]	0.323	0.313	0.239	0.224	0.210	0.178	0.274	0.263	0.237	0.213	0.203	0.155
	[C]	0.256	0.239	0.189	0.171	0.178	0.149	0.226	0.214	0.187	0.149	0.176	0.145
(30,20,15)	[A]	0.182	0.165	0.178	0.142	0.171	0.134	0.178	0.149	0.174	0.139	0.163	0.127
	[B]	0.254	0.237	0.182	0.153	0.178	0.138	0.202	0.185	0.179	0.149	0.174	0.135
	[C]	0.177	0.147	0.171	0.133	0.156	0.121	0.171	0.139	0.169	0.132	0.155	0.120
(50,30,20)	[A]	0.162	0.138	0.148	0.119	0.136	0.111	0.149	0.123	0.147	0.118	0.130	0.107
	[B]	0.177	0.138	0.158	0.128	0.155	0.120	0.168	0.130	0.156	0.120	0.152	0.117
	[C]	0.159	0.125	0.135	0.107	0.130	0.105	0.148	0.122	0.135	0.107	0.129	0.105
(50,40,30)	[A]	0.144	0.111	0.131	0.103	0.125	0.097	0.135	0.103	0.128	0.101	0.115	0.092
	[B]	0.155	0.121	0.135	0.105	0.128	0.103	0.146	0.112	0.133	0.105	0.120	0.097
	[C]	0.142	0.110	0.123	0.099	0.116	0.089	0.131	0.101	0.123	0.095	0.107	0.087
(80,40,20)	[A]	0.122	0.095	0.114	0.092	0.104	0.082	0.115	0.095	0.112	0.087	0.104	0.082
	[B]	0.131	0.101	0.115	0.093	0.107	0.083	0.118	0.095	0.113	0.089	0.106	0.083
	[C]	0.117	0.094	0.111	0.088	0.104	0.082	0.114	0.092	0.106	0.083	0.102	0.081
(80,60,30)	[A]	0.111	0.088	0.104	0.082	0.095	0.077	0.108	0.084	0.101	0.081	0.094	0.074
	[B]	0.114	0.092	0.105	0.087	0.100	0.079	0.112	0.088	0.102	0.082	0.100	0.077
	[C]	0.107	0.084	0.101	0.081	0.094	0.075	0.106	0.084	0.098	0.079	0.088	0.069
Pop-2
(30,15,10)	[A]	0.529	0.499	0.401	0.322	0.342	0.282	0.410	0.376	0.363	0.287	0.260	0.217
	[B]	0.782	0.774	0.501	0.487	0.445	0.428	0.546	0.511	0.495	0.474	0.414	0.364
	[C]	0.524	0.494	0.373	0.294	0.326	0.266	0.381	0.347	0.356	0.275	0.259	0.209
(30,20,15)	[A]	0.371	0.345	0.339	0.267	0.271	0.236	0.359	0.276	0.286	0.261	0.257	0.206
	[B]	0.457	0.428	0.372	0.291	0.275	0.241	0.376	0.343	0.349	0.271	0.259	0.208
	[C]	0.365	0.341	0.330	0.264	0.268	0.224	0.358	0.271	0.284	0.252	0.248	0.206
(50,30,20)	[A]	0.308	0.254	0.267	0.219	0.233	0.192	0.267	0.225	0.249	0.192	0.209	0.169
	[B]	0.330	0.267	0.310	0.235	0.235	0.198	0.312	0.254	0.268	0.223	0.209	0.170
	[C]	0.288	0.231	0.242	0.191	0.233	0.186	0.265	0.218	0.233	0.191	0.208	0.167
(50,40,30)	[A]	0.254	0.213	0.229	0.185	0.220	0.173	0.252	0.204	0.225	0.184	0.203	0.166
	[B]	0.279	0.218	0.241	0.189	0.229	0.184	0.265	0.216	0.231	0.186	0.205	0.167
	[C]	0.250	0.213	0.229	0.184	0.204	0.168	0.244	0.192	0.219	0.172	0.202	0.163
(80,40,20)	[A]	0.243	0.197	0.208	0.172	0.200	0.161	0.239	0.189	0.207	0.165	0.195	0.159
	[B]	0.250	0.209	0.213	0.173	0.201	0.166	0.244	0.191	0.209	0.166	0.200	0.162
	[C]	0.241	0.191	0.204	0.160	0.193	0.157	0.234	0.182	0.201	0.159	0.188	0.154
(80,60,30)	[A]	0.227	0.187	0.189	0.155	0.172	0.136	0.226	0.172	0.174	0.137	0.154	0.128
	[B]	0.234	0.189	0.194	0.158	0.176	0.139	0.231	0.180	0.186	0.147	0.163	0.134
	[C]	0.219	0.171	0.172	0.140	0.165	0.129	0.184	0.151	0.170	0.136	0.146	0.121

,

Table 3. The RMSE (1st Col.) and MAB (2nd Col.) results of $R\left(x\right)$.

$(\mathit{n},{\mathit{m}}_{\mathit{max}},{\mathit{m}}_{\mathit{min}})$	S	MLE		MCMC				MLE		MCMC
		$\tau =\mathbf{0}.\mathbf{5}$						$\tau =\mathbf{1}.\mathbf{5}$
Prior→				1		2				1		2
Pop-1
(30,15,10)	[A]	0.563	0.528	0.481	0.440	0.416	0.382	0.502	0.477	0.473	0.431	0.411	0.334
	[B]	0.572	0.537	0.489	0.444	0.443	0.411	0.572	0.537	0.489	0.440	0.417	0.342
	[C]	0.704	0.685	0.603	0.579	0.530	0.498	0.613	0.582	0.567	0.540	0.521	0.473
(30,20,15)	[A]	0.398	0.330	0.378	0.302	0.355	0.281	0.393	0.318	0.356	0.299	0.355	0.275
	[B]	0.559	0.524	0.449	0.413	0.409	0.342	0.479	0.429	0.433	0.398	0.407	0.334
	[C]	0.406	0.369	0.406	0.333	0.381	0.318	0.406	0.343	0.384	0.318	0.369	0.314
(50,30,20)	[A]	0.361	0.286	0.310	0.243	0.297	0.240	0.327	0.254	0.304	0.241	0.293	0.237
	[B]	0.367	0.315	0.334	0.271	0.310	0.252	0.349	0.277	0.333	0.260	0.297	0.243
	[C]	0.398	0.315	0.355	0.281	0.353	0.275	0.356	0.298	0.354	0.277	0.347	0.268
(50,40,30)	[A]	0.310	0.231	0.282	0.224	0.274	0.213	0.295	0.230	0.280	0.216	0.245	0.197
	[B]	0.358	0.277	0.304	0.238	0.282	0.225	0.319	0.247	0.292	0.237	0.273	0.221
	[C]	0.316	0.237	0.302	0.231	0.278	0.220	0.304	0.237	0.287	0.222	0.261	0.208
(80,40,20)	[A]	0.270	0.214	0.252	0.201	0.237	0.185	0.259	0.208	0.249	0.200	0.230	0.185
	[B]	0.277	0.216	0.257	0.210	0.238	0.186	0.261	0.215	0.255	0.208	0.238	0.185
	[C]	0.302	0.231	0.270	0.213	0.245	0.189	0.278	0.218	0.259	0.210	0.239	0.189
(80,60,30)	[A]	0.229	0.193	0.226	0.184	0.219	0.177	0.228	0.192	0.224	0.180	0.213	0.171
	[B]	0.252	0.214	0.243	0.199	0.231	0.185	0.246	0.208	0.237	0.199	0.228	0.180
	[C]	0.244	0.200	0.237	0.187	0.225	0.185	0.242	0.200	0.229	0.185	0.214	0.175
Pop-2
(30,15,10)	[A]	1.107	1.139	0.876	0.798	0.726	0.594	0.910	0.825	0.729	0.631	0.582	0.471
	[B]	1.118	1.149	0.920	0.828	0.765	0.659	0.945	0.865	0.787	0.668	0.598	0.488
	[C]	1.218	1.778	1.058	1.177	0.819	1.039	1.128	1.477	0.876	1.093	0.625	0.986
(30,20,15)	[A]	0.839	0.782	0.646	0.600	0.556	0.465	0.730	0.617	0.604	0.510	0.522	0.455
	[B]	1.052	0.984	0.867	0.759	0.631	0.552	0.906	0.822	0.724	0.627	0.568	0.470
	[C]	0.852	0.793	0.651	0.621	0.576	0.473	0.741	0.672	0.622	0.541	0.538	0.465
(50,30,20)	[A]	0.599	0.490	0.528	0.435	0.502	0.416	0.558	0.457	0.522	0.431	0.471	0.379
	[B]	0.603	0.510	0.561	0.461	0.508	0.425	0.600	0.492	0.523	0.432	0.474	0.384
	[C]	0.701	0.603	0.602	0.508	0.514	0.432	0.664	0.552	0.529	0.446	0.475	0.386
(50,40,30)	[A]	0.562	0.480	0.516	0.416	0.460	0.380	0.545	0.452	0.493	0.400	0.458	0.370
	[B]	0.597	0.489	0.522	0.425	0.497	0.405	0.557	0.453	0.515	0.422	0.464	0.378
	[C]	0.566	0.482	0.517	0.417	0.495	0.405	0.547	0.452	0.502	0.416	0.460	0.375
(80,40,20)	[A]	0.534	0.437	0.455	0.369	0.439	0.362	0.495	0.406	0.445	0.365	0.429	0.351
	[B]	0.546	0.451	0.472	0.391	0.451	0.373	0.517	0.435	0.451	0.374	0.444	0.367
	[C]	0.562	0.469	0.480	0.393	0.459	0.379	0.518	0.436	0.474	0.392	0.453	0.368
(80,60,30)	[A]	0.477	0.388	0.389	0.318	0.358	0.292	0.420	0.344	0.386	0.309	0.331	0.274
	[B]	0.523	0.431	0.443	0.362	0.416	0.336	0.483	0.396	0.426	0.343	0.373	0.306
	[C]	0.517	0.421	0.432	0.354	0.391	0.310	0.481	0.391	0.397	0.321	0.352	0.291

and

Table 4. The RMSE (1st Col.) and MAB (2nd Col.) results of $h\left(x\right)$.

