Optimum Progressive Data Analysis and Bayesian Inference for Unified Progressive Hybrid INH Censoring with Applications to Diamonds and Gold

Abstract

A novel unified progressive hybrid censoring is introduced to combine both progressive and hybrid censoring plans to allow flexible test termination either after a prespecified number of failures or at a fixed time. This work develops both frequentist and Bayesian inferential procedures for estimating the parameters, reliability, and hazard rates of the inverted Nadarajah–Haghighi lifespan model when a sample is produced from such a censoring plan. Maximum likelihood estimators are obtained through the Newton–Raphson iterative technique. The delta method, based on the Fisher information matrix, is utilized to build the asymptotic confidence intervals for each unknown quantity. In the Bayesian methodology, Markov chain Monte Carlo techniques with independent gamma priors are implemented to generate posterior summaries and credible intervals, addressing computational intractability through the Metropolis—Hastings algorithm. Extensive Monte Carlo simulations compare the efficiency and utility of frequentist and Bayesian estimates across multiple censoring designs, highlighting the superiority of Bayesian inference using informative prior information. Two real-world applications utilizing rare minerals from gold and diamond durability studies are examined to demonstrate the adaptability of the proposed estimators to the analysis of rare events in precious materials science. By applying four different optimality criteria to multiple competing plans, an analysis of various progressive censoring strategies that yield the best performance is conducted. The proposed censoring framework is effectively applied to real-world datasets involving diamonds and gold, demonstrating its practical utility in modeling the reliability and failure behavior of rare and high-value minerals.

1. Introduction

Censoring and life-testing are essential tools in analyzing time-to-event data when full observations are not always available. They enable researchers to make meaningful inferences from incomplete data, which is common in fields such as medicine, engineering, and social sciences. By allowing for efficient and cost-effective analysis, these methods support quicker decision-making without waiting for all outcomes to occur. Ultimately, censoring helps us to use available data wisely in order to improve public health, product reliability, and policy development. Hybrid censoring combines features from both Type-I (time) and Type-II (failure) censoring strategies, and is used when an experiment ends at the earlier of either a set number of failures r or a fixed time limit t. Although such a setup limits the total duration of the test to no more than t, it has a critical shortcoming in that there may be too few failures at t to draw meaningful inferences.

To address this, Childs et al. [1] named this method Type-I hybrid censoring (H-CT1), and introduced an alternative approach called Type-II hybrid censoring (H-CT2) in which the experiment is stopped at $max(Y_{m:m:n},t)$. While H-CT2 ensures that a minimum of m failures is observed, it may result in an extended experimental duration, especially when failures occur infrequently. To mitigate the limitations inherent in both H-CT1 and H-CT2, Chandrasekar et al. [2] proposed two enhanced censoring frameworks: generalized H-CT1 (GH-CT1) and generalized H-CT2 (GH-CT2), known as generalized hybrid censoring schemes. Under the GH-CT1 plan, n units are placed on a life test starting at time zero and three parameters are predetermined: two integers $r,m\in \{1,2,\dots ,n\}$ (where $r⩽m$) representing the required failure counts, and a censoring time $t\in (0,\infty )$. The termination rule is defined as follows: if $Y_{r:m:n}<t$, then the experiment concludes at $min(Y_{m:m:n},t)$; otherwise, it ends at $Y_{r:m:n}$. This formulation enhances H-CT1 by allowing the test to extend beyond t when necessary and ensures that at least r failures are observed. Conversely, GH-CT2 involves placing n units under observation with a fixed failure threshold $r\in \{1,2,\dots ,n\}$ and two prespecified time bounds $t_1,t_2\in (0,\infty )$ such that $t_1<t_2$. The test is stopped at $t_1$ (if $Y_{r:n}<t_1$), at $Y_{r:n}$ (if $t_1<Y_{r:n}<t_2$), or at $t_2$ (if $Y_{r:n}\ge t_2$). Although GH-CT2 modifies H-CT2 to ensure that the experiment concludes within the time interval $[t_1,t_2]$, it may still suffer from the issue of observing few or no failures before $t_2$, akin to the drawback in H-CT1. Despite these improvements, both generalized plans have residual limitations. In GH-CT1, only a single censoring time t is used, which may be insufficient to guarantee observation of m failures. Similarly, in GH-CT2, it remains possible that very few or no failures may occur before reaching $t_2$. To reconcile the strengths and weaknesses of both of these generalized censoring mechanisms, Balakrishnan et al. [3] proposed a unified hybrid censoring (UH-C) approach. Although UH-C effectively integrates the key elements of GH-CT1 and GH-CT2 into a single more flexible experimental framework, it does not provide the flexibility to remove experimental units before the experiment is terminated.

Type-II progressive censoring (P-CT2) is a valuable approach in reliability research, balancing statistical accuracy with practical efficiency; it reduces experimental time and cost, especially in biomedical and reliability studies. On the other hand, P-CT2 offers more informative and flexible data, enhances lifetime parameter estimates, and supports better experimental design under real-world constraints (see Balakrishnan and Cramer [4]). To avoid the inability to remove survival item(s) during the UH-C life test, Górny and Cramer [5] introduced a new progressive UH-C (UPH-CT1) plan by integrating features from both UH-C and P-CT2 plans to enhance flexibility in life-testing experiments. In Section 2, the UPH-CT1 procedure is discussed in detail.

The Nadarajah and Haghighi (NH$\left(\mathit{\theta }\right)$) distribution, where $\mathit{\theta }={(a,b)}^{\top }$ (introduced by Nadarajah and Haghighi [6]), is a flexible lifespan model derived by modifying the exponential distribution to accommodate increasing failure rates. Thanks to the mathematical tractability of the probability density function (PDF), cumulative distribution function (CDF), reliability function (RF), and hazard rate function (HRF) provided by the NH distribution, it has gained considerable attention in statistics and reliability studies. To model heavy-tailed (or decreasing) HRF scenarios by providing a reciprocal counterpart to the original NH distribution, Tahir et al. [7] proposed the inverted NH (INH) distribution. A lifetime random variable Y is said to have an INH $\left(\mathit{\theta }\right)$ distribution if its PDF (say, $g(·)$), CDF (say, $G(·)$), RF (say, ($R(·)$) and HRF (say, $h(·)$) at mission time x are respectively formulated as follows:

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}g(y;\mathit{\theta })=ab{y}^{-2}{\left(1+b{y}^{-1}\right)}^{a-1}exp\left(1-{\left(1+b{y}^{-1}\right)}^a\right);\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}y>0,$$

(1)

$$G(y;\mathit{\theta })=exp(1-{\left(1+b{y}^{-1}\right)}^a);$$

(2)

$$\begin{array}{l} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} R(x;\mathit{\theta })=1-exp\left(1-{\left(1+b{x}^{-1}\right)}^a\right);\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x>0,\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hspace{1em}\hfill \end{array}$$

(3)

                  and

$$h(x;\mathit{\theta })={\displaystyle \frac{ab{x}^{-2}{\left(1+b{x}^{-1}\right)}^{a-1}}{exp\left({\left(1+b{x}^{-1}\right)}^a-1\right)-1}},$$

(4)

where $a>0$ and $b>0$ denote the shape and scale parameters, respectively.

Several shapes of the PDF and HRF of the INH$\left(\mathit{\theta }\right)$ distribution obtained by fixing $b=1$ and taking various configurations of a are displayed in Figure 1. The figure implies that the INH distribution offers several benefits and advantages, particularly in modeling lifetime and reliability data with specific characteristics, including: (i) a flexible and analytically tractable model for lifetime and reliability data, particularly suited for scenarios with decreasing (or upside-down bathtub) hazard rates; (ii) capturing heavy-tailed behavior, making it valuable for modeling extreme or rare events; (iii) producing a better fit in inverse modeling contexts, such as when failure time is inversely related to stress (or strength); and (iv) serving as a generalization of simpler inverse lifetime distributions, providing broader applicability in statistical modeling. In the context of reliability data analysis, several works have analyzed this model; for example, see Elshahhat and Rastogi [8], Elshahhat and Mohammed [9], Abushal [10], Abushal and AL-Zaydi [11], and Azimi and Esmailian [12], among others.

This study presents several novel contributions from both theoretical and applied perspectives. The motivation behind this study arises from the critical need for flexible and information-rich life-testing schemes capable of accommodating the complexities of modern reliability data, particularly in high-value and rare material contexts such as diamond and gold durability studies. Although research efforts in the literature provide a reliable treatment for the INH distribution across various reliability scenarios, they also illuminate the INH’s applicability, especially in the presence of UPH-CT1 data. Benefiting from both the UPH-CT1, which guarantees a minimum number of observed failures while bounding the test within a fixed time duration, and from the INH model, which is highly flexible in capturing the upside-down bathtub and heavy-tailed hazard structures typical in reliability data, this work introduces and develops inference procedures. To achieve this goal using the proposed strategy, we apply both maximum likelihood and Bayesian inferential approaches for the model parameters along with associated reliability measures, with Bayesian computations performed via Markov chain Monte Carlo (MCMC) under gamma priors using the Metropolis–Hastings algorithm. Based on these inferential setups, we also estimate asymptotic and credible intervals for the same unknown subjects. Furthermore, utilizing several precision metrics, a comprehensive Monte Carlo simulation framework evaluates estimator performance across multiple censoring configurations, demonstrating the robustness of Bayesian estimators. Finally, we establish the practical relevance of the proposed methodology through real-life applications to gold and diamond datasets, along with identification of the optimal censoring strategy using multiple objective criteria.

The rest of this paper is structured as follows: Section 2 explores the mechanism of the UPH-CT1 plan; Section 2 and Section 4 introduce the frequentist and Bayes estimators for various parameters and reliability metrics; Section 5 summarizes the simulation results; Section 6 investigates the examination of diamond and gold datasets; Section 7 explores the problem of determining the best design; finally, Section 8 concludes the paper.

2. The UPH-CT1 Plan

Suppose that n identical units are put in a lifetime experiment at start time zero. We fix the integers $r,m\in 1,...,n$ (where $r<m$), two thresholds $t_1<t_2$ (where $t_1,t_2\in (0,\infty )$), and a P-CT2 plan $\mathbf{R}=(R_1,\dots ,R_m)$ that satisfies $n=m+{\sum }_{i=1}^m{R}_i$. It should be noted here that the removal mechanism in such UPH-CT1 data (by Balakrishnan and Cramer [4]) is performed in a manner similar to that of the standard P-CT2 scheme in such a way that one (or more) of the remaining elements (e.g., $R_i$) is removed as soon as the ith failure time is recorded. Next, the experimenter stops the test at termination point (say, $s^{★}$), such as

$$s^{★}=\left\{\begin{array}{ccc}{Y}_{m:m:n}\hfill & & \mathrm{if}{Y}_{m:n}<t_1,\\ t_1\hfill & & \hspace{1em}\hspace{1em}\hspace{1em} \mathrm{if}{Y}_{r:m:n}<t_1<Y_{m:m:n},\\ Y_{r:m:n}\hfill & & \hspace{1em}\hspace{1em}\mathrm{if}{t}_1<Y_{r:m:n}<t_2,\\ t_2\hfill & & \mathrm{if}{t}_2<Y_{r:m:n}.\end{array}\right.$$

Now, suppose that $\mathbf{y}$ is a UPH-CT1 data vector taken from an INH$\left(\mathit{\theta }\right)$ population with PDF (1) and CDF (2); then, the likelihood function (symbolized by $\mathcal{L}(·)$), where $y_i\equiv y_{i:m:n}$, can be formulated as

$$\mathcal{L}\left(\mathit{\theta }|\mathbf{y}\right)\propto \prod _{i=1}^{\mathcal{V}}g(y_i;\mathit{\theta }){\left[R(y_i;\mathit{\theta })\right]}^{R_i}{\left[R(s^{★};\mathit{\theta })\right]}^{R^{★}},$$

(5)

where $d_i$ denotes the number of failures up to $t_i$ (for $i=1,2$):

$$\mathcal{V}=\left\{\begin{array}{c}m;\hfill \\ d_1;\hfill \\ k;\hfill \\ d_2,\hfill \end{array}\right.\mathrm{and}{r}^{★}=\left\{\begin{array}{c}n-m-{\sum }_{i=1}^{m-1}{R}_i,\hfill \\ n-d_1-{\sum }_{i=1}^{d_1}{R}_i,\hfill \\ n-k-{\sum }_{i=1}^{k-1}{R}_i,\hfill \\ n-d_2-{\sum }_{i=1}^{d_2}{R}_i.\hfill \end{array}\right.$$

Górny and Cramer [5] stated that this new plan combines strengths from progressive, time-controlled, and failure-controlled censoring into a unified and efficient design. They also showed that UPH-CT1 overcomes limitations of the GH-CT1 and GH-CT2 plans, which risk either terminating too early with insufficient failure data, or extending excessively beyond acceptable test durations. Thus, UPH-CT1 increases the efficiency of parameter estimation and the overall robustness of statistical inference. In addition, its ability to ensure both a minimum number of failures and a fixed upper time bound enhances the precision of statistical examinations. Its flexibility in accommodating time, cost, and unit attrition constraints makes it highly applicable in diverse fields.

In recent years, several studies have made use of the UPH-CT1 mechanism to explore various statistical issues of parameter estimation, for example Lone et al. [13] for gamma-mixed Rayleigh distributions, Anwar et al. [14] (or inverted exponentiated Rayleigh distributions, Bayoud et al. [15] for Topp–Leone distributions, and Dutta et al. [16], along with references cited therein.

3. Likelihood Inference

This section introduces the MLEs and $(1-\varrho )100\%$ ACIs of the INH parameters a and b as well as the reliability time parameters $R\left(x\right)$ and $h\left(x\right)$.

3.1. Maximum Likelihood Estimators

Suppose that $\{(y_1,R_1),(y_2,R_2),\dots ,(y_{\mathcal{V}},R_{\mathcal{V}})\}$ is a UPH-CT1 sample with size $\mathcal{V}$ from an INH distribution with PDF and CDF given by (1) and (2), respectively; then, the likelihood function (5), where $y_i^{\circ }\equiv y_i^{-1}$, becomes

$$\begin{array}{c}\hfill \mathcal{L}\left(\mathit{\theta }|\mathbf{y}\right)\propto {\left(ab\right)}^{\mathcal{V}}{e}^{-{\sum }_{i=1}^V\psi \left(y_i^{\circ }\right)}\prod _{i=1}^{\mathcal{V}}{\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)}^{a-1}{\left[1-e^{1-\psi \left(y_i^{\circ }\right)}\right]}^{R_i}{\left[1-e^{1-\psi \left(S^{★}\right)}\right]}^{r^{★}},\end{array}$$

(6)

and its logarithm version (say, ${\mathcal{L}}^{★}(·)\propto log\mathcal{L}(·)$) is

$$\begin{array}{cc}\hfill {\mathcal{L}}^{★}& \propto \mathcal{V}log\left(ab\right)-\sum _{i=1}^V\psi \left(y_i^{\circ }\right)\hfill \\ \hfill & +\left(a-1\right)\sum _{i=1}^{\mathcal{V}}log\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)+\sum _{i=1}^{\mathcal{V}}{R}_{i}log\left(1-e^{1-\psi \left(y_i^{\circ }\right)}\right)+r^{★}log\left(1-e^{1-\psi \left(s^{★}\right)}\right),\hfill \end{array}$$

(7)

where $\psi \left(y_i^{\circ }\right)={\left(1+b{y}_i^{\circ }\right)}^a$ and $\psi \left(s^{★}\right)={\left(1+b{s}^{★}\right)}^a$.

