A New Exponential-Type Model Under Unified Progressive Hybrid Censoring: Computational Inference and Its Applications

Abstract

A new odd exponential-type (NOT-Exp) distribution provides a flexible and analytically tractable framework for modeling lifetime data exhibiting non-constant hazard behaviors, including increasing, decreasing, bathtub-shaped, and unimodal forms, which are commonly observed in real-world reliability and survival studies. In this work, a comprehensive inferential methodology is developed for the NOT-Exp model under a unified progressive Type-II hybrid censoring, allowing several traditional censoring designs to be treated as special cases within a single unified structure. The main advantages of the proposed model lie in its ability to capture complex risk dynamics while maintaining mathematical simplicity, making it particularly suitable for censored lifetime data. Classical inference is conducted via maximum likelihood estimation, along with two asymptotic confidence interval constructions based on normal and log-normal approximations for both model parameters and reliability characteristics. In addition, a Bayesian estimation framework is introduced using independent gamma priors and Markov chain Monte Carlo techniques to obtain posterior estimates, credible intervals, and highest posterior density regions. Extensive simulations demonstrate the accuracy, stability, and robustness of the proposed estimators under varying sample sizes, censoring intensities, and prior specifications. Applications to airborne toxicological variation data and bank customer waiting times highlight the practical importance of the methodology, where the NOT-Exp model consistently outperforms twelve competing lifetime distributions according to standard goodness-of-fit criteria. These results confirm that the suggested design gives a strong and versatile tool for analyzing complex censored lifetime data across environmental and service-system applications.

1. Introduction

Recently, Sapkota et al. [1] introduced a new family of exponential-based distributions that are meant to make it easier to model lifetime and survival data in various practical fields. The proposed family extends the classical exponential distribution and can attract a wide range of hazard rate (HR) behaviors, including increasing, decreasing, bathtub-shaped, and unimodal patterns. One of the main strengths of this new family lies in its balance between flexibility and analytical tractability, called the new odd-exponential (NOT-Exp) distribution, which represents a meaningful advance in the modeling of positive continuous data, particularly for lifetime, reliability, and survival analysis. In contrast to the classical exponential distribution, which is limited to a constant HR, the NOT-Exp model incorporates additional shape parameters, enabling it to capture a broad spectrum of failure patterns, including decreasing, increasing, reverse-J, S-shaped, and bathtub forms. In empirical comparisons using real engineering and medical datasets, Sapkota et al. [1] stated that the NOT-Exp model consistently outperforms several well-known competing models based on standard fit criteria, indicating a clear advantage of the NOT-Exp distribution is its adaptability across different application domains.

This flexibility makes the model capable of representing diverse real-world phenomena that standard distributions often cannot capture. They also derived both classical and Bayesian NOT-Exp estimations with complete datasets. Simulation results show that these estimators are stable and efficient, exhibiting decreasing bias and mean square error as sample size increases, which makes the model reliable even for small or complex datasets. In Section 2, we provided the main NOT-Exp distribution functions and investigated their density and failure rate behaviors.

In reliability studies, censoring commonly occurs when only a subset of the sample lifetimes is observable. The statistical literature offers comprehensive treatments of different censoring schemes, including Type-I (time-based), Type-II (failure-based), and hybrid plans, the latter of which integrate elements of both time and failure schemes. A new modified sampling strategy of the traditional hybrid and Type-II progressive censoring (T2-P) plans, by G’orny and Cramer [2], is called a unified progressively Type-II hybrid (UT2-PH) censoring. In practical experiments, examiners often face constraints related to limited testing time and experimental cost. The plan proposed addresses these challenges by allowing the experiment to terminate either after observing a pre-specified number of failures or when a predetermined time limit is reached, while also permitting the progressive removal of surviving units during the experiment. This flexibility makes the scheme more practical and realistic for modern reliability studies, as it enables efficient data collection while controlling experimental duration and resource consumption. Consequently, the UT2–PH censoring scheme provides a more comprehensive and adaptable framework for modeling lifetime data under practical experimental constraints. Within this censoring framework, a random sample of size n is subjected to a life-testing experiment beginning at time zero, with failure times (say, $m_i$ for $i=1,2$) recorded continuously throughout the study period. Let $q_i\in \{1,\dots ,n\},i=1,2$ (where $q_1<q_2$) denote the predetermined numbers of failures, and let ${\mathsf{ȷ}}_i\in (0,\infty ),i=1,2$ (with ${\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2$) represent the corresponding time thresholds. Consider a T2-P censoring scheme $\mathbf{S}=(S_1,\dots ,S_{q_2})$ that satisfies $q_2+{\sum }_{i=1}^{q_2}{S}_i=n$. Following Górny and Cramer [2], it is important to note that, under the UT2-PH censoring design, the removal procedure mirrors that of the standard T2-P scheme in such a way that after the occurrence of the ith failure, a random number $S_i$ of surviving items are withdrawn from the experiment.

Suppose that $\mathbf{Y}$ represents a UT2-PH sample taken from a NOT-Exp population with a lifetime characterized by a probability density function (PDF, $g(·)$) and a cumulative distribution function (CDF, $G(·)$); then, the corresponding likelihood function (LF, $\mathcal{L}(·)$) is

$$\mathcal{L}\left(\mathit{\xi }|\mathbf{x}\right)\propto {\prod }_{i=1}^{\nabla }g(y_i;\mathit{\xi }){[1-G(y_i;\mathit{\xi })]}^{S_i}{[1-G({\mathcal{T}}^{\star };\mathit{\xi })]}^{{\mathcal{S}}^{\star }},$$

(1)

where $y_i$ represents $y_{i:m_2:n}$ for simplicity, $d_i$ denotes the number of failures up to ${\mathsf{ȷ}}_i$ (for $i=1,2$), and

$$\left(\nabla ,{\mathcal{T}}^{\star },{\mathcal{S}}^{\star }\right)=\left\{\begin{array}{ccc}\left(q_2,y_{q_2},n-q_2-{\sum }_{i=1}^{q_2-1}{S}_i)\right)\hfill & & \mathrm{i}\mathrm{f}{Y}_{q_2}<{\mathsf{ȷ}}_1;\hfill \\ \left(d_1,{\mathsf{ȷ}}_1,n-d_1-{\sum }_{i=1}^{d_1}{S}_i)\right)\hfill & & \mathrm{i}\mathrm{f}{Y}_{q_1}<{\mathsf{ȷ}}_1<Y_{q_2}(\mathrm{o}\mathrm{r}{Y}_{q_1}<{\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2);\hfill \\ \left(q_1,y_{q_1},n-q_1-{\sum }_{i=1}^{q_1-1}{S}_i)\right)\hfill & & \mathrm{i}\mathrm{f}{\mathsf{ȷ}}_1<Y_{q_1}<{\mathsf{ȷ}}_2(\mathrm{o}\mathrm{r}{\mathsf{ȷ}}_1<Y_{q_1}<Y_{q_2});\hfill \\ \left(d_2,{\mathsf{ȷ}}_2,n-d_2-{\sum }_{i=1}^{d_2}{S}_i)\right)\hfill & & \mathrm{i}\mathrm{f}{\mathsf{ȷ}}_2<Y_{q_1}.\hfill \end{array}\right.$$

It is worth noticing here that the control mechanism in UT2-PH combines several stopping rules, including a predefined number of failures, progressive removals, and time-based termination. In practice, certain edge cases may arise during sample generation, particularly when the remaining number of test units at the i-th failure (symbolized by ${\mathcal{S}}_i^{\star }$) becomes smaller than the preassigned number of removals ($S_i$) or when termination occurs earlier due to the time constraint ${\mathcal{T}}^{\star }$. Specifically, Algorithm 1 verifies that the number of remaining units is always sufficient to accommodate the specified progressive removals. If we denote by $S_i$ the planned number of removals at the i-th observed failure and by $N_i$ the number of units still under observation just before the i-th failure, then the feasibility condition requires $S_i\le N_i-1$. If $S_i>N_i-1$, the removal number is adjusted to ${\mathcal{S}}_i^{\star }=min(S_i,N_i-1)$, ensuring that the total number of withdrawn units does not exceed the number of surviving items.