$(\mathit{n},{\mathit{m}}_{\mathit{max}},{\mathit{m}}_{\mathit{min}})$	S	MLE		MCMC				MLE		MCMC
		$\tau =\mathbf{0}.\mathbf{5}$						$\tau =\mathbf{1}.\mathbf{5}$
Prior→				1		2				1		2
Pop-1
(30,15,10)	[A]	0.239	0.225	0.228	0.186	0.186	0.172	0.239	0.225	0.205	0.184	0.175	0.154
	[B]	0.226	0.219	0.210	0.184	0.175	0.160	0.226	0.200	0.196	0.178	0.170	0.143
	[C]	0.294	0.286	0.252	0.242	0.222	0.209	0.272	0.243	0.237	0.226	0.218	0.180
(30,20,15)	[A]	0.213	0.209	0.188	0.167	0.175	0.144	0.213	0.173	0.180	0.158	0.167	0.138
	[B]	0.191	0.138	0.159	0.127	0.152	0.122	0.168	0.134	0.157	0.127	0.150	0.117
	[C]	0.201	0.155	0.170	0.140	0.161	0.136	0.170	0.144	0.164	0.136	0.160	0.128
(50,30,20)	[A]	0.179	0.130	0.142	0.115	0.141	0.109	0.153	0.118	0.142	0.111	0.132	0.107
	[B]	0.170	0.122	0.139	0.105	0.131	0.100	0.152	0.108	0.133	0.103	0.126	0.101
	[C]	0.183	0.134	0.152	0.119	0.148	0.117	0.157	0.126	0.150	0.119	0.144	0.114
(50,40,30)	[A]	0.162	0.118	0.137	0.101	0.126	0.095	0.147	0.105	0.130	0.101	0.116	0.093
	[B]	0.140	0.103	0.129	0.094	0.121	0.090	0.134	0.097	0.124	0.094	0.110	0.083
	[C]	0.144	0.110	0.133	0.098	0.124	0.093	0.136	0.101	0.129	0.098	0.114	0.088
(80,40,20)	[A]	0.134	0.099	0.129	0.091	0.109	0.089	0.130	0.092	0.122	0.090	0.101	0.080
	[B]	0.130	0.092	0.117	0.085	0.107	0.079	0.120	0.085	0.110	0.084	0.097	0.078
	[C]	0.134	0.094	0.120	0.088	0.108	0.080	0.125	0.091	0.119	0.088	0.100	0.075
(80,60,30)	[A]	0.110	0.090	0.108	0.084	0.098	0.078	0.110	0.082	0.100	0.084	0.094	0.071
	[B]	0.103	0.082	0.095	0.076	0.087	0.074	0.098	0.078	0.090	0.075	0.066	0.067
	[C]	0.110	0.084	0.104	0.079	0.097	0.076	0.108	0.081	0.100	0.078	0.090	0.068
Pop-2
(30,15,10)	[A]	0.783	0.775	0.553	0.518	0.453	0.460	0.668	0.647	0.509	0.483	0.314	0.437
	[B]	0.532	0.502	0.389	0.354	0.300	0.274	0.404	0.366	0.337	0.283	0.267	0.216
	[C]	0.536	0.506	0.407	0.368	0.313	0.293	0.418	0.383	0.349	0.299	0.279	0.224
(30,20,15)	[A]	0.379	0.353	0.292	0.274	0.256	0.221	0.330	0.300	0.277	0.242	0.234	0.211
	[B]	0.373	0.349	0.279	0.271	0.242	0.213	0.325	0.295	0.277	0.230	0.210	0.205
	[C]	0.465	0.436	0.368	0.350	0.257	0.246	0.402	0.364	0.281	0.281	0.250	0.215
(50,30,20)	[A]	0.321	0.275	0.277	0.229	0.215	0.197	0.306	0.253	0.241	0.204	0.208	0.187
	[B]	0.272	0.226	0.239	0.197	0.212	0.189	0.251	0.209	0.237	0.197	0.180	0.172
	[C]	0.276	0.232	0.256	0.210	0.214	0.194	0.275	0.226	0.240	0.198	0.200	0.179
(50,40,30)	[A]	0.272	0.225	0.236	0.195	0.210	0.185	0.250	0.207	0.231	0.192	0.176	0.162
	[B]	0.258	0.220	0.215	0.189	0.208	0.173	0.245	0.206	0.210	0.183	0.154	0.168
	[C]	0.261	0.220	0.227	0.191	0.209	0.180	0.249	0.207	0.224	0.189	0.157	0.170
(80,40,20)	[A]	0.251	0.203	0.214	0.177	0.200	0.169	0.236	0.198	0.206	0.170	0.132	0.161
	[B]	0.254	0.196	0.207	0.168	0.194	0.164	0.220	0.183	0.198	0.165	0.120	0.158
	[C]	0.258	0.215	0.214	0.179	0.206	0.171	0.237	0.199	0.207	0.178	0.139	0.167
(80,60,30)	[A]	0.240	0.194	0.194	0.163	0.168	0.151	0.217	0.178	0.191	0.154	0.114	0.138
	[B]	0.215	0.175	0.177	0.144	0.145	0.132	0.189	0.155	0.171	0.139	0.105	0.124
	[C]	0.233	0.193	0.189	0.159	0.159	0.140	0.217	0.178	0.178	0.145	0.109	0.131

displayed the simulated values of RMSE and MAB for the estimators of $\gamma$ and the reliability functions $R\left(x\right)$ and $h\left(x\right)$. In parallel,

Table 5. The AIL (1st Col.) and CP (2nd Col.) results of $\gamma$.