As a sequence, the MLEs $\widehat{a}$ and $\widehat{b}$ of a and b, respectively, can be acquired by simultaneously solving the next two normal expressions, which are obtained by obtaining the first partial derivatives of (7) with respect to INH($a,b$), as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathcal{L}}^{★}}{\partial a}}& ={\displaystyle \frac{V}{a}}-\sum _{i=1}^V\psi \left(y_i^{\circ }\right)log\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)+\sum _{i=1}^{V}log\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)\hfill \\ \hfill & -\sum _{i=1}^V{R}_i\left[{\displaystyle \frac{\psi \left(y_i^{\circ }\right)log\left({\psi }^{\frac{1}{a}}\left(y_i^{\circ }\right)\right)e^{-\psi \left(y_i^{\circ }\right)}}{1-e^{1-\psi \left(y_i^{\circ }\right)}}}\right]-r^{★}\left[{\displaystyle \frac{\psi \left(s^{★}\right)log\left({\psi }^{\frac{1}{a}}\left(s^{★}\right)\right)e^{-\psi \left(s^{★}\right)}}{1-e^{1-\psi \left(s^{★}\right)}}}\right]\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathcal{L}}^{★}}{\partial b}}& ={\displaystyle \frac{V}{b}}-a\sum _{i=1}^V{y}_i^{\circ }\psi \left(y_i^{\circ }\right){\psi }^{-{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)+(a-1)\sum _{i=1}^V{\displaystyle \frac{y_i^{\circ }}{1+b{y}_i^{\circ }}}\hfill \\ \hfill & -\sum _{i=1}^V{R}_i\left[{\displaystyle \frac{a{y}_i^{\circ }\psi \left(y_i^{\circ }\right){\psi }^{-\frac{1}{a}}\left(y_i^{\circ }\right)e^{-\psi \left(y_i^{\circ }\right)}}{1-e^{1-\psi \left(y_i^{\circ }\right)}}}\right]-r^{★}\left[{\displaystyle \frac{a{y}_i^{\circ }\psi \left(s^{★}\right){\psi }^{-\frac{1}{a}}\left(s^{★}\right)e^{-\psi \left(s^{★}\right)}}{1-e^{1-\psi \left(s^{★}\right)}}}\right].\hfill \end{array}$$

(9)

Equations (8) and (9) show that the MLEs $\widehat{a}$ and $\widehat{b}$ do not have explicit expressions; for this purpose, we recommend using numerical iterative approaches such as the Newton–Raphson technique to find $\widehat{a}$ and $\widehat{b}$ numerically when the UPH-CT1 dataset is available.

Estimating the functions for the reliability $R\left(x\right)$ and hazard rate $h\left(x\right)$ is crucial in reliability analysis, as they respectively quantify the probability of survival beyond time x and the risk of failure at time x. These indices help practitioners to assess system durability, plan maintenance schedules, and evaluate failure risks under varying operational conditions. Using the invariance property of $\widehat{a}$ and $\widehat{b}$ by plugging these into the formulas for $R\left(x\right)$ and $h\left(x\right)$ provided by (3) and (4), the MLEs of $R\left(x\right)$ and $h\left(x\right)$ (for $x>0$) can be estimated as

$$\widehat{R}(x;\widehat{\mathit{\theta }})=1-exp\left(1-{\left(1+\widehat{b}{x}^{-1}\right)}^{\widehat{a}}\right)$$

and

$$\widehat{h}(x;\widehat{\mathit{\theta }})={\displaystyle \frac{\widehat{a}\widehat{b}{x}^{-2}{\left(1+\widehat{b}{x}^{-1}\right)}^{\widehat{a}-1}}{exp\left({\left(1+\widehat{b}{x}^{-1}\right)}^{\widehat{a}}-1\right)-1}},$$

respectively.

3.2. Approximate Interval Estimators

To estimate the two bounds of $(1-\varrho )100\%$ ACIs for a, b, $R\left(x\right)$, and $h\left(x\right)$, the asymptotic properties of the MLEs $\widehat{a}$, $\widehat{b}$, $\widehat{R}\left(x\right)$, and $\widehat{h}\left(x\right)$ based on the theory of large samples are used. The asymptotic distribution of INH($\widehat{\mathit{\theta }}$) follows a bivariate normal distribution with mean ($\widehat{\mathit{\theta }}$) with a $2\times 2$ variance-covariance (VC) matrix ${\mathcal{F}}^{-1}\left(\mathit{\theta }\right)$.

In practice, the VC information matrix ${\mathcal{F}}^{-1}\left(\mathit{\theta }\right)$ is approximated by ${\widehat{\mathcal{F}}}^{-1}\left(\widehat{\mathit{\theta }}\right)$ as

$$\begin{array}{c}\hfill {\widehat{\mathcal{F}}}^{-1}\left(\widehat{\mathit{\theta }}\right)={\left(\begin{array}{cc}-{\mathcal{F}}_{11}& -{\mathcal{F}}_{12}\\ -{\mathcal{F}}_{21}& -{\mathcal{F}}_{22}\end{array}\right)}_{(\widehat{\mathit{\theta }}=\widehat{\mathit{\theta }})}^{-1}=\left(\begin{array}{cc}{\widehat{ϵ}}_{11}& {\widehat{ϵ}}_{12}\\ {\widehat{ϵ}}_{21}& {\widehat{ϵ}}_{22}\end{array}\right),\end{array}$$

(10)

where the expressions of ${\mathcal{F}}_{ij},i,j=1,2,$ are provided in Appendix A.

Subsequently, the $(1-\varrho )100\%$ two-sided ACIs of a and b are

$$\widehat{a}\pm z_{0.5\varrho }\sqrt{{\widehat{ϵ}}_{11}}\mathrm{and}\widehat{b}\pm z_{0.5\varrho }\sqrt{{\widehat{ϵ}}_{22}},$$

(11)

where $z_{0.5\varrho }$ is the upper $0.5\varrho$th percentile point of the standard Gaussian distribution.

To estimate the $(1-\varrho )100\%$ ACIs of $R\left(x\right)$ and $h\left(x\right)$, it is necessary to estimate the variability associated with their respective estimators. A widely adopted approach for approximating the variance of functions that depend on estimated parameters is the delta method. Utilizing this method, we can express the approximate variances of the MLEs of $R\left(x\right)$ and $h\left(x\right)$ as follows:

$${\widehat{ϵ}}_R\approx \left[{\mathrm{\Psi }}_R{\mathcal{F}}^{-1}\left(\widehat{\mathit{\theta }}\right){\mathrm{\Psi }}_R^{\top }\right]{{|}_{(\mathit{\theta }=\widehat{\mathit{\theta }})}\mathrm{and}{\widehat{ϵ}}_h\approx \left[{\mathrm{\Psi }}_h{\mathcal{F}}^{-1}\left(\widehat{\mathit{\theta }}\right){\mathrm{\Psi }}_h^{\top }\right]|}_{(\mathit{\theta }=\widehat{\mathit{\theta }})},$$

respectively.

To achieves this, we must obtain ${\mathrm{\Psi }}_R=\left({\displaystyle \frac{\partial R}{\partial a}}{\displaystyle \frac{\partial R}{\partial b}}\right){|}_{(\mathit{\theta }=\widehat{\mathit{\theta }})}$ and ${\mathrm{\Psi }}_h=\left({\displaystyle \frac{\partial h}{\partial a}}{\displaystyle \frac{\partial h}{\partial b}}\right){|}_{(\mathit{\theta }=\widehat{\mathit{\theta }})}$ as

$${\displaystyle \frac{\partial R}{\partial a}}={\left(1+bx\right)}^{a}log\left(1+bx\right)exp\left(1-{\left(1+bx\right)}^a\right),$$

$${\displaystyle \frac{\partial R}{\partial b}}=ax{\left(1+bx\right)}^{a-1}exp\left(1-{\left(1+bx\right)}^a\right),$$

$${\displaystyle \frac{\partial h}{\partial a}}=b{(1+bx)}^{a-1}\left[1+alog(1+bx)\right],$$

and

$${\displaystyle \frac{\partial h}{\partial b}}=a{(1+bx)}^{a-1}\left[1+bx(a-1){(1+bx)}^{-1}\right].$$

As a result, at a confidence level of $(1-\varrho )100\%$, the two-sided ACI limits of $R\left(x\right)$ and $h\left(x\right)$ (for $x>0$) are respectively constructed as

$$\widehat{R}\left(x\right)\pm z_{0.5\varrho }\sqrt{{\widehat{ϵ}}_R}\mathrm{and}\widehat{h}\left(x\right)\pm z_{0.5\varrho }\sqrt{{\widehat{ϵ}}_h}.$$

4. Bayesian Inference

Bayesian analysis offers a versatile framework for parameter estimation by integrating historical information or expert opinion via prior distributions. It produces a comprehensive posterior distribution of the parameters, providing a more detailed insight into uncertainty than point estimates alone. This method is very useful when dealing with limited data or merging numerous sources of information. Assuming that the INH$\left(\mathit{\theta }\right)$ parameters are independent and have gamma distributions, namely, $a\sim Gamma({\alpha }_1,{\beta }_1)$ and $b\sim Gamma({\alpha }_2,{\beta }_2)$, this part develops the Bayes estimators and associated BCI estimators of a, b, $R\left(x\right)$, and $h\left(x\right)$. The joint prior PDF (say, $P(·)$) of a and b is

$$P\left(\mathit{\theta }\right)\propto a^{{\alpha }_1-1}{b}^{{\alpha }_2-1}{e}^{-({\beta }_{1}a+{\beta }_{2}b)},a,b>0,$$

(12)

where the hyperparameters $({\alpha }_i,{\beta }_i)>0,i=1,2$ are nonnegative and known beforehand.

It is important to remember here that incorporating the gamma density as a prior for the INH$\left(\mathit{\theta }\right)$ parameters offers considerable practical and theoretical advantages. Its structural adaptability allows it to encode varying levels of prior belief about each unknown subject, depending on the chosen hyperparameters.

From a computational standpoint, the gamma prior often leads to tractable posterior forms, which is particularly beneficial in high-dimensional settings involving multiple parameters, including the current framework. This computational convenience along with its interpretability justifies its common use in Bayesian inference; for more discussion, see Dey et al. [17].

Integrating (6) and (12) into the continuous Bayes theorem, the joint posterior PDF (say, $P^{★}(·)$) of a and b becomes

$$\begin{array}{cc}\hfill P^{★}\left(\mathit{\theta }|\mathbf{y}\right)& \propto a^{V+{\alpha }_1-1}{b}^{V+{\alpha }_2-1}{e}^{-\left({\beta }_{1}a+{\beta }_{2}b+{\sum }_{i=1}^V\psi \left(y_i^{\circ }\right)\right)}\hfill \\ \hfill & \times \prod _{i=1}^V{\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)}^{a-1}{\left[1-e^{1-\psi \left(y_i^{\circ }\right)}\right]}^{R_i}{\left[1-e^{1-\psi \left(S^{*}\right)}\right]}^{r^{*}},\hfill \end{array}$$

(13)

where its normalized term (say, $\mathcal{K}$) is provided by

$$\begin{array}{cc}\hfill \mathcal{K}={\int }_a^{\infty }{\int }_b^{\infty }& a^{V+{\alpha }_1-1}{b}^{V+{\alpha }_2-1}{e}^{-\left({\beta }_{1}a+{\beta }_{2}b+{\sum }_{i=1}^V\psi \left(y_i^{\circ }\right)\right)}\hfill \\ \hfill & \times \prod _{i=1}^V{\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)}^{a-1}{\left[1-e^{1-\psi \left(y_i^{\circ }\right)}\right]}^{R_i}{\left[1-e^{1-\psi \left(S^{*}\right)}\right]}^{r^{*}}\mathrm{d}a\mathrm{d}b.\hfill \end{array}$$

Owing to the analytical complexity of the joint likelihood PDF defined in (6), under the SEL it is not feasible to derive a closed-form Bayesian estimator of a, b, $R\left(x\right)$, or $h\left(x\right)$. Therefore, to approximate the posterior distributions of the INH parameters a and b, which do not have closed-form solutions due to the model’s complexity, we recommend employing the MCMC technique to numerically approximate the corresponding Bayes estimates as well as to construct the associated BCI estimates. At each iteration in the MCMC approach, candidate values for a and b are generated from normal proposal distributions and accepted or rejected based on their likelihood and prior contributions, producing a dependent sample from the joint posterior. To generate samples using the this method, conditional posterior distributions of a and b, symbolized as $P_a^{★}(·)$ and $P_b^{★}(·)$, must first be established as

$$\begin{array}{cc}\hfill P_a^{★}\left(a|b,\mathbf{y}\right)& \propto a^{V+{\alpha }_1-1}{e}^{-\left({\beta }_{1}a+{\sum }_{i=1}^V\psi \left(y_i^{\circ }\right)\right)}\hfill \\ \hfill & \times \prod _{i=1}^V{\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)}^{a-1}{\left[1-e^{1-\psi \left(y_i^{\circ }\right)}\right]}^{R_i}{\left[1-e^{1-\psi \left(S^{*}\right)}\right]}^{r^{*}}\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\hfill P_b^{★}\left(b|a,\mathbf{y}\right)& \propto b^{V+{\alpha }_2-1}{e}^{-\left({\beta }_{2}b+{\sum }_{i=1}^V\psi \left(y_i^{\circ }\right)\right)}\hfill \\ \hfill & \times \prod _{i=1}^V{\left({\psi }^{{\displaystyle \frac{1}{a}}}\left(y_i^{\circ }\right)\right)}^{a-1}{\left[1-e^{1-\psi \left(y_i^{\circ }\right)}\right]}^{R_i}{\left[1-e^{1-\psi \left(S^{*}\right)}\right]}^{r^{*}},\hfill \end{array}$$

(15)

respectively.

Given the complexity of the expressions in (14) and (15), it becomes evident that the full posterior PDFs of a and b do not conform to any standard or analytically tractable form. As such, direct sampling from the conditional densities $P_a^{★}(·)$ and $P_b^{★}(·)$, as defined in (14) and (15), respectively, is impractical using conventional simulation techniques. To circumvent this issue, we employ the Metropolis–Hastings (M-H) algorithm, incorporating a normal proposal kernel. This approach facilitates the generation of posterior samples for a and b, thereby enabling the estimation (both point and credible interval) of a, b, $R\left(x\right)$, and $h\left(x\right)$.

Now, a single UPH-CT1 dataset is generated from the INH(0.5, 1) distribution using a configuration of $(n,r,m)=(100,25,50)$, $(t_1,t_1)=(2,5)$, $R_i=1,i=1,2,\dots ,m)$, $({\alpha }_1,{\alpha }_2)=(2.5,5)$, and ${\beta }_i=5,i=1,2$. Figure 2 illustrates that the marginal posterior samples of the INH parameters a and b, obtained from (14) and (15), respectively, exhibit a distributional pattern that closely resembles the Gaussian (normal) distribution. This also supports our proposal for the same parameters. Algorithm 1 presents the generation steps of the MCMC sampler to simulate samples of a and b from (14) and (15).