Algorithm 1 The MCMC iterative procedure of $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$

1: Inputs: True values $\alpha$, $\gamma$, $\theta$, total iterations ℘, burn-in ${\wp }^{\star }$, remaining size ${\wp }^{\ast }=\wp -{\wp }^{\star }$.   2: Initialization:   3: Set ${\alpha }^{\left(0\right)}=\widehat{\alpha }$, ${\gamma }^{\left(0\right)}=\widehat{\gamma }$, ${\theta }^{\left(0\right)}=\widehat{\theta }$.   4: for $j=1$ to ℘ do   5:     Generate candidate value:   6:     ${\alpha }^{\ast }\sim N(\widehat{\alpha },\widehat{\mathrm{Var}}\left(\widehat{\alpha }\right))$   7:     Compute   8:     ${\mathsf{\ell }}_{\alpha }={\displaystyle \frac{{\mathcal{P}}_{\alpha }^{\star }\left({\alpha }^{\ast }\right|{\gamma }^{(j-1)},{\theta }^{(j-1)},\mathbf{x})}{{\mathcal{P}}_{\alpha }^{\star }\left({\alpha }^{(j-1)}\right|{\gamma }^{(j-1)},{\theta }^{(j-1)},\mathbf{x})}}$   9:     Draw $u\sim U(0,1)$ 10:     if $u\le min(1,{\mathsf{\ell }}_{\alpha })$ then 11:         ${\alpha }^{\left(j\right)}={\alpha }^{\ast }$ 12:     else 13:         ${\alpha }^{\left(j\right)}={\alpha }^{(j-1)}$ 14:     end if 15:     Repeat for ${\gamma }^{\left(j\right)}$ and ${\theta }^{\left(j\right)}$ 16:     Compute $R^{\left(j\right)}\left(x\right)$ and $h^{\left(j\right)}\left(x\right)$ for $x>0$ 17: end for 18: Bayes estimate: 19: ${\displaystyle \tilde{\alpha }=\frac{1}{{\wp }^{\ast }}\sum _{j={\wp }^{\star }+1}^{\wp }{\alpha }^{\left(j\right)}}$ 20: Construction of $100(1-\sigma )\%$ BCIs: 21: Sort $\left\{{\alpha }^{\left(j\right)}\right\}$ for $j={\wp }^{\star }+1,\dots ,\wp$ 22: ${\displaystyle {\mathrm{BCI}}_{95\%}\left(\alpha \right)=\left({\alpha }_{\left(0.5\sigma {\wp }^{\ast }\right)},{\alpha }_{\left({\wp }^{\ast }(1-0.5\sigma )\right)}\right)}$ 23: Construction of $100(1-\sigma )\%$ HPD intervals: 24: ${\displaystyle {\mathrm{HPD}}_{95\%}\left(\alpha \right)=\left({\alpha }_{\left(j^{\ast }\right)},{\alpha }_{(j^{\ast }+{\wp }^{\ast }(1-\sigma ))}\right)}$ 25: where $j^{\ast }$ minimizes 26: ${\displaystyle {\alpha }_{(j+{\wp }^{\ast }(1-\sigma ))}-{\alpha }_{\left(j\right)}}$ 27: Repeat for $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$

From (1), numerous inferential studies have been conducted. For example, Lone et al. [3] investigated the gamma-mixed Rayleigh model, Anwar et al. [4] considered the inverted exponentiated Rayleigh, Bayoud et al. [5] studied the Topp–Leone, Dutta et al. [6] focused on the Burr-III, Mohammed et al. [7] examined the inverse Nadarajah–Haghighi, and Prakash et al. [8] proposed related methodological extensions, among other contributions.

In modern reliability, survival, and risk-analysis studies, real-world lifetime data are rarely observed completely due to cost, ethical, or operational constraints, leading to increasingly complex censoring mechanisms. At the same time, classical lifetime models, most notably the exponential distribution, often fail to adequately capture the diverse HR behaviors encountered in applied fields such as toxicology and banking management. Although the recently proposed NOT-Exp distribution offers remarkable flexibility in modeling increasing, decreasing, bathtub-shaped, and unimodal HRs, its inferential development has so far been limited primarily to complete data settings. Moreover, existing studies have not yet explored the performance of the NOT-Exp model under unified progressive hybrid censoring schemes, nor have they provided a comprehensive comparison between classical and Bayesian inference for both model parameters and reliability measures. Motivated by these gaps, the present study aims to extend the theoretical and inferential framework of the NOT-Exp distribution to more realistic censoring environments, while delivering robust estimation procedures and practical insights through simulation and real data applications in toxicology and banking systems. The key methodological and inferential contributions of this study can be summarized in six main points as follows:

• Various inference frameworks for the NOT-Exp distribution using diverse datasets generated from the UT2-PH plan are proposed.
• Two asymptotic confidence interval approaches, including normal-based and log-normal-based, are systematically developed and compared for both parameters and reliability time functions, addressing positivity and finite-sample limitations.
• A comprehensive Bayesian estimation procedure is proposed using independent gamma priors and Metropolis–Hastings MCMC algorithms, allowing for the estimation of parameters, reliability function (RF), and HR function (HRF) along with credible and HPD intervals.
• An extensive simulation study evaluates estimator accuracy based on different metrics of precision under varying sample sizes, censoring designs, and prior specifications, providing practical guidance on optimal censoring designs.
• The study demonstrates the superior stability and efficiency of Bayesian estimators—particularly HPD intervals—over classical counterparts in heavily censored scenarios.
• Airborne variation and bank waiting time datasets represent two challenging real-world lifetime settings marked by heterogeneity, skewness, non-constant hazard behavior, and realistic censoring, making them ideal benchmarks for advanced reliability modeling.
• Empirical analyses clearly demonstrate the superiority of the NOT-Exp model over twelve competitive lifetime distributions—alpha-power exponential, Weibull exponential, Nadarajah and Haghighi, generalized exponential, and Weibull distributions, among others—consistently yielding better fit and inferential performance across both environmental and banking applications.

It is important to remember here that the proposed analysis is developed under the UT2–PH censoring framework and focuses on the NOT-Exp distribution, and therefore, the results may depend on the assumed model structure and censoring mechanism. In addition, the computational procedures required for maximum likelihood and Bayesian estimation may become more demanding for larger datasets or more complex censoring schemes. Furthermore, although four interval estimation methodologies are considered in this study, they may be extended in future work using resampling-based iterative procedures. These aspects represent potential limitations of the current work and also indicate several promising directions for future research.

The remainder of this article is structured as follows. Section 2 introduces the NOT-Exp model and examines its principal properties. Frequentist and Bayesian inference procedures are described in Section 3 and Section 4, respectively. Section 5 presents an extensive simulation study evaluating the performance of the proposed theoretical methods. Section 6 is devoted to applications involving toxicological and banking management data sets. In Section 7, four optimal methods are also examined. Finally, conclusions are summarized in Section 8.

2. The NOT-Exp Model

A lifetime random variable Y follows the NOT-Exp($\mathit{\xi }$) distribution, where $\mathit{\xi }={(\alpha ,\gamma ,\theta )}^{\top }$ with $\alpha >0$ and $\gamma >0$ are the shape parameters and $\theta$ is the scale parameter, if its CDF and PDF are given, respectively, as

$$G(y;\mathit{\xi })={(1+\alpha )}^{\gamma }{\left[{\displaystyle \frac{\vartheta \left(y\right)}{\alpha +\vartheta \left(y\right)}}\right]}^{\gamma },y>0,$$

(2)

and

$$g(y;\mathit{\xi })=\alpha \gamma \theta {(1+\alpha )}^{\gamma }{e}^{-\theta y}{\displaystyle \frac{\vartheta {\left(y\right)}^{\gamma -1}}{{\left(\alpha +\vartheta \left(y\right)\right)}^{\gamma +1}}},$$

(3)

where $\vartheta \left(y\right)=1-e^{-\theta y}$.

Setting $\alpha \to \infty$ and $\gamma =1$ into (2), the standard exponential distribution with scale parameter $\theta$ is introduced. In addition to analyzing the model parameters, investigating the RF $R(·)$, and HRF $h(·)$, is crucial as it reveals risk and hazard patterns over time, helping to improve prediction, prevention strategies, and overall system safety. From (2) and (3), the corresponding $R(·)$ and $h(·)$ of the NOT-Exp distribution (for $x>0$), respectively, are

$$R(x;\mathit{\xi })=1-{(1+\alpha )}^{\gamma }{\left[{\displaystyle \frac{\vartheta \left(x\right)}{\alpha +\vartheta \left(x\right)}}\right]}^{\gamma },x>0,$$

(4)

and

$$h(x;\mathit{\xi })={\displaystyle \frac{\alpha \gamma \theta {(1+\alpha )}^{\gamma }{e}^{-\theta x}\vartheta {\left(x\right)}^{\gamma -1}}{(\alpha +\vartheta \left(x\right))\left[{(\alpha +\vartheta \left(x\right))}^{\gamma }-{(1+\alpha )}^{\gamma }{\vartheta \left(x\right)}^{\gamma }\right]}}.$$

(5)

Figure 1a displays the PDFs, demonstrating the model’s capability to capture diverse lifetime behaviors through different shapes, ranging from right-skewed to more symmetric distributions. Figure 1b shows the corresponding hazard rate functions, which exhibit multiple characteristic failure rate behaviors common in reliability analysis. For instance, an increasing hazard rate (IHR) is observed when $\alpha =0.1$, $\gamma =0.1$, and $\theta =0.1$, reflecting aging or wear-out effects where the risk of failure grows over time. Conversely, a decreasing hazard rate (DHR) occurs when $\alpha =2.8$, $\gamma =4.5$, and $\theta =1.5$, which model early-life failures or infant mortality phenomena. A bathtub-shaped (BTS) hazard appears when $\alpha =1.5$, $\gamma =2.8$, and $\theta =1.5$, indicating an initial decline in failure rate followed by a period of constant or increasing risk. Additionally, the model captures an upside-down bathtub-shaped (UBTS) hazard when $\alpha =2.5$, $\gamma =0.9$, and $\theta =3.2$, representing systems that experience a peak failure period before stabilization. Overall, the different shapes of the PDF and HRF highlight the versatility of the NOT-Exp model in explaining various phenomena and failure mechanisms observed in practice.