$(\mathit{n},{\mathit{m}}_{\mathit{max}},{\mathit{m}}_{\mathit{min}})$	S	ACI-NA		ACI-NL		BCI				HPD
Prior→						1		2		1		2
Pop-1 ($\tau =0.5$)
(30,15,10)	[A]	0.779	0.931	0.707	0.939	0.639	0.946	0.605	0.949	0.580	0.952	0.517	0.958
	[B]	0.801	0.929	0.755	0.934	0.644	0.945	0.630	0.947	0.593	0.950	0.586	0.951
	[C]	0.746	0.935	0.681	0.941	0.639	0.946	0.576	0.952	0.512	0.959	0.500	0.960
(30,20,15)	[A]	0.677	0.942	0.629	0.947	0.560	0.954	0.557	0.954	0.498	0.960	0.455	0.965
	[B]	0.681	0.941	0.644	0.945	0.576	0.952	0.573	0.952	0.500	0.960	0.459	0.964
	[C]	0.666	0.943	0.617	0.948	0.557	0.954	0.494	0.961	0.477	0.962	0.439	0.966
(50,30,20)	[A]	0.629	0.947	0.541	0.956	0.498	0.960	0.457	0.964	0.435	0.967	0.402	0.970
	[B]	0.664	0.943	0.550	0.955	0.513	0.959	0.492	0.961	0.467	0.963	0.421	0.968
	[C]	0.615	0.948	0.537	0.956	0.467	0.963	0.435	0.967	0.421	0.968	0.399	0.970
(50,40,30)	[A]	0.592	0.950	0.479	0.962	0.436	0.967	0.430	0.967	0.416	0.969	0.370	0.973
	[B]	0.614	0.948	0.487	0.961	0.442	0.966	0.431	0.967	0.420	0.968	0.375	0.973
	[C]	0.574	0.952	0.443	0.966	0.430	0.967	0.410	0.969	0.391	0.971	0.369	0.974
(80,40,20)	[A]	0.541	0.956	0.439	0.966	0.409	0.969	0.390	0.971	0.379	0.973	0.344	0.976
	[B]	0.571	0.953	0.439	0.966	0.419	0.968	0.407	0.970	0.383	0.972	0.362	0.974
	[C]	0.540	0.956	0.436	0.967	0.407	0.970	0.373	0.973	0.369	0.974	0.340	0.976
(80,60,30)	[A]	0.516	0.958	0.426	0.968	0.361	0.974	0.360	0.974	0.328	0.978	0.308	0.980
	[B]	0.540	0.956	0.429	0.967	0.407	0.970	0.364	0.974	0.329	0.978	0.317	0.979
	[C]	0.422	0.968	0.401	0.970	0.333	0.977	0.307	0.980	0.305	0.980	0.297	0.981
Pop-1 ($\tau =1.5$)
(30,15,10)	[A]	0.747	0.935	0.697	0.940	0.623	0.947	0.602	0.950	0.573	0.952	0.509	0.959
	[B]	0.755	0.934	0.719	0.937	0.644	0.945	0.630	0.947	0.593	0.950	0.586	0.951
	[C]	0.732	0.936	0.639	0.946	0.616	0.947	0.573	0.952	0.501	0.960	0.492	0.961
(30,20,15)	[A]	0.636	0.946	0.596	0.950	0.558	0.954	0.556	0.954	0.482	0.961	0.451	0.965
	[B]	0.648	0.945	0.627	0.947	0.575	0.952	0.573	0.953	0.492	0.961	0.454	0.965
	[C]	0.633	0.946	0.560	0.954	0.556	0.954	0.485	0.962	0.459	0.964	0.435	0.967
(50,30,20)	[A]	0.614	0.948	0.507	0.959	0.494	0.961	0.439	0.966	0.435	0.967	0.399	0.970
	[B]	0.614	0.948	0.519	0.958	0.506	0.959	0.484	0.962	0.453	0.965	0.418	0.969
	[C]	0.592	0.950	0.469	0.963	0.452	0.965	0.431	0.967	0.418	0.969	0.392	0.971
(50,40,30)	[A]	0.554	0.954	0.452	0.965	0.431	0.967	0.418	0.968	0.409	0.969	0.368	0.973
	[B]	0.574	0.952	0.454	0.965	0.441	0.966	0.428	0.967	0.413	0.969	0.369	0.972
	[C]	0.508	0.959	0.442	0.966	0.428	0.967	0.405	0.970	0.383	0.972	0.363	0.974
(80,40,20)	[A]	0.473	0.963	0.438	0.966	0.391	0.971	0.382	0.972	0.361	0.974	0.341	0.976
	[B]	0.508	0.959	0.439	0.966	0.410	0.969	0.391	0.971	0.380	0.972	0.356	0.974
	[C]	0.471	0.963	0.434	0.967	0.378	0.973	0.370	0.973	0.356	0.975	0.339	0.977
(80,60,30)	[A]	0.428	0.967	0.362	0.974	0.361	0.974	0.357	0.975	0.318	0.979	0.307	0.980
	[B]	0.462	0.964	0.426	0.968	0.370	0.973	0.363	0.974	0.326	0.978	0.313	0.979
	[C]	0.421	0.968	0.343	0.976	0.321	0.979	0.305	0.980	0.301	0.981	0.297	0.981
Pop-2 ($\tau =0.5$)
(30,15,10)	[A]	1.475	0.917	1.146	0.926	1.013	0.933	0.927	0.937	0.897	0.940	0.861	0.947
	[B]	1.596	0.915	1.320	0.920	1.020	0.933	0.971	0.934	0.922	0.938	0.871	0.939
	[C]	1.360	0.921	1.143	0.929	0.926	0.933	0.916	0.940	0.897	0.948	0.838	0.949
(30,20,15)	[A]	1.301	0.929	1.137	0.934	0.919	0.942	0.847	0.942	0.822	0.949	0.775	0.954
	[B]	1.331	0.929	1.139	0.933	0.920	0.940	0.882	0.941	0.837	0.949	0.776	0.954
	[C]	1.193	0.930	1.099	0.936	0.918	0.942	0.827	0.950	0.803	0.952	0.774	0.956
(50,30,20)	[A]	1.052	0.934	1.021	0.944	0.841	0.949	0.811	0.954	0.765	0.956	0.743	0.960
	[B]	1.126	0.930	1.028	0.943	0.845	0.948	0.811	0.950	0.769	0.953	0.746	0.958
	[C]	1.046	0.936	1.018	0.945	0.817	0.953	0.803	0.956	0.756	0.958	0.741	0.960
(50,40,30)	[A]	1.031	0.939	0.912	0.951	0.800	0.956	0.785	0.957	0.740	0.959	0.719	0.964
	[B]	1.034	0.936	0.945	0.950	0.806	0.956	0.796	0.957	0.753	0.958	0.732	0.964
	[C]	0.995	0.941	0.908	0.955	0.788	0.957	0.715	0.959	0.669	0.961	0.635	0.964
(80,40,20)	[A]	0.952	0.944	0.845	0.956	0.743	0.959	0.703	0.961	0.663	0.963	0.631	0.967
	[B]	0.959	0.941	0.896	0.956	0.780	0.958	0.707	0.960	0.663	0.962	0.632	0.964
	[C]	0.943	0.944	0.813	0.956	0.699	0.960	0.688	0.963	0.631	0.964	0.610	0.967
(80,60,30)	[A]	0.935	0.947	0.779	0.957	0.592	0.965	0.586	0.965	0.568	0.968	0.559	0.971
	[B]	0.939	0.944	0.812	0.957	0.646	0.960	0.608	0.964	0.587	0.968	0.562	0.970
	[C]	0.777	0.958	0.716	0.960	0.565	0.968	0.549	0.971	0.472	0.971	0.455	0.972
Pop-2 ($\tau =1.5$)
(30,15,10)	[A]	1.379	0.921	1.051	0.927	1.008	0.935	0.920	0.937	0.891	0.941	0.855	0.948
	[B]	1.406	0.920	1.052	0.924	1.010	0.933	0.960	0.934	0.916	0.938	0.867	0.939
	[C]	1.328	0.923	1.048	0.933	0.922	0.935	0.900	0.941	0.891	0.949	0.820	0.950
(30,20,15)	[A]	1.185	0.934	1.030	0.938	0.894	0.942	0.830	0.943	0.816	0.950	0.772	0.955
	[B]	1.192	0.932	1.046	0.935	0.895	0.940	0.849	0.941	0.818	0.950	0.772	0.954
	[C]	1.140	0.934	1.028	0.942	0.890	0.943	0.816	0.951	0.802	0.954	0.771	0.956
(50,30,20)	[A]	1.049	0.936	0.956	0.948	0.837	0.950	0.802	0.956	0.754	0.956	0.743	0.960
	[B]	1.050	0.936	0.980	0.947	0.840	0.948	0.802	0.951	0.756	0.954	0.743	0.958
	[C]	1.018	0.939	0.838	0.952	0.806	0.954	0.797	0.957	0.754	0.958	0.741	0.961
(50,40,30)	[A]	0.969	0.943	0.829	0.954	0.788	0.957	0.775	0.958	0.729	0.959	0.705	0.964
	[B]	1.010	0.941	0.836	0.954	0.802	0.956	0.777	0.957	0.743	0.959	0.730	0.964
	[C]	0.964	0.948	0.816	0.956	0.785	0.957	0.696	0.960	0.664	0.962	0.631	0.964
(80,40,20)	[A]	0.944	0.952	0.781	0.956	0.730	0.961	0.684	0.962	0.646	0.965	0.617	0.967
	[B]	0.957	0.948	0.789	0.956	0.750	0.959	0.687	0.961	0.660	0.963	0.621	0.964
	[C]	0.843	0.952	0.709	0.956	0.691	0.963	0.680	0.964	0.617	0.965	0.606	0.967
(80,60,30)	[A]	0.821	0.957	0.640	0.965	0.587	0.965	0.584	0.965	0.566	0.970	0.544	0.971
	[B]	0.833	0.953	0.675	0.957	0.630	0.964	0.601	0.965	0.568	0.969	0.546	0.970
	[C]	0.768	0.958	0.583	0.967	0.563	0.969	0.536	0.971	0.459	0.971	0.447	0.972

,

Table 6. The AIL (1st Col.) and CP (2nd Col.) results of $R\left(x\right)$.