Algorithm 1 MCMC Generation Steps

1: Input: Start with $𝔍=1$   2: Input: Assign starting points of a as $\widehat{a}$ and of b as $\widehat{b}$   3: Output: Get $a^{★}$ from $N(\widehat{a},{\widehat{ϵ}}_{11})$   4: Output: Calculate ${𝓁}_a=min\left[1,{\displaystyle \frac{P_a^{★}\left(a^{★}|b^{𝔍-1},\mathbf{y}\right)}{P_a^{★}\left(a^{𝔍-1}|b^{𝔍-1},\mathbf{y}\right)}}\right]$   5: Output: Get a uniform variate (say, u) from $U(0,1)$   6: Output: Store $a^{★}$   7: As $a^{\left(𝔍\right)}$   8: if  $u\le {𝓁}_a$  then   9:      Else set $a^{(𝔍-1)}$ 10: end if 11: Input: Redo Steps 3–10 for $b^{\left(𝔍\right)}$ from $P_b^{★}\left(b|a,\mathbf{y}\right)$ 12: Output: Obtain $R^{\left(𝔍\right)}$ from (3) and $h^{\left(𝔍\right)}$ from (4) 13: Input: Set $𝔍=𝔍+1$ 14: Input: Redo Steps 1–13 $\mathbb{M}$ times, and remove ${\mathbb{M}}^{•}$ as a burn-in size 15: Output: Compute $$\begin{array}{cc}\tilde{a}& ={\displaystyle \frac{1}{{\mathbb{M}}^{★★}}}{\sum }_{𝔍={\mathbb{M}}^{•}+1}^{\mathbb{M}}{a}^{\left(𝔍\right)},\hfill \\ \tilde{b}& ={\displaystyle \frac{1}{{\mathbb{M}}^{★★}}}{\sum }_{𝔍={\mathbb{M}}^{•}+1}^{\mathbb{M}}{b}^{\left(𝔍\right)},\hfill \\ \tilde{R}\left(x\right)& ={\displaystyle \frac{1}{{\mathbb{M}}^{★★}}}{\sum }_{𝔍={\mathbb{M}}^{•}+1}^{\mathbb{M}}{R}^{\left(𝔍\right)}\left(x\right),\hfill \end{array}$$ and $$\begin{array}{cc}\hfill \tilde{h}\left(x\right)& ={\displaystyle \frac{1}{{\mathbb{M}}^{★★}}}{\sum }_{𝔍={\mathbb{M}}^{•}+1}^{\mathbb{M}}{h}^{\left(𝔍\right)}\left(x\right),\hfill \end{array}$$ where ${\mathbb{M}}^{★★}=\mathbb{M}-{\mathbb{M}}^{•}$ 16: Input: Sort $a^{\left(𝔍\right)}$, $b^{\left(𝔍\right)}$, $R^{\left(𝔍\right)}\left(x\right)$, and $h^{\left(𝔍\right)}\left(x\right)$ (for $𝔍={\mathbb{M}}^{•}+1,{\mathbb{M}}^{•}+2,\dots ,\mathbb{M}$) 17: Output: Compute the $100(1-\varrho )\%$ BCI estimators of a, b, $R\left(x\right)$, and $h\left(x\right)$ as $$\begin{array}{l}\left\{a^{\left[\left(0.5\varrho \right){\mathbb{M}}^{★★}\right]},a^{\left[\left(1-0.5\varrho \right){\mathbb{M}}^{★★}\right]}\right\},\hfill \\ \left\{b^{\left[\left(0.5\varrho \right){\mathbb{M}}^{★★}\right]},b^{\left[\left(1-0.5\varrho \right){\mathbb{M}}^{★★}\right]}\right\},\hfill \\ \left\{R^{\left[\left(0.5\varrho \right){\mathbb{M}}^{★★}\right]}\left(x\right),R^{\left[\left(1-0.5\varrho \right){\mathbb{M}}^{★★}\right]}\left(x\right)\right\},\hfill \end{array}$$ and $$\begin{array}{c}\left\{h^{\left[\left(0.5\varrho \right){\mathbb{M}}^{★★}\right]}\left(x\right),h^{\left[\left(1-0.5\varrho \right){\mathbb{M}}^{★★}\right]}\left(x\right)\right\}\hfill \end{array}$$

5. Monte Carlo Comparisons

This section carries out detailed Monte Carlo experiments to thoroughly evaluate how well the proposed methods work with the new theoretical estimators discussed earlier.

5.1. Simulation Scenarios

To assess the relevant performance of the proposed frequentist and Bayesian estimators for the parameters a and b and the reliability measures $R\left(x\right)$ and $h\left(x\right)$, a comprehensive setup of simulation studies is carried out. Following Algorithm 2, the analysis is conducted by repeating the proposed censoring strategy 1000 times from two different populations, namely, Pop-1:INH(0.5, 1) and Pop-2:INH(1, 2). The simulation results are obtained based on various choices of n, r, m, $t_i,i=1,2,$ and $R_i,i=1,2,\dots ,m$. Additionally, the simulations are conducted across various configurations of the sample size n, number of observed failures m, mission times $t_i$$(i=1,2)$, and progressive censoring $R_i$$(i=1,2,\dots ,m)$. For mission times $x=0.1$ and 1, the true values of $R\left(x\right)$ and $h\left(x\right)$ are known to be $(0.90139,1.64915)$ for Pop-1 and $(0.86466,0.31303)$ for Pop-2, respectively, and are used distinctly for evaluating estimator accuracy. The sample sizes are set to $n\in$ 20 (small), 40 (moderate), and 80 (large), with corresponding threshold times $t_1\in \{0.2,0.6\}$ and $t_2\in \{0.5,1.2\}$ for Pop-1 along with $t_1\in \{1,2\}$ and $t_2\in \{2,5\}$ for Pop-2. To specify the length of filed items of r and m recorded during the life test, for each n we consider two levels of failure percentages (FPs) of r, called $\mathrm{FP}\left\{r\right\}\%=$25% and 50%, as well as two levels of failure percentages of m, called $\mathrm{FP}\left\{m\right\}\%=$50% and 75%. In each case, the life test is terminated according to the conditions discussed in Section 1. Furthermore, to explore the impact of progressive censoring patterns, multiple designs for $R_i$$(i=1,2,\dots ,m)$ for each simulation scenario are performed; see

Table 1. Comparison frameworks in Monte Carlo simulation.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	Design
20[25%]	20[50%]	$(2^5,0^5)$	[A]
		$(0^2,2^5,0^3)$	[B]
		$(0^5,2^5)$	[C]
20[50%]	20[75%]	$(1^5,0^{10})$	[A]
		$(0^5,1^5,0^5)$	[B]
		$(0^{10},1^5)$	[C]
40[25%]	40[50%]	$(2^{10},0^{10})$	[A]
		$(0^5,2^{10},0^5)$	[B]
		$(0^{10},2^{10})$	[C]
40[50%]	40[75%]	$(1^{10},0^{20})$	[A]
		$(0^{10},1^{10},0^{10})$	[B]
		$(0^{20},1^{10})$	[C]
80[25%]	80[50%]	$(2^{20},0^{20})$	[A]
		$(0^{10},2^{20},0^{10})$	[B]
		$(0^{20},2^{20})$	[C]
80[50%]	80[75%]	$(1^{20},0^{40})$	[A]
		$(0^{20},1^{20},0^{20})$	[B]
		$(0^{40},1^{20})$	[C]

.

After a target of 1000 UPH-CT1 simulated samples is reached, the estimation procedures is conducted in $\mathsf{R}$ version 4.2.2 using the $\mathsf{maxLik}$ and $\mathsf{coda}$ packages as provided by Henningsen and Toomet [18] and Plummer [19], respectively. This computational framework enables evaluation of both the maximum likelihood and Bayesian MCMC estimators together with their corresponding 95% ACI/BCI methods for each unknown coefficient. To perform the required Markov computations for the acquired Bayes point (or credible) estimates, we assign $\mathbb{M}=12,000$ and ${\mathbb{M}}^{•}=2000$; in addition, we assign different priors to the values of the hyperparameters $({\alpha }_i,{\beta }_i),i=1,2,$ according to Kundu’s idea [20], as follows:

• For Pop-1:INH(0.5, 1):

  –

  Prior A[PA]: $({\alpha }_1,{\alpha }_2)=(2.5,5)$ and ${\beta }_i=5,i=1,2$,

  –

  Prior B[PB]: $({\alpha }_1,{\alpha }_2)=(5.0,10)$ and ${\beta }_i=10,i=1,2$,

• For Pop-2:INH(1, 2):

  –

  Prior A[PA]: $({\alpha }_1,{\alpha }_2)=(5,10)$ and ${\beta }_i=5,i=1,2$,

  –

  Prior B[PB]: $({\alpha }_1,{\alpha }_2)=(10,20)$ and ${\beta }_i=10,i=1,2$,

Algorithm 2 UPH-CT1 generation steps

1: Input: Set INH$(a,b)$ population values   2: Output: Simulate $\xi$ independent random variables ${\xi }_1,{\xi }_2,\dots ,{\xi }_m$ from uniform $U(0,1)$ distribution   3: Output: Compute ${\zeta }_i={\xi }_i^{{\left(i+{\sum }_{s=m-i+1}^m{R}_s\right)}^{-1}}$ for $i=1,2,\dots ,m$   4: Output: Obtain $U_i=1-{\prod }_{s=m-i+1}^m{\zeta }_s$ for $i=1,2,\dots ,m$   5: Input: Set $y_i=b{[{(1-log\left(u_i\right))}^{{\displaystyle \frac{1}{a}}}-1]}^{-1},i=1,2,\dots ,m$   6: Input: Identify $d_i$ at $t_i$ (for $i=1,2$)   7: Output: Calculate remaining survival units, $r^{★}$:   8: if  $y_r<y_m<t_1<t_2$  then   9:     Set $r^{★}=n-m-{\sum }_{i=1}^{m-1}{R}_i$ 10: end if 11: if $y_r<t_1<y_m<t_2$ (or $y_r<t_1<t_2<y_m$) then 12:     Set $r^{★}=n-d_1-{\sum }_{i=1}^{d_1}{R}_i$ 13: end if 14: if $t_1<y_r<y_m<t_2$ (or $t_1<y_r<t_2<y_m$) then 15:     Set $r^{★}=n-k-{\sum }_{i=1}^{k-1}{R}_i$ 16: end if 17: if  $t_1<t_2<y_r<y_m$  then 18:     Set $r^{★}=n-d_2-{\sum }_{i=1}^{d_2}{R}_i$ 19: end if 20: Output: Get a UPH-CT1 sample: 21: if  $y_r<y_m<t_1<t_2$  then 22:     Stop the test at $y_m$ 23: end if 24: if $y_r<t_1<y_m<t_2$ (or $y_r<t_1<t_2<y_m$) then 25:     Stop the test at $t_1$ 26: end if 27: if $t_1<y_r<y_m<t_2$ (or $t_1<y_r<t_2<y_m$) then 28:     Stop the test at $y_k$ 29: end if 30: if  $t_1<t_2<y_r<y_m$  then 31:     Stop the test at $t_2$ 32: end if

To illustrate the joint posterior region in the parameter space of INH$(a,b)$ provided in (13) and to facilitate the identification of the most plausible parameter values given the observed UPH-CT1 data and prior assumptions, Figure 3 displays the posterior contour plot, which serves as a graphical representation of the joint posterior distribution of model parameters. This contour is plotted using one UPH-CT1 gathered from Pop-i (for $i=1,2$) when $(n,r,m)=(50,10,15)$, $(t_1,t_1)=(2,5)$, and $R_i=1,i=1,2,\dots ,m)$. It exhibits a powerful visual summary of the joint posterior distribution, effectively highlighting the most credible parameter regions along with their uncertainty and interdependence. It is also a good tool when closed-form posterior analysis is impossible.

From a practical computation, the average point estimates (Av.PEs) of a (for instance) are computed based on either frequentist or Bayesian methods as

$$\overline{\check{a}}={\displaystyle \frac{1}{1000}}\sum _{j=1}^{1000}{\check{a}}^{\left[j\right]},$$

where ${\check{a}}^{\left[j\right]}$ denotes the estimate of a from the ith simulated sample.

To assess the accuracy of the fitted point estimators, we rely on two metrics, namely, the root mean squared error (RMSE) and average relative absolute bias (ARAB), defined as

$$\mathrm{RMSE}\left(\check{a}\right)=\sqrt{{\displaystyle \frac{1}{1000}}\sum _{j=1}^{1000}{\left({\check{a}}^{\left[j\right]}-a\right)}^2},$$

$$\mathrm{ARAB}\left(\check{a}\right)={\displaystyle \frac{1}{1000}}\sum _{j=1}^{1000}{\displaystyle \frac{1}{a}}\left|{\check{a}}^{\left[j\right]}-a\right|,$$

respectively.

We also compare the performance of the proposed interval approaches (including 95% ACI/BCI) using the AIL and CP criteria, respectively provided by

$${\mathrm{AIL}}_{95\%}\left(a\right)={\displaystyle \frac{1}{1000}}\sum _{j=1}^{1000}\left({\mathcal{U}}_{\check{a}}^{\left[j\right]}-{\mathcal{L}}_{\check{a}}^{\left[j\right]}\right),$$

$${\mathrm{CP}}_{95\%}\left(a\right)={\displaystyle \frac{1}{1000}}\sum _{j=1}^{1000}\mathbf{1}\left\{a\in \left[{\mathcal{L}}_{\check{a}}^{\left[j\right]},{\mathcal{U}}_{\check{a}}^{\left[j\right]}\right]\right\},$$

where ${\mathcal{L}}_{\check{a}}^{\left[j\right]}$ and ${\mathcal{U}}_{\check{a}}^{\left[j\right]}$ denote the lower and upper bounds of the ith asymptotic (or credible) interval estimate and $\mathbf{1}\{·\}$ is the indicator function, which is equal to one if the true value a lies within the interval and zero otherwise.

5.2. Simulation Results and Discussion

The simulated outcomes of a, b, $R\left(x\right)$, and $h\left(x\right)$ (including RMSEs, ARABs, AILs, and CPs) are delivered in

Table 2. Point estimation results of a.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	MLE			Bayes[PA]			Bayes[PB]			MLE			Bayes[PA]			Bayes[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$									$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	0.765	1.950	1.789	0.445	0.121	0.188	0.430	0.113	0.174	0.888	1.678	1.724	0.440	0.116	0.178	0.433	0.105	0.166
		[B]	0.648	2.157	1.929	0.438	0.124	0.196	0.426	0.116	0.179	0.721	1.779	1.898	0.443	0.122	0.183	0.427	0.112	0.176
		[C]	0.866	1.732	1.635	0.439	0.112	0.175	0.430	0.103	0.164	0.763	1.643	1.481	0.437	0.108	0.171	0.430	0.099	0.163
20[50%]	20[75%]	[A]	0.741	1.657	1.330	0.475	0.100	0.165	0.488	0.096	0.158	0.604	1.318	1.280	0.474	0.097	0.164	0.489	0.095	0.156
		[B]	0.692	1.679	1.480	0.472	0.105	0.171	0.481	0.097	0.161	0.859	1.464	1.394	0.475	0.100	0.167	0.483	0.097	0.159
		[C]	0.642	1.609	1.268	0.475	0.098	0.165	0.488	0.092	0.153	0.582	1.283	1.235	0.472	0.095	0.160	0.487	0.090	0.150
40[25%]	40[50%]	[A]	0.670	1.466	0.540	0.554	0.091	0.143	0.581	0.087	0.133	0.655	1.226	0.458	0.538	0.088	0.143	0.569	0.085	0.132
		[B]	0.714	1.528	0.551	0.563	0.097	0.161	0.587	0.091	0.145	0.680	1.264	0.484	0.554	0.092	0.146	0.576	0.088	0.139
		[C]	0.619	1.052	0.495	0.539	0.090	0.142	0.574	0.083	0.131	0.617	1.018	0.449	0.543	0.088	0.142	0.577	0.082	0.130
40[50%]	40[75%]	[A]	0.719	0.745	0.392	0.509	0.088	0.141	0.532	0.082	0.128	0.674	0.528	0.347	0.511	0.087	0.140	0.535	0.081	0.128
		[B]	0.694	0.907	0.466	0.506	0.089	0.142	0.536	0.082	0.130	0.619	0.710	0.428	0.498	0.087	0.142	0.532	0.082	0.129
		[C]	0.643	0.324	0.357	0.504	0.087	0.141	0.534	0.081	0.125	0.671	0.321	0.341	0.498	0.087	0.139	0.531	0.081	0.124
80[25%]	80[50%]	[A]	0.585	0.145	0.201	0.511	0.081	0.135	0.558	0.080	0.123	0.542	0.137	0.192	0.501	0.080	0.125	0.543	0.079	0.120
		[B]	0.655	0.160	0.219	0.532	0.086	0.140	0.576	0.081	0.123	0.549	0.146	0.203	0.511	0.081	0.131	0.564	0.080	0.121
		[C]	0.543	0.131	0.186	0.498	0.080	0.124	0.553	0.076	0.121	0.547	0.129	0.180	0.502	0.077	0.124	0.556	0.073	0.119
80[50%]	80[75%]	[A]	0.561	0.103	0.144	0.453	0.072	0.120	0.494	0.067	0.111	0.515	0.100	0.143	0.452	0.072	0.112	0.464	0.062	0.106
		[B]	0.591	0.103	0.144	0.452	0.075	0.123	0.546	0.069	0.115	0.516	0.100	0.143	0.436	0.073	0.116	0.456	0.064	0.110
		[C]	0.571	0.100	0.143	0.451	0.072	0.111	0.426	0.064	0.106	0.516	0.098	0.141	0.435	0.064	0.110	0.450	0.062	0.099
			Pop-2
			$(t_1,t_2)=(1,2)$									$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	1.642	2.320	1.918	1.278	0.361	0.303	1.069	0.180	0.153	1.489	1.946	1.866	1.264	0.359	0.303	1.068	0.178	0.148
		[B]	1.874	2.424	2.093	1.274	0.374	0.317	1.056	0.185	0.153	1.641	1.979	1.952	1.277	0.360	0.304	1.056	0.178	0.149
		[C]	1.470	1.928	1.894	1.282	0.358	0.302	1.082	0.176	0.148	1.533	1.921	1.645	1.296	0.355	0.290	1.085	0.175	0.147
20[50%]	20[75%]	[A]	1.641	1.879	1.891	1.037	0.344	0.289	0.873	0.175	0.137	1.483	1.781	1.500	1.031	0.337	0.284	0.876	0.174	0.134
		[B]	1.385	1.275	0.909	1.035	0.196	0.155	0.920	0.141	0.116	1.177	1.111	0.553	1.018	0.196	0.154	0.920	0.139	0.114
		[C]	1.556	1.743	1.744	1.025	0.339	0.267	0.871	0.175	0.133	1.381	1.492	1.277	1.017	0.318	0.251	0.866	0.168	0.130
40[25%]	40[50%]	[A]	1.440	1.628	1.189	1.061	0.264	0.170	0.949	0.152	0.121	1.294	1.378	1.167	1.054	0.210	0.163	0.954	0.152	0.124
		[B]	1.306	1.725	1.679	1.078	0.297	0.177	0.953	0.161	0.128	1.434	1.415	1.223	1.054	0.229	0.166	0.941	0.158	0.126
		[C]	1.252	1.472	1.175	1.045	0.250	0.166	0.959	0.147	0.120	1.190	1.323	1.110	1.053	0.208	0.161	0.953	0.145	0.120
40[50%]	40[75%]	[A]	1.482	1.870	1.787	1.025	0.341	0.288	0.874	0.175	0.137	1.538	1.512	1.427	1.017	0.337	0.283	0.866	0.173	0.132
		[B]	1.295	1.337	1.157	1.041	0.242	0.165	0.915	0.144	0.119	1.355	1.279	1.067	1.011	0.205	0.161	0.911	0.143	0.118
		[C]	1.134	1.314	1.047	1.030	0.232	0.164	0.918	0.142	0.118	1.249	1.238	0.990	1.017	0.203	0.159	0.915	0.141	0.116
80[25%]	80[50%]	[A]	1.236	1.007	0.497	1.274	0.183	0.149	1.121	0.125	0.104	1.431	0.910	0.438	1.282	0.168	0.141	1.119	0.123	0.105
		[B]	1.251	1.202	0.586	1.273	0.195	0.151	1.121	0.127	0.110	1.238	1.023	0.507	1.240	0.176	0.142	1.115	0.124	0.108
		[C]	1.299	0.748	0.419	1.257	0.180	0.143	1.116	0.123	0.104	1.247	0.688	0.370	1.278	0.157	0.139	1.114	0.122	0.102
80[50%]	80[75%]	[A]	1.123	0.525	0.282	0.995	0.131	0.127	0.951	0.122	0.101	1.125	0.495	0.281	0.995	0.131	0.105	0.951	0.122	0.101
		[B]	1.112	0.502	0.279	0.991	0.128	0.110	0.942	0.122	0.100	1.126	0.457	0.269	0.975	0.125	0.101	0.941	0.121	0.099
		[C]	1.106	0.488	0.267	0.983	0.126	0.108	0.945	0.121	0.099	1.121	0.436	0.263	0.948	0.122	0.100	0.931	0.118	0.098