In general, when the sample size is small or when the data are subject to heavy censoring, the available information about the distribution tail may be limited. In such situations, the Fisher information matrix can become nearly singular for some parameter combinations, resulting in relatively flat likelihood regions and increased variability of the parameter estimates. From a structural perspective, the parameter $\theta$ primarily governs the scale of the lifetime distribution, while the parameters $\alpha$ and $\gamma$ control the shape and curvature of the density and failure rate functions. This separation of roles allows the model to generate a wide spectrum of hazard behaviors while maintaining interpretability of the parameters. Similar identifiability considerations have been reported for other flexible lifetime models, including the Weibull, gamma, and generalized exponential distributions when the available data provide limited information about the failure-time distribution. Consequently, while the NOT-Exp model provides considerable flexibility for modeling complex hazard rate patterns, its most reliable performance is expected under moderate sample sizes or realistic censoring schemes where sufficient information is available to distinguish the effects of the model parameters.

3. Likelihood Inference

We derive, in this section, the maximum likelihood estimators (MLEs) of the parameter vector $\mathit{\xi }$ and the associated $R\left(x\right)$ and $h\left(x\right)$ for the NOT-Exp distribution based on UT2-PH censored data. In addition, two different approaches are employed to construct two-sided $(1-\sigma )100\%$ asymptotic confidence intervals (ACIs) for the same unknown quantities.

3.1. Maximum Likelihood Estimators

Let $\mathbf{y}=\left\{Y_i\right\},i=1,2,\dots ,\nabla$, denote a UT2-PH censored sample drawn from the NOT-Exp distribution with parameter vector $\mathit{\xi }$ under a fixed T2-P censoring scheme $\mathbf{S}$. Using (1), (3), and (2), the joint-LF of $\mathit{\xi }$ can be expressed as

$$\begin{array}{cc}\hfill \mathcal{L}\left(\left.\mathit{\xi }\right|\mathbf{y}\right)& \propto {\left(\alpha \gamma \theta \right)}^{\nabla }{\left(1+\alpha \right)}^{\nabla \gamma }{e}^{-\theta {\sum }_{i=1}^{\nabla }{y}_i}\hfill \\ \hfill & \times \prod _{i=1}^{\nabla }{\displaystyle \frac{\vartheta {\left(y_i\right)}^{\gamma -1}}{{\left(\alpha +\vartheta \left(y_i\right)\right)}^{\gamma +1}}}{\left[1-{\left(\eta \left(y_i;\alpha ,\theta \right)\right)}^{\gamma }\right]}^{S_i}{\left[1-{\left(\eta \left(T^{\star };\alpha ,\theta \right)\right)}^{\gamma }\right]}^{S^{\star }},\hfill \end{array}$$

(6)

where $\eta \left(y_i;\alpha ,\theta \right)={\displaystyle \frac{\left(1+\alpha \right)\vartheta \left(y_i\right)}{\left(\alpha +\vartheta \left(y_i\right)\right)}}$ for $i=1,2,\dots ,\nabla$.

The corresponding log-LF (${\mathcal{L}}^{\star }(·)\propto log\mathcal{L}(·)$) becomes

$$\begin{array}{cc}{\mathcal{L}}^{\star }\left(\left.\mathit{\xi }\right|\mathbf{y}\right)\hfill & \propto \nabla log\left(\alpha \gamma \theta \right)+\nabla \gamma log\left(1+\alpha \right)-\theta \sum _{i=1}^{\nabla }{y}_i\hfill \\ \hfill & +\left(\gamma -1\right)\sum _{i=1}^{\nabla }log\left(\vartheta \left(y_i\right)\right)-\left(\gamma +1\right)\sum _{i=1}^{\nabla }log\left(\alpha +\vartheta \left(y_i\right)\right)\hfill \\ \hfill & +\sum _{i=1}^{\nabla }{S}_{i}log\left(1-{\left(\eta \left(y_i;\alpha ,\theta \right)\right)}^{\gamma }\right)+S^{\star }log\left(1-{\left(\eta \left(T^{\star };\alpha ,\theta \right)\right)}^{\gamma }\right).\hfill \end{array}$$

(7)

Differentiating (7) with respect to $\alpha$, $\vartheta$, and $\theta$ yields the following likelihood equations:

$$\begin{array}{c}\hfill {\left.{\displaystyle \frac{\nabla }{\alpha }}+{\displaystyle \frac{\nabla \gamma }{1+\alpha }}-\left(\gamma +1\right)\sum _{i=1}^{\nabla }{\left(\alpha +\vartheta \left(y_i\right)\right)}^{-1}-\sum _{i=1}^{\nabla }{S}_i{A}_i-S^{\star }{A}_T\right|}_{\mathit{\xi }=\widehat{\mathit{\xi }}},\end{array}$$

(8)

$$\begin{array}{c}\hfill {\left.{\displaystyle \frac{\nabla }{\gamma }}+\nabla log\left(1+\alpha \right)+\sum _{i=1}^{\nabla }log\left(\vartheta \left(y_i\right)\right)-\sum _{i=1}^{\nabla }log\left(\alpha +\vartheta \left(y_i\right)\right)-\sum _{i=1}^{\nabla }{S}_i{B}_i-S^{\star }{B}_T\right|}_{\mathit{\xi }=\widehat{\mathit{\xi }}}=0,\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\left.{\displaystyle \frac{\nabla }{\theta }}-\sum _{i=1}^{\nabla }{y}_i+\left(\gamma -1\right)\sum _{i=1}^{\nabla }{\displaystyle \frac{{\vartheta }^{•}\left(y_i\right)}{\vartheta \left(y_i\right)}}-\left(\gamma +1\right)\sum _{i=1}^{\nabla }{\displaystyle \frac{{\vartheta }^{•}\left(y_i\right)}{\left(\alpha +\vartheta \left(y_i\right)\right)}}-\sum _{i=1}^{\nabla }{S}_i{C}_i-S^{\star }{C}_T\right|}_{\mathit{\xi }=\widehat{\mathit{\xi }}}=0,\end{array}$$

(10)

respectively, where

• ${\eta }_i=\eta \left(y_i;\alpha ,\theta \right),{\eta }_T=\eta \left(T^{\star };\alpha ,\theta \right),{\vartheta }^{•}\left(y_i\right)=-y_i{e}^{-\theta y_i}$, $A_i={\displaystyle \frac{\gamma {\eta }_i^{\gamma -1}{\eta }_i^{\circ }}{1-{\eta }_i^{\gamma }}},B_i={\displaystyle \frac{\gamma {\eta }_i^{\gamma }log\left({\eta }_i\right)}{1-{\eta }_i^{\gamma }}},C_i={\displaystyle \frac{\gamma {\eta }_i^{\gamma -1}{\eta }_i^{•}}{1-{\eta }_i^{\gamma }}},{\eta }_i^{\circ }={\displaystyle \frac{\vartheta \left(y_i\right)\left(\vartheta \left(y_i\right)-1\right)}{{\left(\alpha +\vartheta \left(y_i\right)\right)}^2}},{\eta }_i^{•}={\displaystyle \frac{\alpha \left(1+\alpha \right){\vartheta }^{•}\left(y_i\right)}{{\left(\alpha +\vartheta \left(y_i\right)\right)}^2}}.$

It is evident from (8) and (9) that the MLEs $\widehat{\mathit{\xi }}$ of $\mathit{\xi }$ are obtained by solving iteratively. Consequently, an appropriate iterative numerical procedure, such as the Newton–Raphson (NR) algorithm, is required to compute these estimates. In Section 5 and Section 7, where UT2-PH censored samples $\{Y_i,S_i\},i=1,2,\dots ,\nabla$, are considered, the maxLik package in R (version 4.4.2), developed by Henningsen and Toomet [9], is employed to obtain the MLEs. Once the estimates $\widehat{\alpha }$, $\widehat{\gamma }$, and $\widehat{\theta }$ are available, the corresponding MLEs of the RF $\widehat{R}\left(x\right)$ and HRF $\widehat{h}\left(x\right)$ are obtained by replacing $\alpha$, $\gamma$, and $\theta$ with their respective frequentist estimates.

To highlight that the acquired MLEs $\widehat{\alpha }$, $\widehat{\gamma }$, and $\widehat{\theta }$ covered both existence and uniqueness features, Figure 2 presents log-likelihood contour plots of $\alpha$, $\vartheta$, and $\theta$ based on two UT2–PH censored samples generated from NOT-Exp$(0.8,0.5,0.2)$ and NOT-Exp$(1.8,1.5,1.2)$ models when $(q_1,q_{,}n)=(20,50,100)$, $({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1,2)$, and $S_i=1$ for ($i=1,\dots ,q_2$). In both scenarios, the contours display a single, well-defined minimum, indicating good parameter identifiability and a stable estimation problem. Overall, the close alignment of the minima with the true parameter values confirms that the MLEs $\widehat{\alpha }$, $\widehat{\gamma }$, and $\widehat{\theta }$ existed and are unique.