$(\mathit{n},{\mathit{m}}_{\mathit{max}},{\mathit{m}}_{\mathit{min}})$	S	ACI-NA		ACI-NL		BCI				HPD
Prior→						1		2		1		2
Pop-1 ($\tau =0.5$)
(30,15,10)	[A]	0.583	0.953	0.666	0.944	0.519	0.959	0.510	0.960	0.465	0.965	0.454	0.966
	[B]	0.584	0.953	0.711	0.939	0.554	0.956	0.531	0.958	0.511	0.960	0.472	0.964
	[C]	0.588	0.952	0.730	0.937	0.576	0.953	0.543	0.957	0.535	0.958	0.516	0.960
(30,20,15)	[A]	0.501	0.961	0.578	0.953	0.456	0.966	0.441	0.968	0.419	0.970	0.401	0.972
	[B]	0.515	0.960	0.588	0.952	0.512	0.960	0.465	0.965	0.446	0.967	0.418	0.970
	[C]	0.507	0.961	0.581	0.953	0.501	0.961	0.464	0.965	0.445	0.967	0.414	0.971
(50,30,20)	[A]	0.427	0.969	0.491	0.963	0.406	0.971	0.399	0.972	0.384	0.974	0.355	0.977
	[B]	0.457	0.966	0.494	0.962	0.415	0.971	0.414	0.971	0.396	0.973	0.366	0.976
	[C]	0.471	0.965	0.502	0.961	0.455	0.966	0.439	0.968	0.416	0.970	0.384	0.974
(50,40,30)	[A]	0.394	0.973	0.420	0.970	0.388	0.973	0.366	0.976	0.346	0.978	0.325	0.980
	[B]	0.405	0.972	0.442	0.968	0.398	0.972	0.394	0.973	0.380	0.974	0.335	0.979
	[C]	0.404	0.972	0.434	0.969	0.395	0.973	0.383	0.974	0.374	0.975	0.327	0.980
(80,40,20)	[A]	0.368	0.976	0.396	0.973	0.345	0.978	0.322	0.980	0.316	0.981	0.309	0.982
	[B]	0.370	0.975	0.410	0.971	0.355	0.977	0.347	0.978	0.329	0.980	0.312	0.982
	[C]	0.379	0.974	0.417	0.970	0.370	0.975	0.354	0.977	0.343	0.978	0.316	0.981
(80,60,30)	[A]	0.290	0.984	0.375	0.975	0.271	0.986	0.264	0.987	0.259	0.987	0.252	0.988
	[B]	0.368	0.976	0.391	0.973	0.338	0.979	0.309	0.982	0.297	0.983	0.286	0.984
	[C]	0.320	0.981	0.391	0.973	0.316	0.981	0.308	0.982	0.288	0.984	0.258	0.986
Pop-1 ($\tau =1.5$)
(30,15,10)	[A]	0.569	0.954	0.584	0.953	0.518	0.968	0.467	0.965	0.455	0.966	0.446	0.967
	[B]	0.569	0.954	0.633	0.947	0.552	0.956	0.523	0.959	0.484	0.963	0.463	0.965
	[C]	0.588	0.952	0.652	0.945	0.576	0.953	0.543	0.957	0.535	0.958	0.516	0.960
(30,20,15)	[A]	0.498	0.962	0.505	0.961	0.449	0.967	0.436	0.968	0.418	0.970	0.396	0.973
	[B]	0.515	0.960	0.571	0.954	0.512	0.960	0.454	0.966	0.436	0.968	0.413	0.971
	[C]	0.505	0.961	0.541	0.957	0.498	0.962	0.454	0.966	0.419	0.970	0.408	0.971
(50,30,20)	[A]	0.406	0.971	0.429	0.969	0.406	0.972	0.391	0.973	0.381	0.974	0.351	0.977
	[B]	0.449	0.967	0.466	0.965	0.414	0.971	0.401	0.972	0.395	0.973	0.364	0.976
	[C]	0.464	0.965	0.472	0.965	0.447	0.967	0.429	0.969	0.399	0.972	0.381	0.974
(50,40,30)	[A]	0.392	0.973	0.403	0.972	0.372	0.975	0.355	0.977	0.345	0.978	0.320	0.981
	[B]	0.402	0.972	0.417	0.970	0.395	0.973	0.388	0.973	0.377	0.975	0.332	0.979
	[C]	0.396	0.973	0.415	0.971	0.393	0.973	0.380	0.974	0.363	0.976	0.326	0.980
(80,40,20)	[A]	0.367	0.976	0.396	0.973	0.339	0.979	0.320	0.981	0.314	0.981	0.307	0.982
	[B]	0.367	0.976	0.400	0.972	0.354	0.977	0.346	0.978	0.323	0.980	0.308	0.982
	[C]	0.372	0.975	0.400	0.972	0.368	0.976	0.346	0.978	0.323	0.980	0.314	0.981
(80,60,30)	[A]	0.277	0.985	0.313	0.981	0.270	0.986	0.262	0.987	0.259	0.987	0.251	0.988
	[B]	0.362	0.976	0.388	0.973	0.311	0.982	0.304	0.982	0.286	0.984	0.282	0.985
	[C]	0.317	0.981	0.343	0.978	0.309	0.982	0.288	0.984	0.285	0.985	0.257	0.987
Pop-2 ($\tau =0.5$)
(30,15,10)	[A]	0.837	0.949	1.217	0.930	0.832	0.950	0.807	0.951	0.770	0.953	0.744	0.954
	[B]	0.909	0.946	1.322	0.925	0.841	0.949	0.807	0.951	0.780	0.952	0.773	0.953
	[C]	0.919	0.945	1.428	0.920	0.889	0.947	0.857	0.948	0.826	0.950	0.784	0.952
(30,20,15)	[A]	0.826	0.950	1.074	0.938	0.750	0.954	0.738	0.954	0.727	0.955	0.697	0.956
	[B]	0.830	0.950	1.189	0.932	0.799	0.951	0.770	0.953	0.741	0.954	0.701	0.956
	[C]	0.829	0.950	1.166	0.933	0.768	0.953	0.750	0.954	0.739	0.954	0.698	0.956
(50,30,20)	[A]	0.738	0.954	0.919	0.945	0.722	0.955	0.711	0.956	0.686	0.957	0.674	0.957
	[B]	0.768	0.953	0.950	0.944	0.725	0.955	0.712	0.956	0.693	0.957	0.675	0.957
	[C]	0.772	0.953	1.013	0.941	0.725	0.955	0.717	0.955	0.698	0.956	0.679	0.957
(50,40,30)	[A]	0.710	0.956	0.819	0.950	0.678	0.957	0.644	0.959	0.617	0.960	0.586	0.962
	[B]	0.726	0.955	0.853	0.949	0.713	0.956	0.705	0.956	0.681	0.957	0.664	0.958
	[C]	0.724	0.955	0.823	0.950	0.707	0.956	0.698	0.956	0.663	0.958	0.653	0.959
(80,40,20)	[A]	0.648	0.959	0.736	0.954	0.631	0.960	0.615	0.960	0.575	0.962	0.563	0.963
	[B]	0.667	0.958	0.766	0.953	0.649	0.959	0.634	0.959	0.611	0.961	0.582	0.962
	[C]	0.710	0.956	0.810	0.951	0.655	0.958	0.641	0.959	0.611	0.961	0.583	0.962
(80,60,30)	[A]	0.525	0.965	0.706	0.956	0.520	0.965	0.496	0.966	0.437	0.969	0.410	0.970
	[B]	0.612	0.961	0.735	0.954	0.572	0.963	0.555	0.963	0.536	0.964	0.511	0.966
	[C]	0.579	0.962	0.716	0.955	0.540	0.964	0.535	0.964	0.521	0.965	0.508	0.967
Pop-2 ($\tau =1.5$)
(30,15,10)	[A]	0.835	0.949	1.188	0.932	0.813	0.951	0.802	0.951	0.758	0.953	0.711	0.956
	[B]	0.904	0.946	1.234	0.930	0.836	0.949	0.802	0.951	0.775	0.952	0.712	0.955
	[C]	0.912	0.946	1.258	0.928	0.867	0.948	0.831	0.950	0.788	0.952	0.714	0.955
(30,20,15)	[A]	0.804	0.951	1.022	0.940	0.743	0.954	0.732	0.955	0.701	0.956	0.694	0.956
	[B]	0.811	0.951	1.070	0.938	0.773	0.953	0.757	0.953	0.709	0.956	0.697	0.956
	[C]	0.808	0.951	1.063	0.938	0.766	0.953	0.749	0.954	0.701	0.956	0.696	0.956
(50,30,20)	[A]	0.728	0.955	0.758	0.953	0.716	0.955	0.691	0.957	0.678	0.957	0.667	0.958
	[B]	0.764	0.953	0.863	0.948	0.719	0.955	0.694	0.956	0.678	0.957	0.669	0.958
	[C]	0.768	0.953	0.884	0.947	0.719	0.955	0.698	0.956	0.679	0.957	0.669	0.958
(50,40,30)	[A]	0.707	0.956	0.737	0.954	0.656	0.958	0.629	0.960	0.612	0.961	0.582	0.962
	[B]	0.720	0.955	0.756	0.953	0.709	0.956	0.684	0.957	0.668	0.958	0.658	0.959
	[C]	0.717	0.955	0.750	0.954	0.704	0.956	0.672	0.958	0.658	0.958	0.641	0.959
(80,40,20)	[A]	0.636	0.959	0.654	0.958	0.623	0.960	0.575	0.962	0.566	0.963	0.559	0.964
	[B]	0.658	0.958	0.705	0.956	0.647	0.959	0.618	0.960	0.597	0.961	0.569	0.963
	[C]	0.690	0.957	0.713	0.956	0.653	0.959	0.624	0.960	0.609	0.961	0.570	0.962
(80,60,30)	[A]	0.522	0.965	0.536	0.964	0.497	0.966	0.485	0.967	0.425	0.970	0.402	0.971
	[B]	0.585	0.962	0.648	0.959	0.567	0.963	0.548	0.964	0.520	0.965	0.498	0.967
	[C]	0.557	0.963	0.639	0.959	0.538	0.964	0.534	0.964	0.518	0.965	0.460	0.969

and

Table 7. The AIL (1st Col.) and CP (2nd Col.) results of $h\left(x\right)$.