,

Table 3. Point estimation results of b.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	MLE			Bayes[PA]			Bayes[PB]			MLE			Bayes[PA]			Bayes[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$									$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	1.014	1.661	0.924	0.899	0.436	0.341	0.907	0.341	0.297	1.143	1.651	0.891	0.865	0.424	0.337	0.895	0.340	0.296
		[B]	1.141	1.505	0.848	0.922	0.398	0.328	0.911	0.337	0.288	1.044	1.465	0.774	0.955	0.387	0.311	0.931	0.325	0.280
		[C]	1.089	1.609	0.890	0.873	0.409	0.336	0.902	0.332	0.289	1.074	1.492	0.850	0.907	0.389	0.321	0.877	0.329	0.285
20[50%]	20[75%]	[A]	0.989	1.144	0.719	1.031	0.396	0.324	0.878	0.317	0.276	1.018	1.143	0.714	1.004	0.380	0.309	0.870	0.283	0.239
		[B]	0.971	1.036	0.673	1.041	0.386	0.305	0.879	0.282	0.228	1.017	0.995	0.653	1.069	0.366	0.297	0.884	0.266	0.227
		[C]	0.956	1.042	0.679	1.046	0.394	0.313	0.878	0.298	0.241	0.995	1.034	0.676	1.043	0.374	0.298	0.873	0.273	0.234
40[25%]	40[50%]	[A]	1.060	0.874	0.615	1.017	0.365	0.296	0.819	0.262	0.217	1.017	0.814	0.564	1.086	0.346	0.281	0.866	0.259	0.209
		[B]	1.036	0.841	0.556	1.104	0.351	0.294	0.855	0.261	0.209	1.007	0.750	0.525	1.076	0.341	0.278	0.843	0.258	0.209
		[C]	1.028	0.950	0.618	0.967	0.380	0.298	0.782	0.272	0.227	1.045	0.909	0.608	1.022	0.348	0.284	0.822	0.265	0.221
40[50%]	40[75%]	[A]	0.969	0.663	0.495	0.878	0.339	0.277	0.717	0.260	0.202	0.997	0.659	0.491	0.863	0.334	0.276	0.707	0.252	0.197
		[B]	0.977	0.642	0.482	0.900	0.335	0.273	0.716	0.256	0.197	0.993	0.630	0.463	0.928	0.334	0.272	0.721	0.252	0.195
		[C]	0.982	0.695	0.515	0.890	0.339	0.277	0.709	0.260	0.205	0.985	0.692	0.508	0.934	0.338	0.277	0.726	0.258	0.197
80[25%]	80[50%]	[A]	1.023	0.563	0.416	1.059	0.331	0.269	0.798	0.253	0.196	1.013	0.514	0.389	1.085	0.327	0.265	0.871	0.250	0.193
		[B]	1.025	0.529	0.390	1.114	0.329	0.267	0.826	0.248	0.194	1.009	0.487	0.366	1.089	0.324	0.265	0.807	0.248	0.192
		[C]	1.041	0.626	0.460	0.945	0.333	0.270	0.735	0.253	0.196	1.029	0.581	0.428	1.053	0.330	0.265	0.778	0.252	0.193
80[50%]	80[75%]	[A]	1.081	0.456	0.343	1.138	0.318	0.261	1.019	0.234	0.192	1.080	0.450	0.341	1.124	0.317	0.258	1.008	0.223	0.189
		[B]	1.081	0.446	0.341	1.157	0.304	0.246	1.025	0.233	0.191	1.059	0.420	0.326	1.276	0.272	0.223	1.121	0.222	0.185
		[C]	1.082	0.464	0.352	1.124	0.327	0.265	1.008	0.247	0.192	1.076	0.461	0.351	1.265	0.321	0.258	1.077	0.233	0.191
			Pop-2
			$(t_1,t_2)=(1,2)$									$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	2.161	3.352	0.991	1.768	0.664	0.299	2.039	0.388	0.163	2.208	2.805	0.895	1.808	0.655	0.295	2.058	0.384	0.163
		[B]	2.268	3.127	0.936	1.793	0.663	0.296	2.045	0.384	0.159	2.435	2.712	0.892	1.765	0.648	0.290	2.032	0.379	0.158
		[C]	2.042	3.851	1.025	1.728	0.673	0.304	2.023	0.394	0.166	2.391	3.133	0.914	1.724	0.664	0.300	2.012	0.392	0.160
20[50%]	20[75%]	[A]	2.019	2.777	0.830	1.580	0.623	0.262	1.857	0.381	0.157	2.061	2.619	0.800	1.608	0.620	0.253	1.895	0.379	0.153
		[B]	2.004	2.478	0.811	1.585	0.603	0.257	1.862	0.380	0.155	2.061	2.608	0.760	1.608	0.592	0.247	1.895	0.369	0.153
		[C]	2.097	2.874	0.884	1.560	0.660	0.283	1.854	0.383	0.157	2.132	2.702	0.883	1.567	0.637	0.264	1.851	0.379	0.155
40[25%]	40[50%]	[A]	2.018	1.718	0.625	1.792	0.581	0.246	1.937	0.357	0.150	2.076	1.688	0.617	1.817	0.562	0.238	1.930	0.351	0.149
		[B]	2.053	1.695	0.623	1.865	0.567	0.246	1.947	0.356	0.150	2.133	1.658	0.610	1.845	0.556	0.237	1.969	0.349	0.149
		[C]	1.936	1.801	0.647	1.713	0.602	0.254	1.871	0.372	0.152	2.065	1.791	0.636	1.791	0.566	0.240	1.936	0.356	0.151
40[50%]	40[75%]	[A]	1.896	1.612	0.593	1.680	0.544	0.234	1.915	0.348	0.147	1.927	1.602	0.588	1.732	0.529	0.230	1.938	0.345	0.145
		[B]	1.893	1.564	0.583	1.671	0.536	0.232	1.902	0.345	0.145	1.913	1.558	0.552	1.722	0.528	0.229	1.912	0.345	0.144
		[C]	1.936	1.649	0.608	1.653	0.564	0.240	1.907	0.351	0.147	1.954	1.621	0.593	1.723	0.553	0.233	1.919	0.348	0.145
80[25%]	80[50%]	[A]	1.904	1.116	0.441	1.655	0.526	0.220	1.935	0.339	0.143	1.895	1.029	0.407	1.650	0.523	0.212	1.953	0.334	0.141
		[B]	1.884	1.070	0.417	1.698	0.499	0.214	1.967	0.334	0.142	1.916	1.011	0.391	1.653	0.482	0.204	1.961	0.328	0.140
		[C]	1.880	1.180	0.471	1.678	0.531	0.230	1.952	0.344	0.144	1.881	1.152	0.450	1.749	0.527	0.226	1.975	0.337	0.143
80[50%]	80[75%]	[A]	2.116	0.971	0.376	1.849	0.394	0.157	1.963	0.316	0.136	2.112	0.944	0.374	1.902	0.387	0.152	1.974	0.315	0.135
		[B]	2.119	0.964	0.372	1.879	0.383	0.151	1.966	0.315	0.136	2.086	0.924	0.353	2.015	0.375	0.146	2.026	0.310	0.133
		[C]	2.116	1.015	0.393	1.826	0.415	0.168	1.914	0.320	0.138	2.113	0.995	0.376	1.826	0.415	0.168	1.914	0.317	0.137

,

Table 4. Point estimation results of $R\left(x\right)$.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	MLE			Bayes[PA]			Bayes[PB]			MLE			Bayes[PA]			Bayes[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$									$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	0.852	0.132	0.120	0.902	0.115	0.115	0.879	0.083	0.072	0.854	0.122	0.116	0.902	0.114	0.113	0.877	0.081	0.070
		[B]	0.857	0.126	0.115	0.909	0.114	0.109	0.886	0.078	0.069	0.857	0.119	0.111	0.911	0.111	0.107	0.888	0.076	0.067
		[C]	0.862	0.121	0.107	0.914	0.112	0.105	0.892	0.070	0.063	0.861	0.115	0.105	0.915	0.101	0.102	0.891	0.068	0.062
20[50%]	20[75%]	[A]	0.871	0.088	0.082	0.776	0.080	0.080	0.762	0.065	0.061	0.873	0.085	0.080	0.775	0.078	0.079	0.763	0.063	0.060
		[B]	0.874	0.085	0.079	0.775	0.077	0.077	0.763	0.062	0.059	0.875	0.082	0.078	0.777	0.072	0.073	0.765	0.061	0.058
		[C]	0.874	0.083	0.078	0.776	0.076	0.074	0.763	0.060	0.059	0.875	0.081	0.076	0.777	0.074	0.072	0.765	0.056	0.058
40[25%]	40[50%]	[A]	0.857	0.078	0.072	0.823	0.072	0.068	0.810	0.058	0.049	0.859	0.075	0.072	0.828	0.070	0.064	0.815	0.052	0.048
		[B]	0.861	0.069	0.069	0.823	0.067	0.059	0.820	0.055	0.047	0.862	0.067	0.069	0.834	0.065	0.058	0.821	0.051	0.047
		[C]	0.863	0.066	0.066	0.839	0.061	0.055	0.827	0.053	0.047	0.863	0.066	0.066	0.841	0.061	0.055	0.827	0.050	0.046
40[50%]	40[75%]	[A]	0.866	0.065	0.059	0.810	0.059	0.055	0.795	0.051	0.046	0.867	0.062	0.056	0.804	0.059	0.053	0.797	0.050	0.046
		[B]	0.867	0.057	0.054	0.808	0.055	0.051	0.799	0.047	0.044	0.868	0.055	0.053	0.808	0.055	0.051	0.801	0.046	0.043
		[C]	0.868	0.055	0.053	0.805	0.054	0.050	0.798	0.045	0.042	0.868	0.055	0.052	0.807	0.054	0.050	0.799	0.045	0.042
80[25%]	80[50%]	[A]	0.869	0.054	0.052	0.903	0.052	0.049	0.897	0.037	0.035	0.860	0.054	0.051	0.905	0.050	0.048	0.898	0.037	0.034
		[B]	0.863	0.054	0.051	0.908	0.051	0.048	0.894	0.034	0.034	0.862	0.053	0.051	0.903	0.050	0.048	0.897	0.034	0.032
		[C]	0.865	0.053	0.050	0.903	0.050	0.048	0.896	0.034	0.032	0.863	0.051	0.050	0.902	0.049	0.047	0.895	0.033	0.031
80[50%]	80[75%]	[A]	0.861	0.049	0.050	0.823	0.045	0.046	0.818	0.033	0.030	0.861	0.047	0.047	0.823	0.044	0.045	0.818	0.032	0.030
		[B]	0.867	0.047	0.047	0.826	0.043	0.044	0.822	0.032	0.029	0.862	0.045	0.046	0.827	0.043	0.043	0.823	0.031	0.029
		[C]	0.862	0.046	0.046	0.828	0.042	0.042	0.824	0.031	0.026	0.862	0.044	0.042	0.831	0.041	0.042	0.826	0.030	0.026
			Pop-2
			$(t_1,t_2)=(1,2)$									$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	0.910	0.122	0.115	0.806	0.113	0.108	0.798	0.059	0.053	0.909	0.117	0.110	0.812	0.111	0.102	0.803	0.058	0.052
		[B]	0.908	0.130	0.119	0.802	0.121	0.112	0.797	0.060	0.055	0.907	0.125	0.117	0.804	0.119	0.110	0.795	0.060	0.053
		[C]	0.905	0.134	0.123	0.798	0.127	0.119	0.793	0.064	0.056	0.905	0.132	0.121	0.796	0.125	0.116	0.791	0.063	0.055
20[50%]	20[75%]	[A]	0.910	0.081	0.065	0.862	0.072	0.058	0.853	0.056	0.041	0.909	0.078	0.065	0.864	0.071	0.057	0.852	0.056	0.040
		[B]	0.908	0.087	0.071	0.856	0.077	0.063	0.845	0.057	0.051	0.908	0.087	0.071	0.856	0.077	0.063	0.844	0.057	0.051
		[C]	0.909	0.082	0.070	0.863	0.073	0.062	0.852	0.057	0.047	0.909	0.081	0.066	0.862	0.072	0.059	0.852	0.056	0.045
40[25%]	40[50%]	[A]	0.905	0.062	0.052	0.921	0.057	0.048	0.916	0.046	0.036	0.904	0.062	0.052	0.921	0.052	0.046	0.916	0.041	0.035
		[B]	0.904	0.063	0.053	0.918	0.057	0.049	0.913	0.041	0.037	0.904	0.063	0.053	0.920	0.056	0.048	0.914	0.041	0.036
		[C]	0.902	0.064	0.055	0.917	0.058	0.049	0.910	0.043	0.039	0.902	0.063	0.054	0.918	0.057	0.049	0.911	0.042	0.039
40[50%]	40[75%]	[A]	0.907	0.050	0.044	0.865	0.045	0.039	0.859	0.042	0.032	0.906	0.049	0.041	0.865	0.045	0.038	0.858	0.038	0.029
		[B]	0.906	0.054	0.049	0.865	0.050	0.043	0.858	0.044	0.035	0.907	0.053	0.047	0.865	0.048	0.043	0.860	0.040	0.034
		[C]	0.906	0.053	0.047	0.864	0.048	0.040	0.858	0.042	0.033	0.906	0.050	0.043	0.865	0.046	0.039	0.857	0.039	0.032
80[25%]	80[50%]	[A]	0.903	0.042	0.036	0.869	0.039	0.036	0.864	0.033	0.027	0.903	0.041	0.036	0.901	0.036	0.033	0.901	0.027	0.025
		[B]	0.903	0.049	0.040	0.866	0.043	0.037	0.861	0.034	0.029	0.903	0.044	0.037	0.903	0.039	0.036	0.904	0.028	0.027
		[C]	0.902	0.046	0.044	0.868	0.044	0.037	0.864	0.034	0.032	0.902	0.048	0.039	0.902	0.040	0.037	0.900	0.028	0.029
80[50%]	80[75%]	[A]	0.895	0.032	0.024	0.902	0.027	0.021	0.902	0.024	0.021	0.899	0.029	0.024	0.871	0.025	0.021	0.867	0.022	0.020
		[B]	0.896	0.037	0.025	0.901	0.031	0.025	0.900	0.028	0.024	0.899	0.033	0.025	0.869	0.033	0.025	0.865	0.027	0.024
		[C]	0.896	0.035	0.024	0.900	0.028	0.024	0.898	0.026	0.023	0.899	0.031	0.024	0.866	0.027	0.024	0.861	0.024	0.023