3.2. Asymptotic Interval Bounds

The ACI[Norm] method constructs two bounds directly from the normal approximation of the MLE, providing a straightforward and computationally efficient approach for quantifying parameter uncertainty in large samples. In contrast, the ACI[Log-Norm] method applies the normal approximation to a logarithmic transformation of the parameter, improving interval accuracy and ensuring positivity when parameters are inherently non-negative, which is particularly important in reliability and survival analysis. To construct the $100(1-\sigma )\%$ ACI[Norm] estimators for $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$, we make use of the asymptotic properties of their MLEs.

Under the regularity conditions established by Lawless [10], the estimator vector $\widehat{\mathit{\xi }}$ is approximately multivariate normally distributed with mean $\mathit{\xi }$ and covariance structure given by the inverse Fisher information (IFI) matrix ${\mathbf{I}}^{-1}(·)$, where

$$\begin{array}{c}\hfill {\mathbf{I}}^{-1}\left(\widehat{\mathit{\xi }}\right)={\left[\begin{array}{ccc}{\Im }_{11}& {\Im }_{12}& {\Im }_{13}\\ & {\Im }_{22}& {\Im }_{23}\\ & & {\Im }_{33}\end{array}\right]}_{\mathit{\xi }=\widehat{\mathit{\xi }}}^{-1}=\left[\begin{array}{ccc}\widehat{Var}\left(\widehat{\alpha }\right)& \widehat{Cov}(\widehat{\alpha },\widehat{\gamma })& \widehat{Cov}(\widehat{\alpha },\widehat{\theta })\\ & \widehat{Var}\left(\widehat{\gamma }\right)& \widehat{Cov}(\widehat{\gamma },\widehat{\theta })\\ & & \widehat{Var}\left(\widehat{\theta }\right)\end{array}\right],\end{array}$$

(11)

where the components ${\Im }_{ij}$ for $i,j=1,2,3$ are presented in the Supplementary File.

For a UT2-PH sample with $q_2$ observed failures, the MLE $\widehat{\mathit{\xi }}$ satisfies

$$\sqrt{q_2}(\widehat{\mathit{\xi }}-{\mathit{\xi }}_0)\approx -{\left[{\displaystyle \frac{1}{q_2}}{\displaystyle \frac{{\partial }^2{\mathcal{L}}^{\star }}{\partial \mathit{\xi }\partial {\mathit{\xi }}^{\top }}}\right]}^{-1}\left({\displaystyle \frac{1}{\sqrt{q_2}}}{\displaystyle \frac{\partial {\mathcal{L}}^{\star }}{\partial \mathit{\xi }}}\right),$$

(12)

where ${\mathit{\xi }}_0$ is the true parameter vector.

Under regularity conditions,

$$-{\displaystyle \frac{1}{q_2}}{\displaystyle \frac{{\partial }^2{\mathcal{L}}^{\star }}{\partial \mathit{\xi }\partial {\mathit{\xi }}^{\top }}}\stackrel{p}{\to }\mathbf{I}\left({\mathit{\xi }}_0\right).$$

(13)

Consequently, from (12) and (13), it follows that

$$\sqrt{q_2}(\widehat{\mathit{\xi }}-{\mathit{\xi }}_0)\stackrel{d}{\to }\mathcal{N}\left(\mathbf{0},{\mathbf{I}}^{-1}\left({\mathit{\xi }}_0\right)\right),$$

or equivalently,

$$\widehat{\mathit{\xi }}\approx \mathcal{N}\left({\mathit{\xi }}_0,{\displaystyle \frac{1}{q_2}}{\mathbf{I}}^{-1}\left({\mathit{\xi }}_0\right)\right),$$

where ${\mathbf{I}}^{-1}\left({\mathit{\xi }}_0\right)$ is the asymptotic covariance matrix of $\widehat{\mathit{\xi }}$.

Now, the $100(1-\sigma )\%$ ACI[Norm] bounds of $\alpha$, $\gamma$, and $\theta$ are

$$\begin{array}{c}\hfill \left(\widehat{\alpha }\mp z_{0.5\sigma }\sqrt{\widehat{Var}\left(\widehat{\alpha }\right)}\right),\left(\widehat{\gamma }\mp z_{0.5\sigma }\sqrt{\widehat{Var}\left(\widehat{\gamma }\right)}\right),\mathrm{a}\mathrm{n}\mathrm{d}\left(\widehat{\theta }\mp z_{0.5\sigma }\sqrt{\widehat{Var}\left(\widehat{\theta }\right)}\right),\end{array}$$

receptively, where $z_{0.5\sigma }$ is the $0.5\sigma$th standard Gaussian point.

Alternatively, the $100(1-\sigma )\%$ asymptotic confidence intervals [Norm] for $R\left(x\right)$ and $h\left(x\right)$ are obtained by approximating the variances of $\widehat{R}\left(x\right)$ and $\widehat{h}\left(x\right)$ using the delta method (see Greene [11]). The estimated variances, denoted by $\widehat{\mathrm{Var}}\left(\widehat{R}\left(x\right)\right)$ and $\widehat{\mathrm{Var}}\left(\widehat{h}\left(x\right)\right)$, are then approximated as

$$\begin{array}{c}\hfill \widehat{Var}\left(\widehat{R}\left(x\right)\right)={\mathcal{G}}_R^{\top }{\mathbf{I}}^{-1}\left(\widehat{\mathit{\xi }}\right){\mathcal{G}}_R\mathrm{a}\mathrm{n}\mathrm{d}\widehat{Var}\left(\widehat{h}\left(x\right)\right)={\mathcal{G}}_h^{\top }{\mathbf{I}}^{-1}\left(\widehat{\mathit{\xi }}\right){\mathcal{G}}_h,\end{array}$$

receptively, where

• ${\mathcal{G}}_R=\left[{\mathcal{G}}_R^{\alpha }{\mathcal{G}}_R^{\gamma }{\mathcal{G}}_R^{\theta }\right]$ and ${\mathcal{G}}_h=\left[{\mathcal{G}}_h^{\alpha }{\mathcal{G}}_h^{\gamma }{\mathcal{G}}_h^{\theta }\right]$,
• ${\mathcal{G}}_R^{\alpha }=-\gamma {(1+\alpha )}^{-1}{\left(\eta (x;\alpha ,\theta )\right)}^{\gamma }\left[1+{\displaystyle \frac{1+\alpha }{\alpha +\vartheta \left(x\right)}}\right],$
• ${\mathcal{G}}_R^{\gamma }=-{\left(\eta (x;\alpha ,\theta )\right)}^{\gamma }log\left(\eta (x;\alpha ,\theta )\right),$
• ${\mathcal{G}}_R^{\theta }=\gamma {\left(\eta (x;\alpha ,\theta )\right)}^{\gamma -1}{\eta }_i^{•}(x;\alpha ,\theta ),$
• ${\mathcal{G}}_h^{\alpha }=h(x;\mathit{\xi })\left[{\displaystyle \frac{1}{\alpha }}+{\displaystyle \frac{\gamma }{1+\alpha }}-{\displaystyle \frac{1}{\omega }}-{\displaystyle \frac{\gamma {\omega }^{\gamma -1}-\gamma {(1+\alpha )}^{\gamma -1}\vartheta {\left(x\right)}^{\gamma }}{{\omega }^{\gamma }-{(1+\alpha )}^{\gamma }\vartheta {\left(x\right)}^{\gamma }}}\right],$
• ${\mathcal{G}}_h^{\alpha }=h(x;\mathit{\xi })\left[{\displaystyle \frac{1}{\gamma }}+log(1+\alpha )+log\left(\vartheta \left(x\right)\right)-{\displaystyle \frac{{\omega }^{\gamma }log\omega -{(1+\alpha )}^{\gamma }\vartheta {\left(x\right)}^{\gamma }\left[log(1+\alpha )+log\left(\vartheta \right(x\left)\right)\right]}{{\omega }^{\gamma }-{(1+\alpha )}^{\gamma }\vartheta {\left(x\right)}^{\gamma }}}\right],$
• and
• ${\mathcal{G}}_h^{\theta }=h(x;\mathit{\xi })\left[{\displaystyle \frac{1}{\theta }}-x+(\gamma -1){\displaystyle \frac{x{e}^{-\theta x}}{\vartheta \left(x\right)}}-{\displaystyle \frac{x{e}^{-\theta x}}{\omega }}-{\displaystyle \frac{\gamma x{e}^{-\theta x}[{\omega }^{\gamma -1}-{(1+\alpha )}^{\gamma }\vartheta {\left(x\right)}^{\gamma -1}]}{{\omega }^{\gamma }-{(1+\alpha )}^{\gamma }\vartheta {\left(x\right)}^{\gamma }}}\right],$
• with $\omega =(\alpha +\vartheta (x\left)\right).$

A popular limitation of ACI[Norm] is that it may yield negative lower bounds for parameters that are strictly positive. To address this issue, Meeker and Escobar [12] proposed a modification in which the MLEs are first log-transformed before constructing the ACIs. This procedure, known as ACI[Log-Norm], ensures that all confidence limits are positive. While the ACI[Norm] approach can occasionally provide slightly higher coverage probabilities, the ACI[Log-Norm] generally delivers more reliable and interpretable intervals, particularly for moderate or small sample sizes. The enhanced performance arises from the Log-Norm, which reduces skewness in the sampling distribution of the MLEs. Accordingly, the two-sided $100(1-\sigma )\%$ ACI[Log-Norm] for $\alpha$ is given by

$$\widehat{\alpha }exp\left(\mp {\displaystyle \frac{z_{0.5\sigma }}{\widehat{\alpha }}}\sqrt{\widehat{Var}\left(\widehat{\alpha }\right)}\right),$$

and in the same way, the $100(1-\sigma )\%$ ACI[Log-Norm] for $\gamma$, $\theta$, $R\left(x\right)$, or $h\left(x\right)$ can be constructed.