$(\mathit{n},{\mathit{m}}_{\mathit{max}},{\mathit{m}}_{\mathit{min}})$	S	ACI-NA		ACI-NL		BCI				HPD
Prior→						1		2		1		2
Pop-1 ($\tau =0.5$)
(30,15,10)	[A]	0.760	0.932	0.669	0.947	0.607	0.954	0.590	0.958	0.567	0.959	0.503	0.961
	[B]	0.708	0.937	0.621	0.951	0.572	0.958	0.550	0.959	0.497	0.959	0.482	0.962
	[C]	0.781	0.926	0.676	0.944	0.628	0.954	0.615	0.956	0.580	0.958	0.570	0.960
(30,20,15)	[A]	0.627	0.939	0.608	0.952	0.544	0.959	0.539	0.960	0.483	0.962	0.444	0.964
	[B]	0.616	0.945	0.536	0.953	0.507	0.959	0.475	0.962	0.465	0.963	0.426	0.966
	[C]	0.619	0.940	0.575	0.952	0.536	0.959	0.528	0.961	0.482	0.963	0.438	0.965
(50,30,20)	[A]	0.534	0.952	0.497	0.956	0.492	0.962	0.443	0.963	0.419	0.965	0.388	0.966
	[B]	0.532	0.953	0.494	0.960	0.484	0.963	0.433	0.963	0.408	0.966	0.372	0.966
	[C]	0.596	0.948	0.503	0.956	0.500	0.962	0.465	0.962	0.426	0.965	0.408	0.966
(50,40,30)	[A]	0.522	0.957	0.468	0.962	0.438	0.963	0.420	0.964	0.403	0.966	0.354	0.967
	[B]	0.516	0.958	0.430	0.963	0.415	0.963	0.393	0.968	0.386	0.970	0.340	0.972
	[C]	0.521	0.957	0.460	0.962	0.425	0.963	0.415	0.964	0.395	0.966	0.342	0.967
(80,40,20)	[A]	0.513	0.959	0.426	0.963	0.394	0.965	0.391	0.968	0.362	0.970	0.329	0.972
	[B]	0.468	0.961	0.415	0.967	0.384	0.969	0.366	0.969	0.339	0.972	0.326	0.973
	[C]	0.479	0.961	0.420	0.963	0.386	0.966	0.377	0.968	0.349	0.970	0.326	0.973
(80,60,30)	[A]	0.458	0.961	0.410	0.968	0.363	0.971	0.358	0.973	0.313	0.973	0.301	0.975
	[B]	0.393	0.965	0.331	0.973	0.302	0.975	0.282	0.976	0.269	0.976	0.260	0.978
	[C]	0.443	0.961	0.347	0.968	0.332	0.974	0.320	0.974	0.300	0.975	0.268	0.975
Pop-1 ($\tau =1.5$)
(30,15,10)	[A]	0.674	0.936	0.614	0.953	0.593	0.957	0.590	0.959	0.557	0.959	0.492	0.961
	[B]	0.668	0.939	0.575	0.958	0.566	0.958	0.548	0.959	0.482	0.959	0.473	0.963
	[C]	0.693	0.935	0.628	0.953	0.628	0.955	0.615	0.958	0.580	0.958	0.570	0.960
(30,20,15)	[A]	0.609	0.945	0.544	0.958	0.539	0.959	0.525	0.961	0.473	0.963	0.439	0.965
	[B]	0.597	0.948	0.518	0.958	0.502	0.961	0.466	0.963	0.445	0.964	0.419	0.966
	[C]	0.608	0.945	0.538	0.958	0.529	0.960	0.509	0.962	0.472	0.963	0.431	0.965
(50,30,20)	[A]	0.526	0.956	0.496	0.960	0.477	0.962	0.421	0.964	0.417	0.966	0.386	0.967
	[B]	0.522	0.956	0.487	0.962	0.453	0.963	0.410	0.964	0.404	0.966	0.368	0.967
	[C]	0.535	0.955	0.502	0.960	0.495	0.962	0.459	0.963	0.425	0.966	0.404	0.966
(50,40,30)	[A]	0.479	0.957	0.459	0.962	0.427	0.964	0.408	0.965	0.399	0.966	0.351	0.968
	[B]	0.434	0.958	0.417	0.963	0.410	0.964	0.393	0.968	0.365	0.970	0.335	0.972
	[C]	0.461	0.958	0.454	0.962	0.416	0.964	0.408	0.965	0.380	0.967	0.341	0.968
(80,40,20)	[A]	0.427	0.960	0.401	0.964	0.391	0.967	0.391	0.969	0.346	0.971	0.326	0.973
	[B]	0.420	0.963	0.388	0.968	0.368	0.969	0.359	0.970	0.326	0.973	0.312	0.975
	[C]	0.425	0.962	0.390	0.965	0.378	0.967	0.368	0.969	0.328	0.972	0.322	0.974
(80,60,30)	[A]	0.415	0.964	0.368	0.969	0.358	0.972	0.321	0.973	0.301	0.975	0.297	0.976
	[B]	0.347	0.970	0.312	0.974	0.288	0.975	0.275	0.976	0.268	0.977	0.239	0.978
	[C]	0.415	0.965	0.335	0.971	0.331	0.974	0.302	0.974	0.298	0.975	0.268	0.976
Pop-2 ($\tau =0.5$)
(30,15,10)	[A]	1.643	0.912	1.255	0.931	1.039	0.941	0.990	0.944	0.942	0.946	0.896	0.948
	[B]	1.398	0.924	1.090	0.939	0.946	0.946	0.925	0.947	0.918	0.947	0.863	0.950
	[C]	1.514	0.918	1.181	0.934	1.038	0.941	0.944	0.946	0.928	0.947	0.886	0.949
(30,20,15)	[A]	1.339	0.927	1.082	0.939	0.919	0.947	0.870	0.949	0.843	0.951	0.798	0.953
	[B]	1.225	0.932	1.045	0.941	0.915	0.947	0.851	0.950	0.828	0.951	0.779	0.954
	[C]	1.370	0.925	1.087	0.939	0.920	0.947	0.907	0.948	0.861	0.950	0.820	0.952
(50,30,20)	[A]	1.160	0.935	0.998	0.943	0.864	0.950	0.849	0.950	0.790	0.953	0.766	0.954
	[B]	1.045	0.941	0.892	0.948	0.840	0.951	0.826	0.952	0.778	0.954	0.761	0.955
	[C]	1.081	0.939	0.981	0.944	0.860	0.950	0.841	0.951	0.787	0.953	0.764	0.954
(50,40,30)	[A]	0.973	0.944	0.860	0.950	0.837	0.951	0.822	0.952	0.775	0.954	0.753	0.955
	[B]	0.943	0.946	0.843	0.951	0.831	0.951	0.737	0.956	0.675	0.959	0.643	0.960
	[C]	0.957	0.945	0.854	0.950	0.832	0.951	0.808	0.952	0.761	0.955	0.740	0.956
(80,40,20)	[A]	0.891	0.948	0.829	0.951	0.766	0.954	0.725	0.956	0.675	0.959	0.619	0.961
	[B]	0.890	0.948	0.743	0.956	0.709	0.957	0.700	0.958	0.649	0.960	0.615	0.962
	[C]	0.923	0.947	0.840	0.951	0.791	0.953	0.728	0.956	0.675	0.959	0.631	0.961
(80,60,30)	[A]	0.886	0.949	0.735	0.956	0.666	0.959	0.625	0.961	0.617	0.962	0.578	0.964
	[B]	0.787	0.953	0.618	0.962	0.576	0.964	0.564	0.964	0.475	0.968	0.433	0.971
	[C]	0.875	0.949	0.725	0.956	0.608	0.962	0.600	0.962	0.578	0.964	0.568	0.964
Pop-2 ($\tau =1.5$)
(30,15,10)	[A]	1.446	0.922	1.049	0.941	1.002	0.943	0.949	0.946	0.928	0.947	0.892	0.948
	[B]	1.368	0.925	0.954	0.945	0.939	0.946	0.925	0.947	0.914	0.947	0.843	0.951
	[C]	1.419	0.923	1.044	0.941	0.951	0.945	0.928	0.947	0.924	0.947	0.879	0.949
(30,20,15)	[A]	1.219	0.932	0.945	0.946	0.909	0.947	0.854	0.950	0.839	0.951	0.795	0.953
	[B]	1.174	0.935	0.945	0.946	0.903	0.948	0.839	0.951	0.824	0.952	0.766	0.954
	[C]	1.225	0.932	0.947	0.946	0.917	0.947	0.871	0.949	0.841	0.951	0.796	0.953
(50,30,20)	[A]	1.009	0.943	0.902	0.948	0.859	0.950	0.828	0.951	0.780	0.953	0.765	0.954
	[B]	0.978	0.944	0.863	0.950	0.837	0.951	0.822	0.952	0.772	0.954	0.752	0.955
	[C]	0.985	0.944	0.900	0.948	0.856	0.950	0.822	0.952	0.778	0.954	0.764	0.955
(50,40,30)	[A]	0.970	0.945	0.855	0.950	0.824	0.952	0.798	0.953	0.766	0.954	0.735	0.956
	[B]	0.935	0.946	0.840	0.951	0.808	0.952	0.724	0.956	0.674	0.959	0.638	0.961
	[C]	0.940	0.946	0.849	0.950	0.823	0.952	0.797	0.953	0.748	0.955	0.729	0.956
(80,40,20)	[A]	0.869	0.949	0.804	0.953	0.753	0.955	0.711	0.957	0.649	0.960	0.614	0.962
	[B]	0.837	0.951	0.720	0.957	0.708	0.957	0.688	0.958	0.617	0.962	0.610	0.962
	[C]	0.909	0.947	0.813	0.952	0.759	0.955	0.717	0.957	0.662	0.959	0.628	0.961
(80,60,30)	[A]	0.824	0.952	0.696	0.958	0.648	0.960	0.620	0.961	0.583	0.963	0.561	0.964
	[B]	0.693	0.958	0.596	0.963	0.566	0.964	0.552	0.965	0.461	0.969	0.423	0.971
	[C]	0.802	0.953	0.659	0.960	0.603	0.962	0.599	0.962	0.574	0.964	0.560	0.964

present the AIL and CP. A thorough analysis of Table 2, Table 3, Table 4, Table 5, Table 6 and Table 7, we provide the following key methodological insights into the efficiency of the estimators as follows:

• The estimation procedures demonstrate robust performance for $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ across all simulated scenarios. All estimation accuracy improves as the sample size n increases and the total number of censored units ${\sum }_{i=1}^{m_{max}}{S}_i$ decreases, underscoring the impact of sample size and censoring intensity on the quality of inference.
• As the effective sample size $m_{min}$ (or $m_{max}$) increase, the estimation results for $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ improve consistently.
• With increasing threshold level $\tau$, the RMSE, MAB, and AIL values for $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ decrease, whereas their CPs increase.
• When the XLindley($\gamma$) values increase, the following patterns are observed:

  –

  The RMSEs and MABs of the model parameters $\gamma$ increase, whereas those of the reliability measures $R\left(x\right)$ and $h\left(x\right)$ decrease;

  –

  The AILs of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ increase;

  –

  The CPs of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ decrease.

• As $\gamma$ increases, the RMSE and MAB values of $\widehat{\gamma }$, as well as those of $R\left(x\right)$ and $h\left(x\right)$, increase. This behavior can be explained by the increasing sensitivity of the model and reliability measures to $\gamma$ at higher parameter levels. In particular, both $R\left(x\right)$ and $h\left(x\right)$ are nonlinear functions of $\gamma$, and therefore small estimation errors in $\widehat{\gamma }$ are amplified through error propagation, as suggested by the delta method. Moreover, larger values of $\gamma$ lead to steeper likelihood curvature based on G-PH-CS, which increases estimation variability and results in higher RMSE and MAB values for both the model parameter and the reliability measures.
• The use of informative gamma priors substantially enhances the performance of Bayesian estimators, resulting in tighter and more accurate HPD intervals. When prior knowledge is incorporated, Bayesian inference consistently surpasses classical approaches in estimating $\gamma$, $R\left(x\right)$, and $h\left(x\right)$, particularly under informative prior scenarios.
• Since Prior–2 exhibits lower variance compared to Prior–1, Bayesian estimation based on Prior–2 yields more precise posterior estimates of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ for both populations (Pop–$i,i=1,2$), outperforming the results obtained under alternative prior choices.
• Comparison of interval estimation methods reveals the following notes:

  –

  Credible intervals for $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ obtained via BCI/HPD outperform their asymptotic counterparts (ACI-NA/ACI-NL);

  –

  The ACI-NA outperformed ACI-NL for $\gamma$, $R\left(x\right)$, and $h\left(x\right)$, whereas the reverse holds for $R\left(x\right)$;

  –

  For moderate to large effective sample sizes, both ACI-NA and ACI-NL achieve coverage probabilities close to the nominal level.

  –

  For heavy censoring, the ACI-NA intervals tend to undercover, reflecting the increased skewness of the sampling distribution of the MLE. In contrast, the ACI-NL intervals exhibit improved coverage stability and reduced sensitivity to censoring intensity.

  –

  The HPD method consistently outperforms the BCI method for all parameters considered;

  –

  Overall, these findings suggest that while asymptotic normal approximations remain reasonable in moderately censored settings, the ACI-NL approach is preferable under severe censoring. Bayesian intervals consistently provide superior coverage performance, highlighting their robustness in complex censoring environments.

• The MCMC estimates and associated BCI/HPD interval estimates often exhibit improved stability in terms of the smallest RMSE, MAB, and AIL values as well as the highest interval coverages under heavy censoring or small effective sample sizes. However, these advantages are not universal and depend on the choice of prior distribution and the underlying censoring configuration. This fact should not be interpreted as uniformly superior forever, but rather as a complementary inference framework that may offer practical advantages in complex censoring scenarios where asymptotic approximations are less reliable.
• Comparing the censoring schemes listed in Table 1, it is noted that

  –

  For $\gamma$: When the primary objective is precise inference on the distributional parameter $\gamma$, right-censoring schemes (Test–C) are preferable. These schemes retain a larger proportion of early and mid-range failure times, which are most informative about the global scale of the lifetime distribution.

  –

  For $R\left(x\right)$: Left-censoring schemes (Test–A) provide superior performance for estimating $R\left(x\right)$, particularly at moderate time points. Early failures contribute substantial information about survival probabilities, explaining the improved efficiency observed under Test–A designs.

  –

  For $h\left(x\right)$: Middle-censoring schemes (Test–B) are most effective for hazard rate estimation. By preserving failure information around the central portion of the lifetime distribution, these schemes offer a balanced representation of local failure dynamics, which is crucial for accurately capturing hazard behavior.

  –

  In practice, the choice of a censoring scheme should therefore be guided by the practitioner’s primary inferential goal. When multiple objectives are of interest, middle-censoring designs (Test–B) offer a reasonable compromise, delivering stable performance across $\gamma$, $R\left(x\right)$, and $h\left(x\right)$. These guidelines provide actionable insight for designing efficient life-testing experiments under cost and time constraints.

• As the XLindley$\left(\gamma \right)$ parameter grows, the RMSE, MAB, and AIL values of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ increase, while their corresponding CP values decrease.
• Overall, the Bayesian methodology, and in particular the M-H algorithm, demonstrates strong robustness and reliability for conducting inference on both XLindley parameters and reliability measures when the proposed censored dataset is available.

5. Real-World Applications

In this section, two real-world datasets, one on physical studies and the other on engineering, are analyzed to demonstrate the practical utility of the proposed estimation methods. These applications can be represented as

• Physics Application: The monthly rainfall records from the Carrol rain gauge station in New South Wales serve as an important measure of regional climate variability. This information is valuable for tracking long-term weather patterns, guiding land use decisions, and supporting sustainable farming practices. This example records monthly rainfall (in millimeters) observed at the Carrol rain gauge station, New South Wales, Australia, covering the period from January 2000 to February 2007; see Table 8. This dataset was reanalyzed by Alotaibi et al. [25].
• Engineering Application: Analysis of vehicle fatalities in South Carolina offers an important perspective on regional variations in traffic-related mortality. Such examination is essential for enhancing transportation safety measures and allocating resources effectively to reduce preventable loss of life. Table 8 reports the number of vehicle fatalities across (in day) thirty-nine counties in South Carolina during 2012 (www-fars.nhtsa.dot.gov/States); see Mann [26].

Table 8. Data points of physics and engineering applications.

Application	Times
Physics		0.80		0.80	1.80	3.00	4.70	4.80	5.10	5.90	6.40	6.60
		6.70		6.90	7.60	7.70	8.02	9.50	10.1	11.2	12.0	12.7
		13.9		14.0	15.0	15.4	17.3	17.7	17.9	19.4	19.7	20.4
		21.1		22.7	23.8	23.8	24.5	24.5	25.7	27.2	28.6	29.2
		29.7		31.8	31.9	32.0	32.2	32.5	33.1	33.9	36.5	37.2
		37.7		39.4	39.5	41.6	41.6	42.5	44.5	46.3	49.5	49.9
		50.2		50.7	51.6	52.5	53.9	55.2	55.2	55.8	57.2	59.0
		59.7		62.3	62.8	65.8	67.9	71.6	73.7	73.8	75.5	76.1
		84.0		85.7	98.7
Engineering		0.1		0.2	0.3	0.4	0.4	0.5	0.6	0.6	0.8	0.9
		0.9		0.9	0.9	1.0	1.2	1.2	1.3	1.3	1.3	1.4
		1.5		1.6	1.6	1.7	1.7	2.0	2.0	2.2	2.3	2.6
		2.7		3.1	3.3	4.8	4.8	5.0	5.1	5.2	6.8

Table 8. Data points of physics and engineering applications.