,

Table 5. Point estimation results of $h\left(x\right)$.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	MLE			Bayes[PA]			Bayes[PB]			MLE			Bayes[PA]			Bayes[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$									$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	1.786	1.065	0.515	2.470	0.999	0.497	2.460	0.976	0.458	1.780	1.034	0.512	2.435	0.981	0.492	2.489	0.951	0.447
		[B]	1.871	1.179	0.543	2.475	1.038	0.517	2.476	0.994	0.500	1.826	1.135	0.520	2.521	1.012	0.507	2.498	0.992	0.482
		[C]	1.702	0.951	0.484	2.411	0.924	0.458	2.438	0.906	0.420	1.708	0.932	0.478	2.346	0.910	0.441	2.392	0.858	0.405
20[50%]	20[75%]	[A]	1.690	0.877	0.397	1.933	0.716	0.349	2.094	0.598	0.289	1.668	0.859	0.387	1.945	0.709	0.349	2.099	0.598	0.288
		[B]	1.725	0.930	0.415	1.998	0.736	0.361	2.148	0.643	0.307	1.698	0.903	0.399	2.015	0.722	0.356	2.160	0.631	0.303
		[C]	1.652	0.826	0.375	1.940	0.693	0.348	2.087	0.593	0.284	1.644	0.817	0.372	1.916	0.691	0.345	2.089	0.591	0.282
40[25%]	40[50%]	[A]	1.776	0.689	0.344	1.513	0.661	0.293	1.656	0.586	0.279	1.758	0.687	0.338	1.477	0.635	0.284	1.620	0.568	0.270
		[B]	1.733	0.684	0.322	1.477	0.592	0.277	1.600	0.576	0.269	1.733	0.678	0.321	1.427	0.582	0.270	1.559	0.561	0.261
		[C]	1.703	0.678	0.318	1.411	0.572	0.275	1.550	0.552	0.262	1.704	0.676	0.316	1.420	0.570	0.264	1.557	0.551	0.256
40[50%]	40[75%]	[A]	1.674	0.531	0.246	2.055	0.453	0.225	2.201	0.444	0.207	1.674	0.521	0.243	2.053	0.448	0.223	2.219	0.429	0.204
		[B]	1.689	0.535	0.248	2.041	0.471	0.228	2.209	0.454	0.225	1.648	0.524	0.244	2.006	0.471	0.225	2.178	0.453	0.225
		[C]	1.656	0.509	0.239	2.027	0.447	0.225	2.190	0.435	0.207	1.659	0.498	0.233	2.015	0.441	0.219	2.195	0.426	0.203
80[25%]	80[50%]	[A]	1.701	0.429	0.204	1.708	0.411	0.197	1.838	0.393	0.188	1.697	0.421	0.202	1.634	0.399	0.190	1.787	0.393	0.188
		[B]	1.687	0.422	0.203	1.635	0.376	0.180	1.773	0.365	0.174	1.671	0.407	0.190	1.591	0.367	0.175	1.692	0.323	0.152
		[C]	1.671	0.421	0.201	1.598	0.353	0.168	1.732	0.348	0.166	1.685	0.370	0.172	1.617	0.352	0.167	1.759	0.316	0.152
80[50%]	80[75%]	[A]	1.661	0.341	0.163	1.869	0.318	0.154	1.957	0.306	0.148	1.670	0.330	0.157	1.897	0.312	0.150	1.989	0.294	0.141
		[B]	1.668	0.391	0.186	1.897	0.340	0.160	1.989	0.328	0.157	1.666	0.362	0.172	1.806	0.330	0.158	1.920	0.300	0.143
		[C]	1.659	0.321	0.153	1.852	0.308	0.148	1.949	0.290	0.142	1.669	0.316	0.153	1.783	0.301	0.144	1.881	0.280	0.137
			Pop-2
			$(t_1,t_2)=(1,2)$									$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	0.355	0.169	0.400	0.251	0.136	0.378	0.277	0.121	0.332	0.357	0.166	0.394	0.247	0.132	0.366	0.273	0.118	0.321
		[B]	0.368	0.179	0.421	0.264	0.138	0.382	0.287	0.123	0.334	0.359	0.175	0.416	0.264	0.138	0.381	0.289	0.122	0.334
		[C]	0.342	0.149	0.374	0.242	0.135	0.366	0.269	0.121	0.331	0.341	0.144	0.373	0.242	0.132	0.347	0.269	0.118	0.321
20[50%]	20[75%]	[A]	0.339	0.142	0.362	0.426	0.110	0.294	0.416	0.095	0.246	0.333	0.135	0.346	0.426	0.109	0.292	0.416	0.092	0.239
		[B]	0.333	0.133	0.341	0.425	0.107	0.289	0.416	0.092	0.239	0.327	0.125	0.322	0.421	0.105	0.286	0.412	0.091	0.236
		[C]	0.331	0.130	0.334	0.424	0.105	0.278	0.415	0.091	0.235	0.327	0.125	0.322	0.421	0.100	0.270	0.412	0.089	0.227
40[25%]	40[50%]	[A]	0.340	0.114	0.282	0.359	0.100	0.270	0.362	0.084	0.218	0.335	0.108	0.270	0.356	0.095	0.259	0.361	0.083	0.218
		[B]	0.350	0.116	0.293	0.375	0.101	0.274	0.375	0.087	0.221	0.341	0.113	0.283	0.365	0.099	0.264	0.366	0.084	0.221
		[C]	0.332	0.110	0.278	0.348	0.099	0.267	0.355	0.081	0.211	0.329	0.107	0.269	0.347	0.094	0.256	0.354	0.080	0.200
40[50%]	40[75%]	[A]	0.337	0.107	0.274	0.398	0.092	0.244	0.388	0.077	0.202	0.335	0.102	0.261	0.392	0.085	0.224	0.386	0.077	0.200
		[B]	0.335	0.105	0.265	0.393	0.079	0.205	0.384	0.076	0.188	0.332	0.099	0.248	0.387	0.076	0.199	0.381	0.075	0.185
		[C]	0.333	0.095	0.242	0.394	0.072	0.188	0.386	0.071	0.182	0.329	0.094	0.232	0.389	0.071	0.188	0.384	0.070	0.179
80[25%]	80[50%]	[A]	0.333	0.077	0.200	0.278	0.067	0.178	0.275	0.065	0.174	0.331	0.075	0.198	0.274	0.067	0.174	0.271	0.065	0.171
		[B]	0.339	0.089	0.224	0.273	0.071	0.185	0.271	0.069	0.179	0.338	0.085	0.222	0.267	0.070	0.180	0.268	0.069	0.176
		[C]	0.332	0.075	0.199	0.272	0.064	0.172	0.271	0.063	0.167	0.327	0.070	0.180	0.275	0.063	0.171	0.273	0.063	0.166
80[50%]	80[75%]	[A]	0.318	0.063	0.164	0.362	0.062	0.158	0.361	0.059	0.155	0.319	0.062	0.162	0.359	0.060	0.158	0.359	0.058	0.153
		[B]	0.320	0.063	0.167	0.367	0.062	0.165	0.367	0.062	0.164	0.320	0.063	0.165	0.367	0.062	0.164	0.367	0.060	0.160
		[C]	0.317	0.061	0.157	0.359	0.058	0.154	0.359	0.057	0.143	0.319	0.059	0.156	0.349	0.058	0.151	0.353	0.055	0.143

,

Table 6. The 95% interval estimation results of a.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	ACI		BCI[PA]		BCI[PB]		ACI		BCI[PA]		BCI[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$						$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	2.358	0.919	0.395	0.932	0.338	0.936	1.462	0.920	0.361	0.935	0.324	0.936
		[B]	2.423	0.917	0.424	0.932	0.341	0.933	1.713	0.918	0.391	0.932	0.331	0.934
		[C]	2.183	0.924	0.376	0.933	0.285	0.937	1.352	0.926	0.347	0.936	0.321	0.937
20[50%]	20[75%]	[A]	1.765	0.927	0.357	0.937	0.290	0.941	1.258	0.928	0.338	0.937	0.317	0.941
		[B]	2.169	0.943	0.366	0.947	0.326	0.955	1.314	0.946	0.346	0.948	0.320	0.958
		[C]	1.543	0.931	0.351	0.938	0.291	0.942	1.248	0.936	0.338	0.941	0.315	0.942
40[25%]	40[50%]	[A]	1.406	0.937	0.323	0.941	0.252	0.944	1.195	0.940	0.318	0.942	0.288	0.944
		[B]	1.493	0.936	0.344	0.940	0.293	0.943	1.216	0.939	0.327	0.941	0.293	0.944
		[C]	1.395	0.937	0.322	0.941	0.319	0.945	1.152	0.940	0.317	0.942	0.287	0.946
40[50%]	40[75%]	[A]	1.275	0.929	0.318	0.938	0.311	0.941	1.039	0.935	0.310	0.940	0.273	0.942
		[B]	1.326	0.941	0.322	0.945	0.312	0.947	1.105	0.942	0.313	0.945	0.275	0.950
		[C]	0.895	0.943	0.300	0.947	0.245	0.953	0.860	0.944	0.298	0.947	0.265	0.957
80[25%]	80[50%]	[A]	0.484	0.947	0.286	0.949	0.267	0.956	0.461	0.948	0.273	0.951	0.250	0.961
		[B]	0.526	0.944	0.290	0.948	0.271	0.955	0.486	0.946	0.279	0.949	0.259	0.960
		[C]	0.446	0.948	0.283	0.950	0.247	0.959	0.427	0.948	0.260	0.956	0.243	0.961
80[50%]	80[75%]	[A]	0.381	0.949	0.212	0.955	0.209	0.961	0.371	0.950	0.200	0.958	0.201	0.962
		[B]	0.386	0.950	0.218	0.957	0.197	0.962	0.381	0.952	0.212	0.959	0.211	0.963
		[C]	0.376	0.952	0.205	0.958	0.198	0.962	0.354	0.954	0.188	0.960	0.179	0.964
			Pop-2
			$(t_1,t_2)=(1,2)$						$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	3.192	0.897	0.865	0.930	0.531	0.932	3.058	0.916	0.868	0.933	0.526	0.935
		[B]	3.272	0.894	0.884	0.924	0.556	0.932	3.125	0.902	0.878	0.928	0.556	0.933
		[C]	2.976	0.904	0.856	0.933	0.524	0.935	2.891	0.922	0.854	0.936	0.515	0.935
20[50%]	20[75%]	[A]	2.863	0.913	0.836	0.936	0.490	0.942	2.755	0.928	0.813	0.938	0.482	0.936
		[B]	1.721	0.905	0.756	0.935	0.439	0.942	1.688	0.924	0.692	0.936	0.424	0.935
		[C]	2.774	0.923	0.792	0.937	0.485	0.943	2.421	0.928	0.795	0.938	0.471	0.936
40[25%]	40[50%]	[A]	2.536	0.925	0.792	0.940	0.468	0.947	2.192	0.931	0.775	0.941	0.456	0.942
		[B]	2.570	0.925	0.792	0.939	0.469	0.945	2.281	0.930	0.781	0.940	0.465	0.941
		[C]	2.507	0.927	0.788	0.941	0.460	0.948	2.182	0.934	0.774	0.941	0.441	0.942
40[50%]	40[75%]	[A]	2.597	0.932	0.803	0.943	0.487	0.950	2.677	0.940	0.802	0.943	0.476	0.945
		[B]	2.195	0.929	0.785	0.942	0.454	0.949	1.892	0.936	0.749	0.942	0.434	0.944
		[C]	1.827	0.938	0.764	0.945	0.441	0.950	1.769	0.941	0.724	0.945	0.429	0.947
80[25%]	80[50%]	[A]	1.658	0.942	0.701	0.946	0.424	0.951	1.555	0.944	0.627	0.950	0.402	0.950
		[B]	1.721	0.940	0.748	0.946	0.424	0.951	1.599	0.943	0.689	0.949	0.420	0.948
		[C]	1.554	0.942	0.666	0.946	0.422	0.953	1.534	0.946	0.555	0.952	0.397	0.954
80[50%]	80[75%]	[A]	1.451	0.944	0.535	0.947	0.420	0.954	1.497	0.950	0.514	0.953	0.388	0.955
		[B]	1.279	0.944	0.497	0.947	0.412	0.954	1.258	0.949	0.486	0.953	0.382	0.955
		[C]	1.147	0.945	0.482	0.948	0.410	0.955	1.137	0.952	0.463	0.955	0.372	0.956