4. Bayesian Estimation

This section outlines the Bayesian estimation framework for $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$, including the corresponding Bayesian credible intervals (BCIs) and highest posterior density (HPD) estimators. The derivation is based on the assumption that the parameters $\mathit{\xi }$ are a priori stochastically independent, with prior knowledge incorporated through their respective prior distributions.

Among the various prior choices, the gamma conjugate prior is often preferred due to its flexibility, solution convenience, and interpretability (see Kundu [13]). Its analytical simplicity and ability to accommodate diverse forms of prior beliefs make it particularly suitable for modeling positive-valued parameters. In this study, we assume that $\alpha$, $\gamma$, and $\theta$ are a priori independent and follow gamma distributions, such as $\alpha \sim \mathrm{Gamma}(a_1,b_1),\gamma \sim \mathrm{Gamma}(a_2,b_2),\theta \sim \mathrm{Gamma}(a_3,b_3).$

Accordingly, the joint prior density of $\mathit{\xi }$ with hyperparameters $a_i>0$ and $b_i>0$ (for $i=1,2,3$), denoted by $\mathcal{P}(·)$, can be expressed as

$$\mathcal{P}\left(\mathit{\xi }\right)\propto {\alpha }^{a_1-1}{\gamma }^{a_2-1}{\theta }^{a_3-1}{e}^{-\left(b_1\alpha +b_2\gamma +b_3\theta \right)},\alpha ,\gamma ,\theta >0.$$

(14)

In Bayes’ continuous theorem, the joint posterior PDF of $\mathit{\xi }$ (say, ${\mathcal{P}}^{\star }(·)$) is given by

$${\mathcal{P}}^{\star }\left(\left.\mathit{\xi }\right|\mathbf{x}\right)={\mho }^{-1}\mathcal{L}\left(\left.\mathit{\xi }\right|\mathbf{x}\right)\mathcal{P}\left(\mathit{\xi }\right),$$

(15)

where $\mho ={\int }_{\alpha }{\int }_{\gamma }{\int }_{\theta }\mathcal{L}\left(\left.\mathit{\xi }\right|\mathbf{x}\right)\mathcal{P}\left(\mathit{\xi }\right)\mathrm{d}\alpha \mathrm{d}\gamma \mathrm{d}\theta .$

The joint posterior PDF (15) for the NOT-Exp parameters $\alpha$, $\gamma$, and $\theta$, from (6) and (14), becomes

$$\begin{array}{cc}\hfill {\mathcal{P}}^{\star }\left(\left.\mathit{\xi }\right|\mathbf{x}\right)& \propto {\alpha }^{\nabla +a_1-1}{\gamma }^{\nabla +a_2-1}{\theta }^{\nabla +a_3-1}{\left(1+\alpha \right)}^{\nabla \gamma }{e}^{-\left(b_1\alpha +b_2\gamma +b_3^{\ast }\theta \right)}\hfill \\ \hfill & \times \prod _{i=1}^{\nabla }{\displaystyle \frac{\vartheta {\left(y_i\right)}^{\gamma -1}}{{\left(\alpha +\vartheta \left(y_i\right)\right)}^{\gamma +1}}}{\left[1-{\left(\eta \left(y_i;\alpha ,\theta \right)\right)}^{\gamma }\right]}^{S_i}{\left[1-{\left(\eta \left(T^{\ast };\alpha ,\theta \right)\right)}^{\gamma }\right]}^{S^{\ast }},\hfill \end{array}$$

(16)

where $b_3^{\ast }=b_3+{\sum }_{i=1}^{\nabla }{y}_i.$

Under the squared error loss (SEL) criterion, the Bayes estimator of any function $\beta \left(\mathit{\xi }\right)$ is defined as its posterior mean. Let $\Xi$ represent the SEL function; then the corresponding Bayes estimator $\tilde{\beta }\left(\mathit{\xi }\right)$ can be expressed as

$$\tilde{\beta }\left(\mathit{\xi }\right)={\int }_0^{\infty }{\int }_0^{\infty }{\int }_0^{\infty }\beta \left(\mathit{\xi }\right)\mathcal{L}\left(\mathit{\xi }\mid \mathbf{x}\right)\mathcal{P}\left(\mathit{\xi }\right)\mathrm{d}\alpha \mathrm{d}\gamma \mathrm{d}\theta .$$

Due to the nonlinear and analytically intractable structure of the posterior distribution given in (16), closed-form expressions for the Bayes estimators of $\alpha$, $\gamma$, and $\theta$, $R\left(x\right)$, and $h\left(x\right)$ are not attainable. Consequently, Markov chain Monte Carlo (MCMC) techniques are adopted to obtain approximate Bayes estimates as well as to construct BCI/HPD intervals.

To implement this procedure, the full conditional posterior probability density functions of the NOT-Exp model parameters $\alpha$, $\gamma$, and $\theta$ are first derived and are presented as follows:

$${\mathcal{P}}_{\alpha }^{\star }\left(\alpha |\gamma ,\theta ,\mathbf{x}\right)\propto {\alpha }^{\nabla +a_1-1}{\left(1+\alpha \right)}^{\nabla \gamma }{e}^{-b_1\alpha }{\prod }_{i=1}^{\nabla }{\left(\alpha +\vartheta \left(y_i\right)\right)}^{-\left(\gamma +1\right)}{\Psi }_i^{S_i}{\Psi }_T^{S^{\ast }},$$

(17)

$${\mathcal{P}}_{\gamma }^{\star }\left(\gamma |\alpha ,\theta ,\mathbf{x}\right)\propto {\gamma }^{\nabla +a_2-1}{\left(1+\alpha \right)}^{\nabla \gamma }{e}^{-b_2\gamma }{\prod }_{i=1}^{\nabla }\vartheta {\left(y_i\right)}^{\gamma -1}{\left(\alpha +\vartheta \left(y_i\right)\right)}^{-\left(\gamma +1\right)}{\Psi }_i^{S_i}{\Psi }_T^{S^{\ast }},$$

(18)

and

$${\mathcal{P}}_{\theta }^{\star }\left(\theta |\alpha ,\gamma ,\mathbf{x}\right)\propto {\theta }^{\nabla +a_3-1}{e}^{-b_3^{\ast }\theta }{\prod }_{i=1}^{\nabla }\vartheta {\left(y_i\right)}^{\gamma -1}{\left(\alpha +\vartheta \left(y_i\right)\right)}^{-\left(\gamma +1\right)}{\Psi }_i^{S_i}{\Psi }_T^{S^{\ast }},$$

(19)

respectively, where ${\Psi }_i=\left[1-{\left(\eta \left(y_i;\alpha ,\theta \right)\right)}^{\gamma }\right]$ and ${\Psi }_T=\left[1-{\left(\eta \left(T^{\ast };\alpha ,\theta \right)\right)}^{\gamma }\right]$.

It can be observed that the full conditional PDFs ${\mathcal{P}}_{ϵ}^{\star }$, ${\mathcal{P}}_{\gamma }^{\star }$, and ${\mathcal{P}}_{\theta }^{\star }$ of $\alpha$, $\gamma$, and $\theta$, given in (17)–(19), respectively, do not correspond to any known standard distributions and lack closed-form representations. Nevertheless, the diagnostic plots displayed in Figure 3 suggest that these conditional posterior distributions are well approximated by normal densities. Accordingly, parameter estimation and the construction of BCI/HPD intervals are carried out using the Metropolis–Hastings (MH) algorithm. The MH procedure, outlined in Algorithm 1, is employed as an effective MCMC sampling method.

5. Monte Carlo Comparisons

This section presents a detailed simulation study designed to evaluate the finite-sample performance of the proposed theoretical estimators for $\alpha$, $\gamma$, and $\theta$, along with $R\left(x\right)$ and $h\left(x\right)$ discussed earlier.