Application	Times
Physics		0.80		0.80	1.80	3.00	4.70	4.80	5.10	5.90	6.40	6.60
		6.70		6.90	7.60	7.70	8.02	9.50	10.1	11.2	12.0	12.7
		13.9		14.0	15.0	15.4	17.3	17.7	17.9	19.4	19.7	20.4
		21.1		22.7	23.8	23.8	24.5	24.5	25.7	27.2	28.6	29.2
		29.7		31.8	31.9	32.0	32.2	32.5	33.1	33.9	36.5	37.2
		37.7		39.4	39.5	41.6	41.6	42.5	44.5	46.3	49.5	49.9
		50.2		50.7	51.6	52.5	53.9	55.2	55.2	55.8	57.2	59.0
		59.7		62.3	62.8	65.8	67.9	71.6	73.7	73.8	75.5	76.1
		84.0		85.7	98.7
Engineering		0.1		0.2	0.3	0.4	0.4	0.5	0.6	0.6	0.8	0.9
		0.9		0.9	0.9	1.0	1.2	1.2	1.3	1.3	1.3	1.4
		1.5		1.6	1.6	1.7	1.7	2.0	2.0	2.2	2.3	2.6
		2.7		3.1	3.3	4.8	4.8	5.0	5.1	5.2	6.8

Prior to applying the theoretical estimation procedures, the suitability of the XLindley distribution was examined using the BC dataset, summarized in Table 8. This preliminary assessment utilized the Kolmogorov–Smirnov ($\mathsf{KS}$) test along with its corresponding p-value to evaluate the goodness of the model; see the code script reported in Appendix B. In addition, the MLE of $\gamma$, along with its standard error (SE) and 95% ACI-NA/ACI-NL intervals (including the interval widths (IWs)), was computed (see

Table 9. Fit results of the XLindley model from physics and engineering datasets.

Data	MLE		95% ACI-NA			$\mathbf{KS}$
			95% ACI-NL
	Est.	SE	Low.	Upp.	IW	Statistic	$\mathbf{P}$ -Value
Physics	0.0559	0.0043	0.0474	0.0644	0.0170	0.0825	0.6248
			0.0480	0.0651	0.0171
Engineering	0.6896	0.0845	0.5234	0.8558	0.3324	0.1148	0.6835
			0.5419	0.8775	0.3356

). The results demonstrate that the XLindley distribution offers an excellent fit to both the physics and engineering datasets.

Using graphical visualization for conveying model-fitting, Figure 5 shows diagnostic plots for both datasets, including (a) empirical and fitted survival functions $R\left(y\right)$, (b) probability–probability (PP) plot, (c) quantile–quantile (QQ) plot, (d) empirical and fitted scaled–TTT lines, (e) likelihood contour map, and (f) boxplot embedded within a violin representation. Figure 5a–c confirms that the XLindley distribution adequately characterizes the observed data in both physics and engineering applications. Moreover, Figure 5d indicates an increasing hazard function, consistent with the XLindley theoretical shapes. The contour maps in Figure 5e further confirm the existence and uniqueness of the MLE $\widehat{\gamma }$ obtained from the physics and engineering datasets, justifying its use as an initial value in subsequent numerical routines to enhance computational efficiency and estimation accuracy. Finally, Figure 5f shows that the physics dataset exhibits moderate right skewness, while the engineering dataset exhibits strong right skewness. For clarity and better visualization of the fit results, the numerical summaries previously reported in Table 9 are presented graphically in Figure 5.

To calculate the acquired estimates of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ for the full physics and engineering datasets, G-PH-CS samples were generated under different specifications of $\mathbf{S}$ and $\tau$, with $(m_{min},m_{max})=(20,,33)$ for the physics data and $(15,,20)$ for the engineering data (see

Table 10. Different G-PH-CS samples from physics and engineering datasets.

Data	Sample	S	$\mathit{\tau }\left(\mathit{d}\right)$	${\mathcal{S}}^{*}$	${\mathcal{T}}^{*}$	Censored Times
Physics	$\mathsf{S}\left[1\right]$	$(5^{10},0^{23})$	30(13)	0	49.5	0.80, 1.80, 3.00, 4.70, 7.60, 9.50, 11.2, 17.9, 19.4, 22.7,
						25.7, 28.6, 29.2, 33.9, 39.4, 39.5, 41.6, 42.5, 46.3, 49.5
	$\mathsf{S}\left[2\right]$	$(0^{12},5^{10},0^{11})$	80(30)	3	80	0.80, 0.80, 1.80, 3.00, 4.70, 4.80, 5.10, 5.90, 6.40, 6.60,
						6.70, 6.90, 7.60, 10.1, 13.9, 17.7, 25.7, 31.9, 33.1, 37.7,
						39.4, 42.5, 49.5, 50.2, 57.2, 59.0, 59.7, 73.7, 75.5, 76.1
	$\mathsf{S}\left[3\right]$	$(0^{23},5^{10})$	45(33)	0	44.5	0.80, 0.80, 1.80, 3.00, 4.70, 4.80, 5.10, 5.90, 6.40, 6.60,
						6.70, 6.90, 7.60, 7.70, 8.20, 9.50, 10.1, 11.2, 12.0, 12.7,
						13.9, 14.0, 15.0, 15.4, 17.7, 19.7, 29.2, 32.2, 36.5, 37.7,
						39.4, 41.6, 44.5
Engineering	$\mathsf{S}\left[1\right]$	$(2^9,0^{11})$	2.0(11)	0	3.3	0.1, 0.3, 0.5, 0.6, 0.8, 0.9, 0.9, 1.0, 1.3, 1.5,
						1.7, 2.0, 2.3, 2.7, 3.3
	$\mathsf{S}\left[2\right]$	$(0^5,2^9,0^6)$	5.0(18)	2	5.0	0.1, 0.2, 0.3, 0.4, 0.4, 0.5, 0.6, 0.9, 0.9, 1.0,
						1.4, 1.6, 1.7, 2.0, 2.3, 2.7, 3.1, 4.8
	$\mathsf{S}\left[3\right]$	$(0^{11},2^9)$	3.2(20)	0	3.1	0.1, 0.2, 0.3, 0.4, 0.4, 0.5, 0.6, 0.6, 0.8, 0.9,
						0.9, 0.9, 1.2, 1.4, 1.5, 1.6, 2.0, 2.3, 2.7, 3.1

). Since no prior information on the XLindley$\left(\gamma \right)$ parameter was available for these datasets, Bayesian estimation was performed using ${\Im }^{\circ }=5,000$ and ${\Im }^{•}=30,000$.

In Table 10, both point estimates (with SEs) and interval estimates (with IWs) of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ were obtained via maximum likelihood and Bayesian approaches (see

Table 11. Estimates of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ from physics and engineering datasets.

Sample	Par.	MLE		MCMC		ACI-NA			BCI
						ACI-NL			HPD
				Est.	SE	Low.	Upp.	IW	Low.	Upp.	IW
Physics Data
$\mathsf{S}\left[1\right]$	$\gamma$	0.0524	0.0077	0.0522	0.0061	0.0374	0.0674	0.0300	0.0405	0.0645	0.0240
						0.0393	0.0698	0.0304	0.0401	0.0640	0.0239
	$R\left(x\right)$	0.9426	0.0125	0.9516	0.0099	0.9181	0.9671	0.0489	0.9306	0.9695	0.0389
						0.9185	0.9674	0.0489	0.9323	0.9707	0.0384
	$h\left(x\right)$	0.6797	0.0036	0.0141	0.0029	0.6727	0.6868	0.0141	0.0090	0.0202	0.0112
						0.6727	0.6868	0.0141	0.0086	0.0197	0.0111
$\mathsf{S}\left[2\right]$	$\gamma$	0.0340	0.0040	0.0340	0.0037	0.0262	0.0418	0.0156	0.0270	0.0414	0.0143
						0.0270	0.0428	0.0158	0.0269	0.0412	0.0143
	$R\left(x\right)$	0.9426	0.0048	0.9776	0.0044	0.9332	0.9520	0.0188	0.9682	0.9855	0.0173
						0.9332	0.9520	0.0188	0.9686	0.9858	0.0172
	$h\left(x\right)$	0.6797	0.0014	0.0066	0.0013	0.6770	0.6825	0.0055	0.0043	0.0093	0.0050
						0.6770	0.6825	0.0055	0.0043	0.0092	0.0050
$\mathsf{S}\left[3\right]$	$\gamma$	0.0428	0.0047	0.0428	0.0043	0.0336	0.0520	0.0185	0.0347	0.0513	0.0167
						0.0345	0.0531	0.0186	0.0344	0.0510	0.0166
	$R\left(x\right)$	0.9426	0.0067	0.9660	0.0061	0.9295	0.9557	0.0263	0.9533	0.9770	0.0237
						0.9295	0.9558	0.0263	0.9543	0.9777	0.0234
	$h\left(x\right)$	0.6797	0.0019	0.0100	0.0018	0.6759	0.6835	0.0076	0.0068	0.0136	0.0068
						0.6759	0.6835	0.0076	0.0066	0.0134	0.0068
Engineering Data
$\mathsf{S}\left[1\right]$	$\gamma$	0.6508	0.1242	0.6477	0.1040	0.4073	0.8942	0.4869	0.4540	0.8600	0.4059
						0.4477	0.9460	0.4983	0.4514	0.8563	0.4049
	$R\left(x\right)$	0.9426	0.0110	0.9595	0.0092	0.9210	0.9642	0.0432	0.9404	0.9761	0.0357
						0.9213	0.9644	0.0432	0.9420	0.9775	0.0355
	$h\left(x\right)$	0.6797	0.1150	0.4166	0.0960	0.4543	0.9051	0.4508	0.2438	0.6174	0.3736
						0.4879	0.9470	0.4591	0.2297	0.6008	0.3711
$\mathsf{S}\left[2\right]$	$\gamma$	0.5780	0.0989	0.5768	0.0848	0.3842	0.7718	0.3876	0.4177	0.7500	0.3323
						0.4134	0.8082	0.3949	0.4154	0.7460	0.3306
	$R\left(x\right)$	0.9426	0.0085	0.9657	0.0073	0.9258	0.9593	0.0335	0.9505	0.9790	0.0286
						0.9260	0.9595	0.0335	0.9509	0.9793	0.0284
	$h\left(x\right)$	0.6797	0.0887	0.3516	0.0760	0.5058	0.8536	0.3478	0.2141	0.5110	0.2969
						0.5263	0.8779	0.3516	0.2111	0.5059	0.2948
$\mathsf{S}\left[3\right]$	$\gamma$	0.5467	0.0874	0.5470	0.0779	0.3754	0.7181	0.3427	0.4037	0.7079	0.3042
						0.3996	0.7480	0.3484	0.3980	0.6996	0.3016
	$R\left(x\right)$	0.9426	0.0075	0.9683	0.0066	0.9280	0.9572	0.0292	0.9543	0.9801	0.0258
						0.9281	0.9573	0.0292	0.9550	0.9806	0.0255
	$h\left(x\right)$	0.6797	0.0772	0.3251	0.0688	0.5284	0.8310	0.3026	0.2030	0.4710	0.2680
						0.5441	0.8492	0.3051	0.1984	0.4631	0.2647