,

Table 7. The 95% interval estimation results of b.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	ACI		BCI[PA]		BCI[PB]		ACI		BCI[PA]		BCI[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$						$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	2.813	0.913	1.483	0.932	0.904	0.938	2.796	0.904	1.456	0.933	0.891	0.940
		[B]	2.600	0.915	1.441	0.934	0.888	0.939	2.069	0.924	1.403	0.934	0.880	0.940
		[C]	2.770	0.913	1.446	0.932	0.891	0.938	2.400	0.912	1.442	0.934	0.887	0.939
20[50%]	20[75%]	[A]	1.893	0.928	1.396	0.935	0.870	0.940	1.848	0.935	1.386	0.937	0.837	0.942
		[B]	1.733	0.929	1.333	0.940	0.839	0.942	1.714	0.939	1.283	0.941	0.818	0.943
		[C]	1.789	0.928	1.384	0.937	0.852	0.940	1.739	0.936	1.353	0.939	0.821	0.942
40[25%]	40[50%]	[A]	1.658	0.940	1.276	0.942	0.817	0.945	1.444	0.941	1.269	0.942	0.816	0.946
		[B]	1.473	0.940	1.260	0.943	0.810	0.947	1.306	0.942	1.259	0.943	0.807	0.948
		[C]	1.724	0.939	1.310	0.941	0.837	0.943	1.610	0.940	1.270	0.941	0.817	0.946
40[50%]	40[75%]	[A]	1.254	0.943	1.203	0.945	0.781	0.952	1.215	0.945	1.198	0.945	0.768	0.954
		[B]	1.234	0.944	1.192	0.947	0.757	0.953	1.193	0.947	1.123	0.947	0.734	0.956
		[C]	1.254	0.942	1.251	0.944	0.791	0.951	1.254	0.943	1.245	0.945	0.769	0.951
80[25%]	80[50%]	[A]	1.172	0.948	1.056	0.952	0.722	0.956	1.160	0.949	0.972	0.954	0.716	0.959
		[B]	1.155	0.948	0.976	0.956	0.705	0.959	1.138	0.950	0.944	0.956	0.691	0.960
		[C]	1.211	0.947	1.179	0.948	0.744	0.956	1.181	0.948	1.060	0.950	0.724	0.958
80[50%]	80[75%]	[A]	1.138	0.949	0.935	0.958	0.656	0.958	1.123	0.954	0.900	0.958	0.655	0.961
		[B]	1.138	0.949	0.922	0.958	0.651	0.958	0.983	0.954	0.830	0.961	0.645	0.962
		[C]	1.154	0.948	0.957	0.956	0.662	0.957	1.138	0.953	0.901	0.958	0.657	0.961
			Pop-2
			$(t_1,t_2)=(1,2)$						$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	3.717	0.898	2.443	0.927	1.343	0.936	2.707	0.919	2.380	0.929	1.324	0.936
		[B]	3.628	0.911	2.435	0.929	1.336	0.937	2.563	0.920	2.346	0.931	1.311	0.937
		[C]	3.727	0.905	2.467	0.923	1.359	0.936	2.881	0.917	2.451	0.928	1.336	0.937
20[50%]	20[75%]	[A]	2.864	0.932	2.187	0.936	1.292	0.939	2.237	0.933	2.133	0.940	1.274	0.940
		[B]	2.746	0.937	2.137	0.938	1.284	0.941	2.171	0.936	2.098	0.941	1.267	0.941
		[C]	3.037	0.936	2.258	0.934	1.302	0.940	2.386	0.929	2.134	0.936	1.280	0.941
40[25%]	40[50%]	[A]	2.308	0.940	2.076	0.942	1.263	0.943	2.150	0.941	1.618	0.944	1.253	0.947
		[B]	2.256	0.941	2.072	0.942	1.254	0.949	2.141	0.941	1.535	0.944	1.252	0.952
		[C]	2.311	0.940	2.092	0.941	1.269	0.941	2.161	0.941	1.734	0.942	1.260	0.941
40[50%]	40[75%]	[A]	2.045	0.944	1.819	0.945	1.249	0.956	2.032	0.944	1.368	0.947	1.215	0.957
		[B]	2.012	0.946	1.775	0.946	1.245	0.956	1.995	0.944	1.308	0.948	1.187	0.958
		[C]	2.107	0.942	1.931	0.944	1.254	0.954	2.034	0.944	1.443	0.945	1.249	0.957
80[25%]	80[50%]	[A]	1.884	0.949	1.511	0.951	1.170	0.958	1.826	0.949	1.170	0.956	1.086	0.960
		[B]	1.828	0.950	1.416	0.957	1.166	0.958	1.532	0.951	1.168	0.959	1.057	0.960
		[C]	1.886	0.946	1.586	0.950	1.177	0.957	1.871	0.948	1.185	0.954	1.104	0.958
80[50%]	80[75%]	[A]	1.498	0.953	1.360	0.954	1.148	0.959	1.497	0.953	1.149	0.960	0.973	0.961
		[B]	1.490	0.954	1.347	0.956	1.138	0.961	1.487	0.954	1.146	0.961	0.948	0.964
		[C]	1.527	0.953	1.415	0.953	1.155	0.958	1.510	0.952	1.164	0.959	1.004	0.960

,

Table 8. The 95% interval estimation results of $R\left(x\right)$.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	ACI		BCI[PA]		BCI[PB]		ACI		BCI[PA]		BCI[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$						$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	0.284	0.923	0.263	0.924	0.259	0.927	0.280	0.924	0.260	0.925	0.259	0.927
		[B]	0.279	0.916	0.257	0.920	0.248	0.922	0.273	0.919	0.255	0.921	0.246	0.922
		[C]	0.276	0.920	0.245	0.922	0.240	0.923	0.267	0.922	0.243	0.923	0.239	0.924
20[50%]	20[75%]	[A]	0.257	0.934	0.236	0.936	0.235	0.937	0.239	0.935	0.236	0.937	0.207	0.939
		[B]	0.234	0.926	0.228	0.929	0.220	0.932	0.229	0.929	0.225	0.930	0.201	0.933
		[C]	0.220	0.931	0.212	0.933	0.210	0.935	0.220	0.932	0.212	0.935	0.199	0.935
40[25%]	40[50%]	[A]	0.219	0.943	0.199	0.944	0.191	0.946	0.214	0.944	0.195	0.945	0.187	0.946
		[B]	0.205	0.939	0.194	0.939	0.177	0.941	0.198	0.939	0.192	0.940	0.175	0.941
		[C]	0.195	0.942	0.191	0.943	0.172	0.944	0.194	0.942	0.186	0.943	0.172	0.944
40[50%]	40[75%]	[A]	0.193	0.947	0.190	0.947	0.170	0.948	0.190	0.947	0.186	0.947	0.169	0.949
		[B]	0.187	0.944	0.186	0.945	0.166	0.947	0.186	0.944	0.184	0.946	0.166	0.947
		[C]	0.185	0.945	0.178	0.946	0.165	0.948	0.184	0.945	0.175	0.947	0.164	0.948
80[25%]	80[50%]	[A]	0.158	0.959	0.143	0.959	0.134	0.961	0.153	0.959	0.139	0.960	0.132	0.961
		[B]	0.153	0.956	0.139	0.957	0.125	0.958	0.149	0.956	0.134	0.957	0.125	0.958
		[C]	0.146	0.957	0.139	0.959	0.125	0.960	0.145	0.957	0.133	0.959	0.121	0.961
80[50%]	80[75%]	[A]	0.139	0.963	0.123	0.965	0.121	0.966	0.136	0.964	0.122	0.965	0.111	0.967
		[B]	0.141	0.961	0.125	0.962	0.119	0.963	0.127	0.961	0.120	0.963	0.108	0.964
		[C]	0.132	0.961	0.122	0.963	0.105	0.964	0.124	0.963	0.117	0.964	0.103	0.965
			Pop-2
			$(t_1,t_2)=(1,2)$						$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	0.317	0.910	0.263	0.917	0.194	0.923	0.303	0.912	0.257	0.917	0.185	0.928
		[B]	0.320	0.909	0.268	0.915	0.210	0.922	0.311	0.910	0.265	0.915	0.194	0.924
		[C]	0.327	0.906	0.279	0.911	0.220	0.918	0.324	0.907	0.276	0.912	0.204	0.920
20[50%]	20[75%]	[A]	0.269	0.923	0.235	0.926	0.175	0.934	0.265	0.924	0.223	0.928	0.169	0.935
		[B]	0.287	0.918	0.244	0.922	0.189	0.929	0.272	0.921	0.243	0.923	0.179	0.930
		[C]	0.270	0.922	0.239	0.923	0.188	0.930	0.270	0.923	0.239	0.924	0.174	0.931
40[25%]	40[50%]	[A]	0.178	0.943	0.169	0.947	0.144	0.949	0.176	0.946	0.167	0.947	0.143	0.950
		[B]	0.180	0.943	0.171	0.946	0.144	0.948	0.178	0.945	0.170	0.947	0.144	0.949
		[C]	0.182	0.940	0.173	0.945	0.151	0.947	0.181	0.944	0.171	0.945	0.151	0.948
40[50%]	40[75%]	[A]	0.148	0.950	0.140	0.956	0.132	0.958	0.144	0.951	0.136	0.956	0.126	0.958
		[B]	0.161	0.946	0.143	0.954	0.141	0.955	0.154	0.948	0.143	0.954	0.137	0.955
		[C]	0.150	0.948	0.140	0.955	0.136	0.957	0.149	0.950	0.140	0.956	0.132	0.957
80[25%]	80[50%]	[A]	0.128	0.956	0.122	0.961	0.118	0.964	0.124	0.961	0.120	0.961	0.098	0.964
		[B]	0.130	0.955	0.123	0.961	0.122	0.963	0.125	0.957	0.122	0.961	0.110	0.963
		[C]	0.135	0.953	0.125	0.960	0.128	0.962	0.131	0.954	0.125	0.960	0.118	0.962
80[50%]	80[75%]	[A]	0.111	0.963	0.105	0.967	0.101	0.969	0.107	0.966	0.103	0.967	0.087	0.969
		[B]	0.119	0.960	0.113	0.964	0.107	0.966	0.113	0.963	0.111	0.964	0.094	0.967
		[C]	0.112	0.962	0.108	0.966	0.104	0.968	0.110	0.965	0.105	0.966	0.090	0.968

and

Table 9. The 95% interval estimation results of $h\left(x\right)$.

$\mathit{n}\mathbf{\left[}\mathbf{FP}\mathbf{\right\{}\mathit{r}\mathbf{\}}\mathbf{\%}\mathbf{]}$	$\mathit{n}\left[\mathbf{FP}\right\{\mathit{m}\}\%]$	R	ACI		BCI[PA]		BCI[PB]		ACI		BCI[PA]		BCI[PB]
			Pop-1
			$(t_1,t_2)=(0.2,0.5)$						$(t_1,t_2)=(0.6,1.2)$
20[25%]	20[50%]	[A]	3.330	0.914	2.219	0.927	2.164	0.930	3.187	0.915	2.170	0.928	2.157	0.930
		[B]	3.469	0.905	2.255	0.926	2.213	0.927	3.484	0.907	2.241	0.926	2.181	0.928
		[C]	3.116	0.920	2.176	0.928	2.122	0.930	2.915	0.922	2.157	0.929	2.121	0.930
20[50%]	20[75%]	[A]	2.798	0.927	2.141	0.931	2.060	0.931	2.733	0.928	2.136	0.931	2.042	0.931
		[B]	2.965	0.923	2.155	0.929	2.087	0.931	2.887	0.923	2.145	0.930	2.073	0.931
		[C]	2.676	0.930	2.132	0.931	2.049	0.932	2.618	0.931	2.126	0.931	2.028	0.931
40[25%]	40[50%]	[A]	2.439	0.937	1.821	0.939	1.749	0.942	2.285	0.939	1.785	0.942	1.747	0.942
		[B]	2.127	0.942	1.798	0.940	1.744	0.945	2.075	0.942	1.784	0.942	1.690	0.945
		[C]	1.980	0.942	1.750	0.942	1.724	0.948	1.946	0.943	1.747	0.944	1.678	0.949
40[50%]	40[75%]	[A]	1.880	0.944	1.742	0.945	1.589	0.951	1.835	0.945	1.715	0.948	1.542	0.951
		[B]	1.917	0.943	1.746	0.942	1.685	0.950	1.893	0.944	1.743	0.944	1.663	0.951
		[C]	1.833	0.945	1.719	0.947	1.517	0.953	1.773	0.947	1.693	0.949	1.503	0.953
80[25%]	80[50%]	[A]	1.579	0.948	1.368	0.951	1.315	0.956	1.508	0.950	1.332	0.957	1.258	0.960
		[B]	1.423	0.950	1.311	0.954	1.228	0.959	1.386	0.952	1.311	0.959	1.228	0.963
		[C]	1.327	0.952	1.246	0.956	1.206	0.960	1.302	0.953	1.239	0.962	1.159	0.964
80[50%]	80[75%]	[A]	1.252	0.955	1.233	0.957	1.166	0.963	1.234	0.957	1.178	0.962	1.092	0.965
		[B]	1.313	0.952	1.244	0.956	1.173	0.962	1.298	0.955	1.225	0.961	1.150	0.964
		[C]	1.234	0.956	1.197	0.958	1.125	0.964	1.203	0.958	1.174	0.962	1.067	0.967
			Pop-2
			$(t_1,t_2)=(1,2)$						$(t_1,t_2)=(2,5)$
20[25%]	20[50%]	[A]	0.566	0.921	0.325	0.928	0.281	0.931	0.543	0.921	0.324	0.928	0.280	0.932
		[B]	0.616	0.914	0.340	0.925	0.286	0.930	0.561	0.919	0.338	0.925	0.284	0.930
		[C]	0.549	0.922	0.321	0.929	0.279	0.932	0.540	0.924	0.320	0.929	0.278	0.932
20[50%]	20[75%]	[A]	0.537	0.925	0.316	0.929	0.259	0.937	0.516	0.925	0.316	0.930	0.256	0.937
		[B]	0.506	0.927	0.306	0.933	0.256	0.938	0.505	0.930	0.302	0.933	0.253	0.938
		[C]	0.499	0.930	0.304	0.933	0.255	0.938	0.482	0.931	0.300	0.933	0.253	0.938
40[25%]	40[50%]	[A]	0.403	0.942	0.258	0.943	0.238	0.943	0.391	0.942	0.253	0.943	0.235	0.944
		[B]	0.450	0.938	0.276	0.939	0.242	0.941	0.422	0.939	0.271	0.939	0.241	0.941
		[C]	0.377	0.941	0.253	0.944	0.244	0.946	0.373	0.943	0.251	0.944	0.231	0.948
40[50%]	40[75%]	[A]	0.375	0.944	0.244	0.946	0.230	0.948	0.355	0.944	0.239	0.947	0.230	0.949
		[B]	0.354	0.944	0.236	0.948	0.228	0.950	0.348	0.945	0.233	0.948	0.228	0.951
		[C]	0.349	0.944	0.223	0.950	0.229	0.951	0.339	0.944	0.230	0.949	0.228	0.952
80[25%]	80[50%]	[A]	0.290	0.948	0.194	0.954	0.189	0.956	0.286	0.951	0.190	0.955	0.186	0.956
		[B]	0.322	0.946	0.202	0.953	0.197	0.954	0.305	0.948	0.201	0.953	0.191	0.956
		[C]	0.273	0.950	0.193	0.954	0.183	0.957	0.256	0.953	0.185	0.957	0.178	0.958
80[50%]	80[75%]	[A]	0.241	0.954	0.173	0.957	0.171	0.959	0.238	0.955	0.172	0.958	0.170	0.959
		[B]	0.257	0.953	0.183	0.955	0.178	0.959	0.254	0.953	0.183	0.957	0.178	0.958
		[C]	0.238	0.954	0.170	0.957	0.169	0.960	0.222	0.956	0.162	0.959	0.167	0.960

. To distinguish, Table 2, Table 3, Table 4 and Table 5 list the Av.PEs (in the first column), RMSEs (second column), and ARABs (third column), whereas Table 6, Table 7, Table 8 and Table 9 list the AILs (in the first column) and CPs (second column) of a, b, $R\left(x\right)$, and $h\left(x\right)$, respectively. In terms of the smallest RMSE, ARAB, and AIL values and highest CPs, some statements about the estimates’ behavior can be presented as follows:

• The obtained estimates of a, b, $R\left(x\right)$, and $h\left(x\right)$ consistently demonstrate enhanced performance, yielding results of greater statistical efficiency.
• With increasing n (r or m), both the Bayesian and classical approaches produce stable and accurate estimates for a, b, $R\left(x\right)$, and $h\left(x\right)$. A similar improvement in estimation quality is evident when the total number of removals ${\sum }_{i=1}^m{R}_i$ is minimized.
• As the threshold values of $t_i,i=1,2$ grow, we observe that the RMSE, ARAB, and AIL values for all parameters and reliability metrics tend to decrease, while the associated CP values tend to increase.
• In both the Pop-1 and Pop-2 groups, Bayesian estimation using PB outperforms PA, as the former exhibits reduced prior variance, resulting in more stable and reliable posterior summaries for all parameters.
• Bayesian estimators of a, b, $R\left(x\right)$, and $h\left(x\right)$ calculated by the Pop-1 and Pop-2 groups provide superior results compared to their frequentist counterparts, largely due to the incorporation of gamma prior information.
• When the true values of INH$(a,b)$ increased, we noted for the Pop-1 and Pop-2 groups that:

  –

  The RMSE and ARAB associated with the frequentist (or Bayesian) estimates of a, b, $R\left(x\right)$, and $h\left(x\right)$ increase.

  –

  The AILs derived from ACI (or BCI) procedures also correspondingly increase; conversely, their CPs tend to decrease.

• Comparing the three censoring designs reported in Table 1, for both Pop-1 and Pop-2 groups it can be noted that the most efficient estimates of a and $h\left(x\right)$ are obtained using the right P-CT2 ‘Design [C]’, while those of of $R\left(x\right)$ are obtained using the left P-CT2 ‘Design [A]’ and those of b are obtained using the middle P-CT2 ‘Design B]’.
• In summary, when analyzing data generated from the UPH-CT1 strategy, the Bayesian approach based on Markovian iterations is strongly recommended for precise inference on INH model parameters and associated reliability measures.

6. Rare Minerals Data Analysis

This section aims to demonstrate the practical adaptability of the proposed inferential procedures by applying them to two empirical case studies drawn from real-world economy science contexts. For both real datasets, the MLE computations, including the ACIs, take 5 s; in comparison, the Bayesian MCMC procedure with credible intervals takes about 20–25 s per dataset on an HP laptop with a Core(TM) i5-5200U processor and 8.00 gigabytes of RAM.