5.1. Simulation Framework

An extensive simulation framework is implemented to examine the empirical behavior of the estimators under a range of UT2-PH censoring scenarios. Key design factors are systematically varied, including the censoring thresholds ${\mathsf{ȷ}}_i(i=1,2)$, the overall sample size n, the corresponding effective sample sizes $q_i(i=1,2)$, and the progressive design $\mathbf{S}$. In each setting, 1000 Monte Carlo samples are generated from two NOT-Exp lifetime distributions: Set-1 with parameters NOT-Exp$(0.8,0.5,0.2)$ and Set-2 with NOT-Exp$(1.8,1.5,1.2)$. For a fixed time point $x=0.1$, from Set-i for $i=1,2$, the true values of the RF $R\left(x\right)$ are 0.7915 and 0.9327, and those of the HRF are 1.2725 and 0.9589, respectively, which serve as reference values for performance evaluation.

For clarification, the performance of all estimators of the parameters $\alpha$, $\gamma$, and $\theta$, as well as the reliability measures $R\left(x\right)$ and $h\left(x\right)$, is evaluated by assuming that their true parameter values lie within the theoretical parameter space of the NOT-Exp distribution without loss of generality. The parameter configurations were not chosen arbitrarily; instead, they were selected to represent distinct lifetime behaviors commonly encountered in reliability analysis. To illustrate the characteristics of the selected parameter sets, Figure 4 presents the PDF and HRF shapes of the NOT-Exp population for Set-i $(i=1,2)$. The resulting PDF shapes exhibit decreasing and unimodal behaviors, while the associated HRFs display distinct patterns. In particular, Set–1 exhibits a decreasing HRF, which is typically associated with early-life failures or infant mortality. In contrast, Set–2 generates an upside-down (unimodal) HRF shape, representing situations in which the failure rate initially increases, reaches a peak, and then gradually declines over time. All density and failure rate shapes depicted in Figure 4 match the original theoretical shapes shown in Figure 1. By considering these contrasting hazard behaviors, the simulation design provides a realistic and informative framework for assessing the finite-sample performance of the proposed estimators under different reliability environments.

Taking $n\in \{30,50,80\}$ with ${\mathsf{ȷ}}_1\in \{0.2,0.5\}$ for Set-1, and (1, 1.5) for Set-2, and ${\mathsf{ȷ}}_2\in \{0.5,0.8\}$ for Set-1, and (1.5, 2.5) for Set-2. Additional variation is introduced through different effective sample sizes $q_i(i=1,2)$ and censoring patterns $\mathbf{S}$. For instance, the censoring scheme $\mathcal{S}:(0,0,1,1,1)$ is denoted compactly as $(0^{\left[2\right]},1^{\left[3\right]})$ (see

Table 1. Removal designs in Monte Carlo experiments.

Design	$({\mathit{q}}_1,{\mathit{q}}_2,\mathit{n})$	Design	$({\mathit{q}}_1,{\mathit{q}}_2,\mathit{n})$
	$(5,10,30)$		$(15,20,30)$
D[11]	$\left(5^{\left[4\right]},0^{\left[6\right]}\right)$	D[14]	$\left(5^{\left[2\right]},0^{\left[18\right]}\right)$
D[12]	$\left(0^{\left[3\right]},5^{\left[4\right]},0^{\left[3\right]}\right)$	D[15]	$\left(0^{\left[9\right]},5^{\left[2\right]},0^{\left[9\right]}\right)$
D[13]	$\left(0^{\left[6\right]},5^{\left[4\right]}\right)$	D[16]	$\left(0^{\left[18\right]},5^{\left[2\right]}\right)$
	$(10,20,50)$		$(20,40,50)$
D[21]	$\left(5^{\left[6\right]},0^{\left[14\right]}\right)$	D[24]	$\left(5^{\left[2\right]},0^{\left[38\right]}\right)$
D[22]	$\left(0^{\left[7\right]},5^{\left[6\right]},0^{\left[7\right]}\right)$	D[25]	$\left(0^{\left[19\right]},5^{\left[2\right]},0^{\left[19\right]}\right)$
D[23]	$\left(0^{\left[14\right]},5^{\left[6\right]}\right)$	D[26]	$\left(0^{\left[38\right]},5^{\left[2\right]}\right)$
	$(30,50,80)$		$(40,60,80)$
D[31]	$\left(5^{\left[10\right]},0^{\left[20\right]}\right)$	D[34]	$\left(5^{\left[4\right]},0^{\left[56\right]}\right)$
D[32]	$\left(0^{\left[10\right]},5^{\left[10\right]},0^{\left[10\right]}\right)$	D[35]	$\left(0^{\left[28\right]},5^{\left[4\right]},0^{\left[28\right]}\right)$
D[33]	$\left(0^{\left[20\right]},5^{\left[10\right]}\right)$	D[36]	$\left(0^{\left[56\right]},5^{\left[4\right]}\right)$

). Collectively, these configurations are intended to investigate the impact of censoring intensity and sampling design on estimator accuracy, precision, and numerical stability. A detailed description of the simulation and estimation procedures is provided in Algorithm 2.

Algorithm 2 Simulate UT2-PH censored datasets from NOT-Exp$\left(\mathit{\xi }\right)$

1: Input: Parameter vector $\mathit{\xi }$ of the NOT-Exp distribution, sample size n, censoring scheme $\mathbf{S}$   2: Input: Generate $q_2$ independent random variables $W_1,W_2,\dots ,W_{q_2}\sim U(0,1)$   3: for $i=1,2,\dots ,q_2$ do   4:     Input: Compute ${\mho }_i=W_i^{{\left(i+{\sum }_{d=q_2-i+1}^{q_2}{S}_d\right)}^{-1}}$   5: end for   6: for $i=1,2,\dots ,q_2$ do   7:     Input: Set $U_i=1-{\prod }_{d=q_2-i+1}^{q_2}{\mho }_d$   8: end for   9: Output: Obtain failure times $Y_i=G^{-1}(U_i;\mathit{\xi })$ 10: Input: Identify $q_i$ at ${\mathsf{ȷ}}_i$ (for $i=1,2$) 11: Output: Calculate ${\mathcal{S}}^{\star }$: 12: if $y_{q_1}<y_{q_2}<{\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2$ then 13:     Set ${\mathcal{S}}^{\star }=n-q_2-{\sum }_{i=1}^{q_2-1}{S}_i$ 14: end if 15: if $y_{q_1}<{\mathsf{ȷ}}_1<y_{q_2}<{\mathsf{ȷ}}_2$ (or $y_{q_1}<{\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2<y_{q_2}$) then 16:     Set ${\mathcal{S}}^{\star }=n-q_1-{\sum }_{i=1}^{q_1}{S}_i$ 17: end if 18: if ${\mathsf{ȷ}}_1<y_{q_1}<y_{q_2}<{\mathsf{ȷ}}_2$ (or ${\mathsf{ȷ}}_1<y_{q_1}<{\mathsf{ȷ}}_2<y_{q_2}$) then 19:     Set ${\mathcal{S}}^{\star }=n-q_1-{\sum }_{i=1}^{q_1-1}{S}_i$ 20: end if 21: if ${\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2<y_{q_1}<y_{q_2}$ then 22:     Set ${\mathcal{S}}^{\star }=n-q_2-{\sum }_{i=1}^{q_2}{S}_i$ 23: end if 24: Output: Stop and get a UT2-PH censored dataset: 25: if $y_{q_1}<y_{q_2}<{\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2$ then 26:     Stop the test at $y_{q_2}$ 27: end if 28: if $y_{q_1}<{\mathsf{ȷ}}_1<y_{q_2}<{\mathsf{ȷ}}_2$ (or $y_{q_1}<{\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2<y_{q_2}$) then 29:     Stop the test at ${\mathsf{ȷ}}_1$ 30: end if 31: if ${\mathsf{ȷ}}_1<y_{q_1}<y_{q_2}<{\mathsf{ȷ}}_2$ (or ${\mathsf{ȷ}}_1<y_{q_1}<{\mathsf{ȷ}}_2<y_{q_2}$) then 32:     Stop the test at $y_{q_1}$ 33: end if 34: if ${\mathsf{ȷ}}_1<{\mathsf{ȷ}}_2<y_{q_1}<y_{q_2}$ then 35:     Stop the test at ${\mathsf{ȷ}}_2$ 36: end if

For $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, or $h\left(x\right)$, the Bayesian point estimates are obtained together with their associated BCI and HPD estimates using an MCMC approach. Beyond setting $\wp =$ 12,000 and ${\wp }^{\star }=2000$, the objective Bayesian analysis is then based on ${\wp }^{\ast }=10,000$ posterior samples to estimate $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$. All numerical analyses are conducted using R (v4.2.2). The MLEs and 95% ACIs, obtained via both Norm and Log-Norm approaches, are computed using the ‘maxLik’ package (v1.5-2.2) (Henningsen and Toomet [9]). Subsequently, all Bayesian point estimates and the associated 95% BCI and HPD intervals are derived using the ‘coda’ package (v0.19-4.1) (Plummer et al. [14]). Two sets of informative gamma priors are assumed for the parameter vector $\mathit{\xi }$. The corresponding hyperparameters ${\upsilon }_i$ and $v_i$ (for $i=1,2,3$) are selected according to the hyperparameter elicitation idea proposed by Kundu [13]. Accordingly, the hyperparameter values of ${\upsilon }_i$ and $v_i$ (for $i=1,2,3$) are specified as follows:

• Set-1 [Prior-A]: $({\nu }_1,{\nu }_2,{\nu }_3)=(4,2.5,1)$ and $v_i=5,i=1,2,3$;
• Set-1 [Prior-B]: $({\nu }_1,{\nu }_2,{\nu }_3)=(8,5,2)$ and $v_i=10,i=1,2,3$,
• Set-2 [Prior-A]: $({\nu }_1,{\nu }_2,{\nu }_3)=(9,7.5,6)$ and $v_i=5,i=1,2,3$;
• Set-2 [Prior-B]: $({\nu }_1,{\nu }_2,{\nu }_3)=(18,15,12)$ and $v_i=10,i=1,2,3$.