). For consistency, results are reported at $x=5$ for the physics data and $x=0.1$ for the engineering data. The findings indicate that Bayesian estimates—particularly those derived from the Markovian sampling process—consistently outperform their frequentist counterparts, yielding smaller standard errors and narrower intervals, thereby demonstrating greater inferential efficiency.

To illustrate the existence and uniqueness of the proposed MLEs $\widehat{\gamma }$, Figure 6 displays contour plots of the log-likelihood equation of $\gamma$ under the censoring schemes $\mathsf{S}\left[i\right],i=1,2,3$, generated in Table 10. Across different values of $\gamma$ and for all G-PH-CS datasets in Table 10, Figure 6 confirms that the estimates $\widehat{\gamma }$ exist and are unique. These results are matched with the same in Table 11 and further suggest that the values of $\widehat{\gamma }$ obtained for each sample can serve as effective initial values in subsequent Bayesian iterations. To evaluate the Markov convergence chain samples of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$, trace and posterior density plots are presented in Figure 7. The subplots in Figure 7 are based on $\mathsf{S}\left[1\right]$ from the physics and engineering datasets as a representative case. Figure 7 shows that the 30,000 residual MCMC-based iterations for each parameter have sufficient convergence. They also reveal that the posterior estimates of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ behave symmetrically, negatively skewed, and positively skewed, respectively. Furthermore, descriptive statistics—including the mean, mode, first three quartiles, standard deviation (SD), and skewness—of $\gamma$, $R\left(x\right)$, and $h\left(x\right)$ are summarized in

Table 12. Summary for Bayesian iterations of $\gamma$ from physics and engineering datasets.

Sample	Par.	Mean	Mode	Quartiles			SD	Skew.
				1 st	2nd	3rd
Physics Data
$\mathsf{S}\left[1\right]$	$\gamma$	0.05216	0.05128	0.04799	0.05210	0.05617	0.00611	0.11557
	$R\left(x\right)$	0.95157	0.95335	0.94527	0.95204	0.95854	0.00994	−0.33606
	$h\left(x\right)$	0.01413	0.01362	0.01213	0.01400	0.01594	0.00286	0.33337
$\mathsf{S}\left[2\right]$	$\gamma$	0.03404	0.03370	0.03153	0.03397	0.03648	0.00368	0.08947
	$R\left(x\right)$	0.97758	0.97816	0.97475	0.97785	0.98070	0.00444	−0.32888
	$h\left(x\right)$	0.00663	0.00646	0.00572	0.00655	0.00745	0.00129	0.31824
$\mathsf{S}\left[3\right]$	$\gamma$	0.04279	0.04306	0.03986	0.04271	0.04560	0.00428	0.12103
	$R\left(x\right)$	0.96603	0.96586	0.96215	0.96636	0.97031	0.00611	−0.33757
	$h\left(x\right)$	0.00997	0.01002	0.00873	0.00988	0.01109	0.00176	0.33005
Engineering Data
$\mathsf{S}\left[1\right]$	$\gamma$	0.64775	0.62064	0.57593	0.64351	0.71640	0.10398	0.23778
	$R\left(x\right)$	0.95950	0.96203	0.95350	0.96001	0.96593	0.00918	−0.32872
	$h\left(x\right)$	0.41664	0.38979	0.34929	0.41082	0.47900	0.09603	0.35552
$\mathsf{S}\left[2\right]$	$\gamma$	0.57677	0.50690	0.51819	0.57285	0.63272	0.08482	0.22793
	$R\left(x\right)$	0.96572	0.96630	0.96097	0.96619	0.97085	0.00731	−0.33210
	$h\left(x\right)$	0.35163	0.28854	0.29831	0.34653	0.40087	0.07603	0.35280
$\mathsf{S}\left[3\right]$	$\gamma$	0.54704	0.50177	0.49347	0.54431	0.59737	0.07791	0.22422
	$R\left(x\right)$	0.96828	0.97222	0.96407	0.96864	0.97291	0.00663	−0.33701
	$h\left(x\right)$	0.32506	0.28413	0.27702	0.32117	0.36861	0.06878	0.35585

. All calculated metrics in Table 12 confirm and validate the convergence trends seen in Figure 7.

Again based on $\mathsf{S}\left[1\right]$ from the physics and engineering datasets as a representative case, in addition to trace plots depicted in Figure 7, convergence of the MCMC chains was formally assessed using the Gelman–Rubin potential scale reduction factor based on multiple parallel chains with dispersed initial values; see Figure 8. The PSRF values were all close to one, confirming satisfactory convergence and adequate mixing of the MCMC samples.

6. Conclusions, Practical Recommendations, and Future Research

This study developed a comprehensive and flexible reliability inference framework for the XLindley distribution under the G-PH-CS, addressing both theoretical and practical challenges commonly encountered in censored lifetime experiments. The main findings, recommendations, and potential research of the study can be summarized as follows:

6.1. Conclusions

Likelihood-based inference was established, and MLEs along with their ACIs were derived for the model parameter, reliability function, and hazard rate. In parallel, a Bayesian inference framework based on gamma priors was formulated, providing coherent uncertainty quantification and complementing the classical approach.

Extensive simulation studies demonstrated that both classical and Bayesian estimators perform reliably across a wide range of experimental configurations, including different threshold times, sample sizes, censoring limits $(m_{min},m_{max})$, and censoring schemes. The estimators consistently exhibited low bias, small mean squared error, reasonable interval lengths, and coverage probabilities close to nominal levels, confirming their robustness under repeated sampling.

Applications to real datasets from physics and engineering further illustrated that the proposed procedures are not only theoretically sound but also effective for modeling real-life failure mechanisms under realistic censoring constraints. By embedding the XLindley distribution within the G-PH-CS framework, the study broadened the applicability of hybrid censoring methodologies and provided practitioners with a flexible and reliable tool for lifetime data analysis.

An additional advantage of the proposed censoring framework is its inherent robustness to extreme observations. By terminating the experiment once at least $m_{min}$ failures are observed—even when $m_{max}$ failures are initially planned—and the G-PH-CS naturally limits the influence of excessively large lifetimes, which are often regarded as outliers in reliability applications. This design-based robustness yields stable likelihood-based inference without explicitly invoking robust likelihood formulations.

6.2. Practical Recommendations

From a practical standpoint, the results provide actionable guidance for designing efficient life-testing experiments under G-PH-CS. The simulation findings indicate that no single censoring scheme is universally optimal; rather, the choice should be guided by the practitioner’s primary inferential objective and operational constraints.

When the main goal is accurate estimation of the model parameter, right-censoring schemes are recommended due to their ability to retain informative early and mid-range failure times while reducing testing duration. For inference on the reliability function, left-censoring designs are preferable, as early failures carry substantial information about survival probabilities. Estimation of the hazard rate benefits most from middle-censoring schemes, which preserve failure information around the central portion of the lifetime distribution and provide balanced local information.

In real testing environments, censoring strategies are rarely selected solely to maximize statistical efficiency. Instead, practitioners must balance inferential precision against experimental cost, testing time, and logistical considerations. Middle-censoring designs often offer a reasonable compromise, delivering stable inferential performance across multiple targets while maintaining manageable experimental costs.

6.3. Future Research Directions

Several promising directions emerge from the present work. First, although the current study focuses on statistical efficiency, future research could formalize the design of censoring schemes by incorporating explicit cost functions and efficiency criteria, leading to optimal cost–efficiency trade-offs under G-PH-CS.

Second, the inherent robustness of the proposed censoring framework motivates further extensions that explicitly integrate robust likelihood techniques, such as M-estimation or divergence-based approaches, to enhance performance in the presence of severe outliers.

Additional extensions may include multivariate lifetime models, accelerated life-testing experiments, competing risks, and more complex system reliability structures. Exploring Bayesian model comparison, hierarchical prior formulations, and adaptive censoring strategies also constitutes valuable future work.

Overall, the proposed framework provides a solid methodological foundation for reliability analysis under generalized progressive hybrid censoring and opens several avenues for further theoretical development and practical innovation.