6.1. Diamond Data

The diamond data from Southwest Africa (Namibia) are representive of a valuable economic resource, reflecting the scale and quality of mineral output from one of the region’s most vital industries. Accurate analysis of these data enables the optimization of extraction strategies, revenue forecasting, and informed investment decisions within the mining sector. Moreover, understanding the size distribution of diamonds contributes to market pricing, resource valuation, and long-term economic planning at both the national and corporate levels. Due to its real-world relevance, this application analyzes 25 diamond-sized items from a large mining area in southwest Africa; see

Table 10. Sizes (in carats) of 25 diamonds.

9	39	358	257.5	137	69.5	40.5	28	20.5	16.5
7.5	7	2.5	4.5	2	2	3	2	1	1.5
5	7	3	1	2

. This dataset was first introduced by Maurya and Goyal [21] and illustrated later by Alqasem et al. [22].

First, before developing the acquired theoretical estimators, we evaluate the suitability of the INH distribution in capturing the behavior of the diamond dataset. To achieve this goal, the MLEs (along with their standard errors (SEs)) and 95% ACIs (along with their interval widths (IWs)) of the parameters a and b are computed; see

Table 11. Fit results for the INH model from the diamond data.

Par.	MLE (SE)	95% ACI			$\mathbf{KS}$ (p-Value)
		Low.	Upp.	IW
a	0.6075 (0.2807)	0.0574	1.1575	1.1001	0.1339 (0.7614)
b	9.0211 (8.6243)	0.0000	25.924	25.924

. Next, in addition to obtaining the fitted MLE values of $\widehat{a}$ and $\widehat{b}$ from the entire diamond data, we calculate the Kolmogorov–Smirnov ($\mathsf{KS}$) statistic and its associated $p$-value; see also Table 11. Because the resulting $p$-value considerably exceeds the nominal significance ($\varrho =0.05$), there is no statistical evidence against the null hypothesis, revealing that the INH distribution adequately fits the observed diamond data.

Comprehensive diagnostic visualization provides multifaceted insights into model adequacy, and enables validation of fit; for this purpose, Figure 4 offers a comprehensive diagnostic visualization of the INH model’s performance, encompassing four distinct graphical tools: (a) the contour map of the log-likelihood; (b) the empirical and estimated reliability lines; (c) the empirical/estimated probability–probability (PP) lines; and (d) the empirical/estimated quantile–quantile (QQ) lines. Figure 4a provides strong empirical support for the existence and uniqueness of the calculated MLEs $\widehat{a}$ and $\widehat{b}$ from the diamond data. Now, the estimates $\widehat{a}\approx 0.6075$ and $\widehat{b}\approx 9.0211$ can be recommended as reliable initialization values for subsequent analytical implementations involving these data.

Additionally, the subplots in Figure 4b–d indicate that the estimated lines are very close to the corresponding empirical lines and support the same numerical fit outcomes listed in Table 11.

From the full diamond dataset, four artificial UPH-CT1 samples (symbolized by $\mathbb{S}\left[i\right],i=1,2,3,4,$) are constructed under $(r,m)=(8,15)$ and varying configurations of $t_i$ ($i=1,2$) and $R_i$ ($i=1,2,\dots ,m$), as presented in

Table 12. Different UPH-CT1 samples from diamond data.

Sample	R	${\mathit{t}}_1\left({\mathit{d}}_1\right)$	${\mathit{t}}_2\left({\mathit{d}}_2\right)$	${\mathit{R}}^{\mathbf{★}}$	${\mathit{s}}^{\mathbf{★}}$	Sample
$\mathcal{S}\left[1\right]$	($2^5$,$0^{10}$)	5(6)	70(15)	0	69.5	1, 1.5, 2, 2.5, 3, 4.5, 5, 7.5, 9, 16.5, 20.5, 28, 39, 40.5, 69.5
$\mathcal{S}\left[2\right]$	($0^5$,$2^5$,$0^5$)	41(13)	45(13)	2	41	1, 2, 2.5, 3, 5, 7, 7.5, 9, 16.5, 20.5, 28, 39, 40.5
$\mathcal{S}\left[3\right]$	($2^2$,$0^{10}$,$2^3$)	6(4)	30(8)	13	28	1, 2, 3, 5, 9, 16.5, 20.5, 28
$\mathcal{S}\left[4\right]$	($0^{10}$,$2^5$)	5(4)	10(6)	19	10	1, 2, 2.5, 4.5, 7.5, 9

. Due to the absence of prior information regarding the INH$(a,b)$ model for the given data, Bayesian inference is conducted via Algorithm 1, employing a total of $\mathbb{M}=50,000$ iterations with a burn-in of ${\mathbb{M}}^{•}=10,000$, assuming a joint non-informative gamma prior by setting ${\alpha }_i=0$ and ${\beta }_i=0,i=1,2$. Consequently,

Table 13. Estimates of a, b, $R\left(x\right)$, and $h\left(x\right)$ from the diamond data.

Sample	Par.	MLE		Bayes		95% ACI			95% BCI
		Est.	SE	Est.	SE	Low.	Upp.	IW	Low.	Upp.	IW
$\mathsf{S}\left[1\right]$	a	0.67067	0.24097	0.66530	0.06510	0.19838	1.14295	0.94458	0.53854	0.79434	0.25580
	b	9.75932	7.13384	9.75653	0.09977	0.22276	23.7414	23.5186	9.56011	9.95334	0.39323
	$R\left(3\right)$	0.80608	0.06808	0.79830	0.04952	0.67265	0.93950	0.26685	0.69265	0.88407	0.19142
	$h\left(3\right)$	0.10861	0.03425	0.10936	0.01255	0.04149	0.17574	0.13425	0.08369	0.13270	0.04901
$\mathsf{S}\left[2\right]$	a	0.45927	0.07094	0.45406	0.05053	0.32022	0.59831	0.27810	0.35701	0.55123	0.19421
	b	34.2369	8.35052	34.2351	0.10061	17.8702	50.6036	32.7334	34.0394	34.4327	0.39326
	$R\left(3\right)$	0.88691	0.05270	0.87586	0.04933	0.78362	0.99020	0.20658	0.76705	0.95065	0.18360
	$h\left(3\right)$	0.05707	0.01494	0.05837	0.01218	0.02777	0.08636	0.05858	0.03515	0.08162	0.04648
$\mathsf{S}\left[3\right]$	a	0.27127	0.03422	0.27094	0.00960	0.20419	0.33834	0.13415	0.25218	0.28991	0.03773
	b	312.559	8.39951	312.557	0.10062	296.096	329.022	32.9255	312.360	312.755	0.39493
	$R\left(3\right)$	0.92081	0.04456	0.91967	0.01267	0.83347	1.00814	0.17466	0.89304	0.94254	0.04949
	$h\left(3\right)$	0.02724	0.00888	0.02735	0.00249	0.00984	0.04464	0.03480	0.02250	0.03226	0.00976
$\mathsf{S}\left[4\right]$	a	0.31628	0.03962	0.31599	0.00968	0.23862	0.39394	0.15531	0.29706	0.33512	0.03806
	b	151.613	12.0308	151.611	0.10055	128.033	175.193	47.1600	151.414	151.809	0.39468
	$R\left(3\right)$	0.91621	0.04491	0.91536	0.01122	0.82820	1.00423	0.17603	0.89201	0.93596	0.04395
	$h\left(3\right)$	0.03290	0.01012	0.03299	0.00250	0.01305	0.05274	0.03969	0.02810	0.03792	0.00982

reports both the maximum likelihood and Bayesian point estimates (with their corresponding SEs) along with asymptotic and credible intervals (including their IWs) for the parameters a and b and reliability functions $R\left(x\right)$ and $h\left(x\right)$ evaluated at $x=3$. The findings reveal that Bayesian estimators consistently yield smaller SEs compared to the competitive maximum likelihood results, indicating greater estimation precision. Additionally, the lowest IW values confirm that the BCIs exhibit superior efficiency relative to the classical ACIs.

To validate the existence and uniqueness properties of the MLEs for parameters a and b of the INH$(a,b)$ distribution, Figure 5 presents log-likelihood contour plots under various configurations of $(a,b)$. These contours confirm that the MLEs $\widehat{a}$ and $\widehat{b}$ exist and are unique, supporting the numerical results reported in Table 13. Consequently, these estimates are recommended as effective initial values when executing Bayesian estimation procedures for each replicate in $\mathcal{S}\left[i\right],i=1,2,3,4$. To assess the convergence behavior of the MCMC sampling process, trace and posterior density plots for the remaining ${\mathbb{M}}^{\circ }=40,000$ simulated values of a, b, $R\left(x\right)$, and $h\left(x\right)$ based on (for instance) sample $\mathcal{S}\left[1\right]$ are jointly displayed in Figure 6. These diagnostics confirm that the M-H algorithm provided in Algorithm 1 converges satisfactorily, yielding credible and stable posterior estimates for the INH parameters $(a,b)$ and reliability measures $\left(R\right(x),h(x\left)\right)$ of interest. Moreover, the posterior distributions for a, b, and $h\left(x\right)$ exhibit high degrees of symmetry, while those of $R\left(x\right)$ are highly negative skewed. The autocorrelation function (ACF) depicted in Figure 7 also assesses the independence of the simulated MCMC variates after burn-in of a, b, $R\left(x\right)$, and $h\left(x\right)$, showing that their sample estimates are independently satisfactorily.

To further support these findings,

Table 14. Vital statistical summary for 40,000 MCMC iterations of a, b, $R\left(x\right)$, and $h\left(x\right)$ from the diamond data.

$\mathcal{S}\left[\mathit{i}\right]$	Par.	Mean	Mode	${\mathbb{Q}}_1$	${\mathbb{Q}}_2$	${\mathbb{Q}}_3$	St.D	Skewness
$\mathcal{S}\left[1\right]$	a	0.66530	0.62915	0.62166	0.66522	0.70885	0.06488	0.02901
	b	9.75653	9.73979	9.69007	9.75613	9.82312	0.09974	0.00423
	$R\left(3\right)$	0.79830	0.77325	0.76735	0.80187	0.83303	0.04890	−0.41703
	$h\left(3\right)$	0.10936	0.11679	0.10108	0.10972	0.11809	0.01253	−0.18035
$\mathcal{S}\left[2\right]$	a	0.45406	0.45264	0.41986	0.45393	0.48863	0.05026	0.00343
	b	34.2351	34.1035	34.1670	34.2350	34.3035	0.10059	0.02031
	$R\left(3\right)$	0.87586	0.88020	0.84727	0.88193	0.91150	0.04807	−0.73553
	$h\left(3\right)$	0.05837	0.05269	0.04980	0.05841	0.06678	0.01211	0.00420
$\mathcal{S}\left[3\right]$	a	0.27094	0.26799	0.26441	0.27093	0.27737	0.00959	0.00946
	b	312.557	312.406	312.489	312.557	312.625	0.10060	0.01986
	$R\left(3\right)$	0.91967	0.91641	0.91150	0.92037	0.92847	0.01262	−0.32504
	$h\left(3\right)$	0.02735	0.02705	0.02566	0.02732	0.02903	0.00248	0.04774
$\mathcal{S}\left[4\right]$	a	0.31599	0.31300	0.30940	0.31600	0.32249	0.00967	0.00731
	b	151.611	151.460	151.543	151.611	151.679	0.10054	0.01928
	$R\left(3\right)$	0.91536	0.91229	0.90804	0.91589	0.92313	0.01119	−0.27155
	$h\left(3\right)$	0.03299	0.03377	0.03130	0.03297	0.03468	0.00250	0.03472

summarizes seven key descriptive measures: the mean, mode, first, second, and third quartiles (${\mathbb{Q}}_1$, ${\mathbb{Q}}_2$, ${\mathbb{Q}}_3$), standard deviation (Std.D), and skewness for the MCMC draws of a, b, $R\left(x\right)$, and $h\left(x\right)$. The numerical summaries align well with the visual evidence in Figure 6, reinforcing the robustness and reliability of the posterior estimates.

Figure 8 depicts the ACI/BCI boundaries that indicate the performance of the reliability indices $R\left(x\right)$ and $h\left(x\right)$ over all data points in each UPH-CT1 sample taken from the diamond dataset. The figure shows that the fitted interval estimates of $R\left(x\right)$ and $h\left(x\right)$ generated by the BCI approach have shorter IWs than the competing ACI method, verifying the numerical data provided in Table 13.

6.2. Gold Data

In 2020, the top 100 central banks held a substantial share of global gold reserves, highlighting their strategic role in financial stability. These holdings, typically measured in thousands of tons, serve as safeguards against currency volatility and geopolitical risks, functioning not only as a hedge against currency depreciation and geopolitical uncertainty but also as a tool for reinforcing monetary stability. The composition and magnitude of these holdings reflect diverse economic strategies in which gold functions as both a store of value and a strategic reserve asset. Given the important impact of global gold holdings on the world economy, this application analyzes the quantities of global gold reserves (in hundreds of tons) for 100 countries; see

Table 15. Global gold reserves in 100 countries.

81.335	33.585	28.140	24.518	24.365	22.985	19.483	10.40	8.460	7.604
6.125	5.048	4.311	4.236	3.826	3.681	3.375	3.231	3.103	2.868
2.816	2.800	2.442	2.287	2.274	1.736	1.612	1.563	1.537	1.297
1.257	1.254	1.254	1.199	1.166	1.141	1.045	1.036	1.020	0.964
0.945	0.798	0.790	0.786	0.666	0.647	0.617	0.553	0.535	0.513
0.504	0.490	0.435	0.425	0.408	0.389	0.378	0.365	0.347	0.317
0.271	0.258	0.221	0.219	0.219	0.215	0.140	0.139	0.131	0.124
0.120	0.109	0.102	0.087	0.082	0.080	0.076	0.073	0.069	0.069
0.068	0.067	0.058	0.047	0.047	0.046	0.042	0.039	0.032	0.031
0.030	0.028	0.022	0.021	0.020	0.019	0.018	0.016	0.015	0.014

.

Following the same numerical and graphical goodness-of-fit tools utilized in Section 6.1, we examine whether or not the INH lifespan distribution is a suitable model to fit the gold data. For this purpose, using Table 15, the results in

Table 16. Fit results for the INH model from the gold data.

Par.	MLE (SE)	95% ACI			$\mathbf{KS}$ (p-Value)
		Low.	Upp.	IW
a	0.2783 (0.0309)	0.2177	0.3389	0.1212	0.0747 (0.6331)
b	2.7343 (0.9930)	0.7880	4.6806	3.8926

state that the INH model satisfactorily fits the global gold datasets and confirms that the investigated data are suitable for illustrating the frequentist and Bayesian estimators under study.

Figure 9a shows that the fitted likelihood estimates $\widehat{a}\cong 0.2783$ and $\widehat{b}\cong 2.7343$ (provided in Table 16) exist and are unique. Accordingly, for any forthcoming inferential procedures or computational routines utilizing the gold dataset, it is advisable to adopt these derived estimates as initial parameter inputs. The subplots in Figure 9b,c support the same conclusion.

To explore the performance of the proposed INH estimators, four UPH-CT1 datasets, denoted as $\mathbb{S}\left[i\right],i=1,2,3,4$, are generated based on the full gold dataset under a fixed sampling scheme with size $(r,m)=(30,50)$, varying threshold values of $t_i$ ($i=1,2$), and progressive censoring values $R_i$ ($i=1,\dots ,m$); see

Table 17. Different UPH-CT1 samples from the gold data.