To investigate the robustness of the Bayesian estimation procedure with respect to prior assumptions, a sensitivity analysis was conducted using several alternative prior specifications. This analysis is particularly relevant in censored lifetime models, where the posterior inference may depend on the degree of prior information. For illustration, from 1000 UT2-PH censored samples generated from the NOT-Exp$(0.8,0.5,0.2)$ distribution (as an example) when $({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.2,0.5)$ and the removal design $D\left[11\right]$, the posterior distributions of the parameters $\alpha$, $\gamma$, and $\theta$ were then examined under four different prior settings:

(i)

Informative (Prior-A): $\alpha \sim \mathrm{Gamma}(4,5)$, $\gamma \sim \mathrm{Gamma}(2.5,5)$, and $\theta \sim \mathrm{Gamma}(1,5)$;

(ii)

Overdispersed: $\alpha \sim \mathrm{Gamma}(0.8,1)$, $\gamma \sim \mathrm{Gamma}(0.5,1)$, and $\theta \sim \mathrm{Gamma}(0.2,1)$;

(iii)

Weakly informative: $\alpha \sim \mathrm{Gamma}(0.4,0.5)$, $\gamma \sim \mathrm{Gamma}(0.25,0.5)$, and $\theta \sim \mathrm{Gamma}(0.1,0.5)$;

(iv)

Improper: $\alpha ,\gamma ,\theta \sim \mathrm{Uniform}(0.01,10)$.

Figure 5 presents the resulting posterior summaries under these prior specifications. The results indicate that the posterior means remain relatively stable across different priors, suggesting that the data provide sufficient information for reliable inference. As expected, the informative prior yields the narrowest BCIs and the smallest posterior standard deviations, reflecting increased estimation precision. In contrast, the weakly informative and nearly noninformative priors lead to slightly wider credible intervals, while the overdispersed prior produces the most diffuse posterior distributions.

Now, for computational implementation, let ${\zeta }_{\varsigma }$ denote a generic parameter, with ${\zeta }_1=\alpha$, ${\zeta }_2=\gamma$, ${\zeta }_3=\theta$, ${\zeta }_4=R\left(x\right)$, and ${\zeta }_5=h\left(x\right)$. For $\varsigma =1,2,\dots ,5,$ (number of parameters) and $i=1,2,\dots ,1000$ (number of replications), the average point estimate (APE) is given by

$$\mathrm{APE}\left({\ddot{\zeta }}_{\varsigma }\right)={\displaystyle \frac{1}{1000}}{\sum }_{i=1}^{1000}{\ddot{\zeta }}_{\varsigma }^{\left(i\right)},$$

where ${\ddot{\zeta }}_{\varsigma }^{\left(i\right)}$ denotes the estimated value of ${\zeta }_{\varsigma }$ from the ith replicate. The performance of the point estimators is evaluated using two widely adopted precision metrics: the root mean squared error (RMSE) and the mean relative absolute bias (MRAB), defined as follows:

$$\mathrm{RMSE}\left({\ddot{\zeta }}_{\varsigma }\right)=\sqrt{{\displaystyle \frac{1}{1000}}{\sum }_{i=1}^{1000}{\left({\ddot{\zeta }}_{\varsigma }^{\left(i\right)}-{\zeta }_{\varsigma }\right)}^2},$$

(20)

and

$$\mathrm{MRAB}\left({\ddot{\zeta }}_{\varsigma }\right)={\displaystyle \frac{1}{1000}}{\sum }_{i=1}^{1000}{\displaystyle \frac{1}{{\zeta }_{\varsigma }}}\left|{\ddot{\zeta }}_{\varsigma }^{\left(i\right)}-{\zeta }_{\varsigma }\right|,$$

(21)

respectively, where ${\ddot{\zeta }}_{\varsigma }^{\left(i\right)}$ is the estimate obtained from the ith replication.

For interval evaluations, the performance is evaluated using the average interval length (AIL) and the coverage probability (CP). Denoting the lower and upper bounds of the ith interval as ${\mathcal{L}}_{{\ddot{\zeta }}_{\varsigma }^{\left(i\right)}}$ and ${\mathcal{U}}_{{\ddot{\zeta }}_{\varsigma }^{\left(i\right)}}$, respectively, the AIL is computed as

$${\mathrm{AIL}}_{95\%}\left({\zeta }_{\varsigma }\right)={\displaystyle \frac{1}{1000}}\sum _{i=1}^{1000}\left({\mathcal{U}}_{{\ddot{\zeta }}_{\varsigma }^{\left(i\right)}}-{\mathcal{L}}_{{\ddot{\zeta }}_{\varsigma }^{\left(i\right)}}\right),$$

(22)

and the CP is

$${\mathrm{CP}}_{95\%}\left({\zeta }_{\varsigma }\right)={\displaystyle \frac{1}{1000}}\sum _{i=1}^{1000}{\aleph }_{\left({\mathcal{L}}_{{\ddot{\zeta }}_{\varsigma }^{\left(i\right)}},{\mathcal{U}}_{{\ddot{\zeta }}_{\varsigma }^{\left(i\right)}}\right)}\left({\zeta }_{\varsigma }\right),$$

(23)

where $\aleph (·)$ is the indicator function, equal to one if ${\zeta }_{\varsigma }$ lies within the interval and zero otherwise.

5.2. Simulation Results and Interpretations

Table A1. The APE (1st Col.), RMSE (2nd Col.), and MRAB (3rd Col.) results of $\alpha$.