Sample	R	${\mathit{t}}_1\left({\mathit{d}}_1\right)$	${\mathit{t}}_2\left({\mathit{d}}_2\right)$	${\mathit{r}}^{\mathbf{★}}$	${\mathit{s}}^{\mathbf{★}}$	Sample
$\mathcal{S}\left[1\right]$	($2^{25}$,$0^{25}$)	35(50)	40(40)	0	33.585	0.014, 0.021, 0.022, 0.030, 0.032, 0.042, 0.047, 0.073, 0.076, 0.080,
						0.102, 0.109, 0.120, 0.131, 0.215, 0.219, 0.221, 0.271, 0.365, 0.378,
						0.389, 0.408, 0.425, 0.490, 0.504, 0.513, 0.553, 0.647, 0.790, 0.798,
						0.945, 0.964, 1.036, 1.045, 1.166, 1.254, 1.257, 1.537, 1.563, 1.612,
						2.287, 2.800, 3.231, 4.311, 5.048, 10.400, 22.985, 24.365, 28.140, 33.585
$\mathcal{S}\left[2\right]$	($0^{12}$,$2^{25}$,$0^{13}$)	30(40)	35(40)	10	30	0.014, 0.022, 0.028, 0.031, 0.047, 0.047, 0.069, 0.082, 0.087, 0.102,
						0.139, 0.140, 0.219, 0.221, 0.271, 0.317, 0.365, 0.378, 0.425, 0.490,
						0.504, 0.553, 0.617, 0.798, 1.045, 1.141, 1.254, 1.254, 1.257, 2.442,
						2.800, 2.868, 3.231, 3.375, 4.236, 4.311, 5.048, 19.483, 24.518, 28.140
$\mathcal{S}\left[3\right]$	($2^{12}$,$0^{25}$,$2^{13}$)	5(25)	25(30)	46	22.985	0.014, 0.032, 0.042, 0.046, 0.067, 0.068, 0.073, 0.087, 0.102, 0.124,
						0.140, 0.215, 0.219, 0.221, 0.389, 0.553, 1.036, 1.166, 1.297, 1.612,
						2.274, 2.800, 2.816, 3.375, 3.681, 6.125, 7.604, 8.460, 19.483, 22.985
$\mathcal{S}\left[4\right]$	($0^{25}$,$2^{25}$)	4(26)	10(20)	80	10	0.014, 0.031, 0.058, 0.219, 0.408, 0.435, 0.490, 0.666, 0.945, 1.020,
						1.254, 1.257, 1.563, 2.442, 2.800, 2.816, 4.236, 4.311, 7.604, 8.460

. In light of the prior knowledge for the INH$(a,b)$ distribution in the context of the considered gold dataset, Bayesian estimation is implemented through the sampling-based procedure outlined in Algorithm 1. A total of $\mathbb{M}=50,000$ posterior samples are drawn, following a burn-in period of ${\mathbb{M}}^{•}=10,000$ iterations; for the prior specification, a non-informative joint gamma prior is adopted by setting ${\alpha }_i=0$ and ${\beta }_i=0$ for $i=1,2$. The estimation outcomes summarized in Table 13 include both classical (maximum likelihood) and Bayesian (MCMC) estimates for the model parameters a and b as well as the reliability measures $R\left(x\right)$ and $h\left(x\right)$ at $x=0.1$, together with their respective SE values. The empirical evidence highlights the superior precision of Bayesian estimators, as reflected in smaller SE values relative to their MLE counterparts. Moreover, the narrower widths observed for the BCIs further underscore their greater inferential efficiency when compared to traditional ACIs.

The contour subplots in Figure 10 confirm that the MLEs $\widehat{a}$ and $\widehat{b}$ exist and are unique, supporting the numerical results reported in

Table 18. Estimates of a, b, $R\left(x\right)$, and $h\left(x\right)$ from the gold data.

Sample	Par.	MLE		Bayes		95% ACI			95% BCI
		Est.	SE	Est.	SE	Low.	Upp.	IW	Low.	Upp.	IW
$\mathsf{S}\left[1\right]$	a	0.31838	0.04181	0.31815	0.00896	0.23643	0.40033	0.16390	0.30046	0.33577	0.03531
	b	3.62221	1.62491	3.61795	0.10001	0.43745	6.80698	6.36953	3.42236	3.81488	0.39253
	$R\left(0.1\right)$	0.88504	0.02670	0.88411	0.01212	0.83271	0.93737	0.10466	0.85924	0.90665	0.04742
	$h\left(0.1\right)$	1.27304	0.22026	1.27664	0.07411	0.84133	1.70474	0.86341	1.13195	1.42193	0.28998
$\mathsf{S}\left[2\right]$	a	0.18790	0.01285	0.18758	0.00779	0.16270	0.21309	0.05039	0.17236	0.20272	0.03036
	b	67.5509	6.08757	67.5492	0.10044	55.6195	79.4823	23.8629	67.3536	67.7474	0.39380
	$R\left(0.1\right)$	0.90951	0.02514	0.90793	0.01589	0.86024	0.95878	0.09855	0.87446	0.93591	0.06145
	$h\left(0.1\right)$	0.63514	0.09997	0.63809	0.06184	0.43921	0.83108	0.39187	0.51935	0.75977	0.24043
$\mathsf{S}\left[3\right]$	a	0.12303	0.00829	0.12279	0.00636	0.10679	0.13927	0.03248	0.11047	0.13535	0.02487
	b	3532.49	8.38876	3532.49	0.10031	3516.05	3548.93	32.8833	3532.29	3532.68	0.39300
	$R\left(0.1\right)$	0.92771	0.02275	0.92551	0.01795	0.88312	0.97230	0.08918	0.88697	0.95612	0.06915
	$h\left(0.1\right)$	0.34771	0.06437	0.35047	0.04915	0.22155	0.47387	0.25232	0.25629	0.44766	0.19138
$\mathsf{S}\left[4\right]$	a	0.14742	0.00906	0.14719	0.00668	0.12966	0.16518	0.03552	0.13433	0.16031	0.02599
	b	4531.11	5.93167	4531.10	0.10025	4519.48	4542.73	23.2517	4530.91	4531.30	0.39259
	$R\left(0.1\right)$	0.97888	0.00997	0.97760	0.00783	0.95935	0.99842	0.03907	0.96010	0.98972	0.02962
	$h\left(0.1\right)$	0.15449	0.04998	0.15821	0.03714	0.05653	0.25244	0.19591	0.09285	0.23563	0.14278

. Consequently, these estimates are recommended as effective initial values when executing Bayesian estimation procedures for each replicate in $\mathcal{S}\left[i\right],i=1,2,3,4$. Figure 11 shows that the posterior distributions for a, b, $R\left(x\right)$, and $h\left(x\right)$ exhibit similar performance to that depicted in Figure 6. Figure 12, based on sample $\mathcal{S}\left[1\right]$ (for example), indicates the independence of a, b, $R\left(x\right)$, and $h\left(x\right)$ and shows that their estimates are efficient and stable. The numerical results in

Table 19. Vital statistical summary for 40,000 MCMC iterations of a, b, $R\left(x\right)$, and $h\left(x\right)$ from the gold data.

$\mathcal{S}\left[\mathit{i}\right]$	Par.	Mean	Mode	${\mathbb{Q}}_1$	${\mathbb{Q}}_2$	${\mathbb{Q}}_3$	St.D	Skewness
$\mathcal{S}\left[1\right]$	a	0.31815	0.32166	0.31212	0.31815	0.32416	0.00896	0.00814
	b	3.61795	3.45842	3.55006	3.61786	3.68573	0.09993	0.02293
	$R\left(0.1\right)$	0.88411	0.88408	0.87616	0.88456	0.89251	0.01209	−0.19668
	$h\left(0.1\right)$	1.27664	1.29319	1.22654	1.27644	1.32670	0.07402	−0.00102
$\mathcal{S}\left[2\right]$	a	0.18758	0.18232	0.18231	0.18760	0.19287	0.00778	−0.01016
	b	67.5492	67.5460	67.4810	67.5490	67.6172	0.10043	0.02836
	$R\left(0.1\right)$	0.90793	0.89793	0.89781	0.90892	0.91910	0.01581	−0.36524
	$h\left(0.1\right)$	0.63809	0.64911	0.59574	0.63752	0.67983	0.06177	0.05264
$\mathcal{S}\left[3\right]$	a	0.12279	0.12483	0.11846	0.12276	0.12716	0.00636	0.00889
	b	3532.49	3532.45	3532.42	3532.49	3532.55	0.10029	0.02149
	$R\left(0.1\right)$	0.92551	0.93253	0.91436	0.92697	0.93842	0.01781	−0.50245
	$h\left(0.1\right)$	0.35047	0.33381	0.31604	0.34979	0.38361	0.04907	0.09293
$\mathcal{S}\left[4\right]$	a	0.14719	0.14646	0.14266	0.14717	0.15173	0.00668	0.01336
	b	4531.10	4531.00	4531.04	4531.10	4531.17	0.10023	0.02339
	$R\left(0.1\right)$	0.97760	0.97781	0.97311	0.97860	0.98322	0.00772	−0.79624
	$h\left(0.1\right)$	0.15821	0.15982	0.13176	0.15590	0.18199	0.03695	0.36601

support the same findings listed in Table 18 and similar facts shown in Figure 11. Figure 13 reveals that the estimated bounds of $R\left(x\right)$ and $h\left(x\right)$ developed by the BCI approach have shorter IWs than the competing ACI method, verifying the numerical data provided in Table 18.

7. Optimal Progressive Design

In reliability experimentation, determining the most informative censoring mechanism among several competing designs is a critical issue; another key objective is to identify the scheme that maximizes the inferential precision regarding the underlying unknown parameters. These competing designs comprise different selections of the censoring sets $(R_1,\dots ,R_m)$ based on prefixed values of n and m. As such, considerable effort on the part of the statistical community has been devoted to developing methods for selecting optimal progressive censoring strategies. The essence of this task lies in evaluating which progressive censoring configuration yields the greatest insight into the underlying model parameters; for additional details, see Ng et al. [23], Pradhan and Kundu [24], Sen et al. [25], Ashour et al. [26], and Nassar and Elshahhat [27], among others.

Below, we list a comparative analysis of several well established optimality criteria commonly employed in the literature, helping to guide the selection of the most effective P-CT2, $\mathbf{R}$ sampling design:

• Maximizing the trace of observed FI items, denoted by ${\mathbb{C}}_1$, as

  $${\mathbb{C}}_1=max\left[{\sum }_{i=1}^2{\mathcal{F}}_{ii}\left(\mathit{\theta }\right)\right];$$
• Minimizing the trace of observed VC items ${\mathcal{F}}_{ii}^{-1},i=1,2$, denoted by ${\mathbb{C}}_2$, as

  $${\mathbb{C}}_2=min\left[{\sum }_{i=1}^2{\mathcal{F}}_{ii}^{-1}\left(\mathit{\theta }\right)\right];$$
• Minimizing the determinant of observed VC items ${\mathcal{F}}_{ii}^{-1},i=1,2$, denoted by ${\mathbb{C}}_3$, as

  $${\mathbb{C}}_3=min\left[det\left({\mathcal{F}}_{ij}^{-1}\left(\mathit{\theta }\right)\right),i,j=1,2\right];$$
• Minimizing the MLE of the logarithm of $\gamma$th quantile, denoted by ${\mathbb{C}}_4$, as

  $${\mathbb{C}}_4=min\left[\Omega {\mathcal{F}}^{-1}{\Omega }^{\top }\right],$$

  where $\Omega =\left[{\Omega }_a{\Omega }_b\right]$ and $\overline{\gamma }=1-log\left(\gamma \right)$ such that

  $${\Omega }_a=\left[{\displaystyle \frac{{\overline{\gamma }}^{\frac{1}{a}}log\left(\overline{\gamma }\right)}{a^2({\overline{\gamma }}^{\frac{1}{a}}-1)}}\right]\mathrm{and}{\Omega }_b={\displaystyle \frac{1}{b}}.$$

It is worth highlighting that the evaluation of all the proposed criteria (${\mathbb{C}}_i,i=1,2,3,4,$) is carried out based on the fitted frequentist point estimates of the INH$(a,b)$ model parameters developed from the proposed real-world datasets analyzed in Section 6. Using the proposed censored samples ($\mathbb{S}\left[i\right],i=1,2,3,4$) created from the diamond and gold datasets reported in Table 12 and Table 17, respectively,

Table 20. Fitted criteria of optimal P-CT2.

Sample	${\mathbb{C}}_1$	${\mathbb{C}}_2$	${\mathbb{C}}_3$	${\mathbb{C}}_4$
$\mathit{\gamma }\to$				0.3	0.6	0.9
Diamond Data
$\mathcal{S}\left[1\right]$	137.163	50.9498	0.37145	1.36213	14.2477	569.595
$\mathcal{S}\left[2\right]$	295.351	69.7363	0.23611	4.11730	27.0794	810.458
$\mathcal{S}\left[3\right]$	655.131	70.5529	0.22094	48.5344	460.554	11297.7
$\mathcal{S}\left[4\right]$	856.090	144.743	0.08241	20.9461	161.392	3940.41
Gold Data
$\mathcal{S}\left[1\right]$	2448.32	2.64208	0.00108	0.00528	0.13424	10.0250
$\mathcal{S}\left[2\right]$	6374.19	37.0587	0.00581	0.08528	2.13306	113.481
$\mathcal{S}\left[3\right]$	12177.5	70.3713	0.00483	6.19312	896.123	76507.6
$\mathcal{S}\left[4\right]$	14567.7	35.1848	0.00289	50.1443	2894.48	155652.5

lists the estimated criteria ${\mathbb{C}}_i,i=1,2,3,4,$ to select the optimal P-CT2 plan. As a result, from Table 20 we notice the following:

(i)

From the Diamond Data:

• According to criteria ${\mathbb{C}}_i,i=1,3$, the left P-CT2 (used in $\mathbb{S}\left[1\right]$) is more optimal than the others;
• According to criteria ${\mathbb{C}}_i,i=2,4$, the right P-CT2 (used in $\mathbb{S}\left[4\right]$) is more optimal than the others.

(ii)

From the Gold Data:

• According to criterion ${\mathbb{C}}_1$, the right P-CT2 (used in $\mathbb{S}\left[4\right]$) is more optimal than the others;
• According to criteria ${\mathbb{C}}_i,i=2,3,4$, the left P-CT2 (used in $\mathbb{S}\left[1\right]$) is more optimal than the others.

This empirical evaluation based on the two real-world industrial datasets provides strong evidence supporting the suitability of the INH distribution as an effective model for describing the underlying data patterns. The optimal censoring plans generated from the diamond and gold datasets align closely with outcomes from the Monte Carlo evaluations, supporting their reliability. Furthermore, several insights emerge regarding the operational flexibility of the UPH-CT1 censoring scheme:

(1)

It demonstrates considerable adaptability, particularly in scenarios where the termination of the test is primarily driven by the number of observed failures;

(2)

It allows researchers to maintain control over the experiment’s duration;

(3)

It facilitates the extraction of reliable and informative estimates for both the reliability function and the hazard rate without the need to observe the entire sample, offering significant reductions in time, cost, and manpower.

8. Concluding Remarks

Unified progressive hybrid censoring was developed to integrate the strengths of progressive, hybrid, and unified hybrid censoring plans by simultaneously controlling the number of failures and the censoring time while allowing the progressive removal of surviving units at each failure point. Considering this censoring design, this has study presented a comprehensive analysis of the INH model, which is a robust lifetime model capable of capturing complex hazard behaviors, including heavy-tailed and upside-down bathtub shapes. To achieve this goal, we have conducted a comprehensive statistical framework through two inferential paradigms, namely, maximum likelihood estimation and a Bayesian framework employing independent gamma priors and Metropolis–Hastings-based MCMC simulations. Asymptotic confidence intervals were developed using the delta method and estimated Fisher information, while Bayesian credible intervals were constructed from posterior MCMC iterations. A key contribution lies in the optimal design assessment, with multiple progressive schemes evaluated under various censoring thresholds, failure percentages, and sample sizes. Simulation experiments under two INH populations demonstrate that the results for both estimation procedures are consistent, with the Bayesian estimators outperforming their frequentist counterparts across several metrics of precision. To highlight the ability of theoretical results under such censored data to reflect practical conditions, two real applications from the rare minerals sector, namely, diamonds and gold, have been analyzed in order to demonstrate the feasibility of the studied methodologies. Furthermore, optimal plans that demonstrate adaptability to different control strategies have been identified through the analysis outcomes of the diamond and gold datasets. Based on numerical analysis, we recommend using the Bayesian MCMC-based approach when the sample size is small ($n\le 40$) or the number of observed failures is limited ($m<20$), as the asymptotic normality assumptions required for classical confidence intervals may not hold. Conversely, with sufficient data availability ($n\ge 40$ and $m\ge 20$), both approaches perform comparably, although the MLE-based approach may be favored for its computational speed and ease of implementation. In summary, this study introduces a novel censoring design and dual inferential strategy, then validates it through simulations, enabling more flexible reliability analyses. Future research may explore multivariate settings, competing risk frameworks, or nonparametric adaptations.