Design	MLE			Bayes							MLE			Bayes
Prior →				A			B							A			B
Set-1→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.2, 0.5)										(0.5, 0.8)
D[11]	1.2772	2.9741	2.7879	0.8350	0.1551	0.1396	0.7884	0.1259	0.1258		1.1454	2.8863	1.8690	0.8335	0.1511	0.1390	0.7889	0.1259	0.1254
D[12]	1.2571	3.1494	2.9283	0.8332	0.1583	0.1417	0.7888	0.1276	0.1271		1.2350	3.0336	2.1479	0.8339	0.1523	0.1408	0.7863	0.1271	0.1268
D[13]	1.3629	2.8863	2.6221	0.8343	0.1532	0.1366	0.7886	0.1254	0.1243		1.2112	2.8060	1.8140	0.8360	0.1503	0.1359	0.7883	0.1254	0.1227
D[14]	1.2405	2.6044	2.3147	0.8180	0.1354	0.1237	0.8080	0.1002	0.0456		0.9691	2.5686	1.6931	1.8373	0.1331	0.1222	1.7913	0.0977	0.0452
D[15]	1.5520	2.6620	2.4707	0.8122	0.1355	0.1239	0.8073	0.1019	0.0464		1.3029	2.5786	1.8082	1.8383	0.1348	0.1223	1.7905	0.1003	0.0457
D[16]	0.9711	2.5008	2.2170	0.8188	0.1323	0.1230	0.8077	0.1001	0.0452		0.9522	2.3814	1.6913	1.8378	0.1303	0.1170	1.7914	0.0971	0.0450
D[21]	1.0739	2.3809	1.7634	0.8120	0.1215	0.0989	0.8117	0.0753	0.0452		1.1446	2.2534	1.0750	0.8138	0.1097	0.0785	0.8089	0.0475	0.0448
D[22]	0.9184	2.3061	1.6389	0.8155	0.1200	0.0979	0.8074	0.0732	0.0451		0.8986	2.1739	0.9861	0.8120	0.1053	0.0743	0.8077	0.0461	0.0446
D[23]	0.8890	2.2981	1.2159	0.8155	0.1192	0.0970	0.8059	0.0715	0.0450		1.1411	2.1159	0.8967	0.8144	0.1047	0.0726	0.8062	0.0455	0.0443
D[24]	0.6148	1.7106	0.7023	0.6971	0.1070	0.0675	0.7746	0.0453	0.0435		0.9433	1.5749	0.5484	0.7009	0.0695	0.0675	0.7769	0.0441	0.0434
D[25]	1.0067	2.1143	0.7796	0.7027	0.1082	0.0677	0.7767	0.0458	0.0449		1.0498	1.9588	0.6354	0.6980	0.0718	0.0677	0.7771	0.0450	0.0441
D[26]	0.9674	1.5622	0.5616	0.6975	0.1004	0.0664	0.7762	0.0449	0.0429		0.9562	1.4622	0.5195	0.6973	0.0685	0.0664	0.7734	0.0440	0.0427
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.5, 1.0)										(1.5, 2.5)
D[31]	0.9744	1.2093	0.4562	0.6945	0.0695	0.0410	0.7785	0.0376	0.0334		1.0893	0.7367	0.3029	0.7008	0.0439	0.0407	0.7752	0.0369	0.0332
D[32]	1.1437	1.5507	0.5290	0.6976	0.0696	0.0417	0.7777	0.0381	0.0339		1.0045	0.9488	0.4641	0.6948	0.0439	0.0414	0.7744	0.0371	0.0343
D[33]	0.9658	1.1479	0.3660	0.6937	0.0685	0.0407	0.7755	0.0376	0.0312		0.9286	0.3870	0.2812	0.6929	0.0437	0.0394	0.7775	0.0367	0.0311
D[34]	0.7548	0.8885	0.2984	0.8531	0.0406	0.0374	0.6781	0.0301	0.0299		1.1275	0.3670	0.2253	0.8531	0.0381	0.0302	0.6781	0.0300	0.0255
D[35]	1.0928	0.7685	0.1869	0.8521	0.0397	0.0371	0.6797	0.0289	0.0282		0.8719	0.2771	0.1577	0.8521	0.0375	0.0291	0.6797	0.0287	0.0235
D[36]	0.8703	0.6207	0.1403	0.8532	0.0394	0.0368	0.6791	0.0281	0.0278		0.9624	0.2692	0.1319	0.8532	0.0368	0.0283	0.6791	0.0279	0.0225
Set-2→
D[11]	1.9381	3.3185	2.1412	1.8373	0.1312	0.1239	1.7913	0.0701	0.0524		1.9430	3.2365	1.7890	1.8398	0.1290	0.1235	1.7902	0.0676	0.0512
D[12]	2.9228	3.4079	2.7127	1.8383	0.1338	0.1251	1.7905	0.0704	0.0548		2.3526	3.3670	1.9491	1.8374	0.1306	0.1243	1.7909	0.0698	0.0544
D[13]	1.9045	3.2133	1.8166	1.8378	0.1304	0.1237	1.7914	0.0697	0.0514		1.8402	3.1954	1.7398	1.8382	0.1286	0.1228	1.7902	0.0659	0.0504
D[14]	2.1887	3.0398	1.6327	1.8131	0.1110	0.1002	1.8133	0.0599	0.0421		1.9271	2.8420	1.3867	1.8136	0.1101	0.0977	1.8123	0.0537	0.0402
D[15]	2.1614	3.0870	1.6752	1.8140	0.1113	0.1100	1.8128	0.0654	0.0393		2.2061	2.9438	1.6721	1.8120	0.1104	0.0989	1.8120	0.0540	0.0374
D[16]	1.3417	2.6599	1.5576	1.8136	0.1101	0.0979	1.8121	0.0536	0.0401		1.8342	2.4696	1.3096	1.8136	0.1097	0.0928	1.8121	0.0536	0.0387
D[21]	1.2409	1.9893	1.5492	1.8121	0.1097	0.0544	1.8113	0.0388	0.0370		2.2976	1.7951	1.2159	1.8127	0.1093	0.0414	1.8140	0.0371	0.0348
D[22]	1.9805	1.7474	1.5245	1.8138	0.1097	0.0530	1.8132	0.0370	0.0326		2.4415	1.6295	1.1003	1.8130	0.1091	0.0383	1.8132	0.0369	0.0311
D[23]	2.3902	1.6472	1.5097	1.8138	0.1089	0.0517	1.8125	0.0347	0.0318		1.3902	1.5738	1.0767	1.8138	0.1089	0.0382	1.8125	0.0347	0.0302
D[24]	1.9208	1.5056	1.4640	1.7091	0.0739	0.0513	1.7911	0.0328	0.0290		1.9713	1.4874	0.7723	1.7104	0.0734	0.0380	1.7916	0.0325	0.0276
D[25]	2.3758	1.5374	1.4908	1.7101	0.0753	0.0514	1.7916	0.0335	0.0308		2.1656	1.5240	1.0580	1.7094	0.0735	0.0381	1.7916	0.0327	0.0291
D[26]	1.9608	1.4900	1.4258	1.7091	0.0724	0.0494	1.7911	0.0319	0.0276		1.9986	1.4790	0.7711	1.7091	0.0720	0.0379	1.7911	0.0318	0.0268
D[31]	1.9036	1.4184	1.3298	1.7094	0.0614	0.0311	1.7920	0.0285	0.0261		1.7226	1.4011	0.7008	1.7102	0.0591	0.0306	1.7911	0.0278	0.0247
D[32]	2.2524	1.4794	1.3453	1.7076	0.0653	0.0328	1.7898	0.0295	0.0270		2.1264	1.4588	0.7454	1.7086	0.0605	0.0311	1.7898	0.0295	0.0256
D[33]	1.8883	1.4011	1.1096	1.7094	0.0583	0.0306	1.7911	0.0279	0.0246		1.8883	1.3703	0.5757	1.7094	0.0551	0.0301	1.7911	0.0263	0.0235
D[34]	1.8804	1.3121	0.9870	1.8582	0.0557	0.0299	1.6747	0.0231	0.0189		1.8676	1.2416	0.5116	1.8576	0.0510	0.0283	1.6737	0.0215	0.0171
D[35]	2.0725	1.3659	1.0793	1.8564	0.0579	0.0305	1.6743	0.0257	0.0234		2.0128	1.3121	0.5338	1.8562	0.0531	0.0299	1.6733	0.0227	0.0227
D[36]	1.9583	1.2299	0.8029	1.8596	0.0527	0.0296	1.6747	0.0210	0.0171		1.9686	0.8806	0.4935	1.8579	0.0505	0.0279	1.6745	0.0200	0.0161

,

Table A2. The APE (1st Col.), RMSE (2nd Col.), and MRAB (3rd Col.) results of $\gamma$.

Design	MLE			Bayes							MLE			Bayes
Prior →				A			B							A			B
Set-1→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.2, 0.5)										(0.5, 0.8)
D[11]	0.6541	1.1524	0.4277	0.5068	0.2232	0.1416	0.4397	0.1183	0.0864		0.6395	1.0684	0.3913	0.5076	0.2232	0.1406	0.4405	0.1183	0.0847
D[12]	0.6341	0.8521	0.4029	0.5076	0.2175	0.1406	0.4386	0.1154	0.0841		0.6191	0.8177	0.3650	0.5068	0.2175	0.1398	0.4396	0.1154	0.0838
D[13]	0.6346	1.2807	0.4408	0.5091	0.2344	0.1487	0.4353	0.1242	0.0881		0.6208	1.2098	0.4066	0.5101	0.2344	0.1483	0.4350	0.1242	0.0879
D[14]	0.5804	0.7982	0.3088	0.4672	0.1637	0.1398	0.4612	0.0895	0.0831		0.8357	0.7925	0.2322	0.4589	0.1625	0.1398	0.4361	0.0889	0.0822
D[15]	0.5512	0.5746	0.2432	0.4760	0.1601	0.1394	0.4648	0.0881	0.0819		0.8448	0.5375	0.1871	0.5078	0.1600	0.1379	0.4277	0.0879	0.0818
D[16]	0.5777	0.7860	0.2705	0.4731	0.1608	0.1398	0.4636	0.0884	0.0822		0.7715	0.5886	0.2190	0.4494	0.1603	0.1394	0.4420	0.0879	0.0819
D[21]	0.5466	0.4431	0.1596	0.4803	0.1574	0.0877	0.4650	0.0824	0.0798		0.5443	0.4030	0.1591	0.4798	0.1557	0.0867	0.4619	0.0818	0.0504
D[22]	0.5423	0.4128	0.1573	0.4856	0.1546	0.0863	0.4645	0.0809	0.0753		0.5432	0.3865	0.1558	0.4850	0.1458	0.0851	0.4643	0.0779	0.0498
D[23]	0.5512	0.5419	0.1602	0.4666	0.1596	0.0891	0.4575	0.0870	0.0819		0.5387	0.4242	0.1597	0.4664	0.1566	0.0879	0.4561	0.0867	0.0530
D[24]	0.5510	0.3865	0.1509	0.5796	0.0823	0.0753	0.5261	0.0720	0.0533		0.5424	0.3283	0.1442	0.5804	0.0762	0.0722	0.5258	0.0591	0.0496
D[25]	0.5442	0.2380	0.1375	0.5800	0.0813	0.0711	0.5266	0.0697	0.0508		0.5395	0.2217	0.1369	0.5801	0.0753	0.0704	0.5263	0.0585	0.0491
D[26]	0.5480	0.3033	0.1476	0.5798	0.0823	0.0716	0.5256	0.0700	0.0510		0.5443	0.2531	0.1415	0.5787	0.0754	0.0708	0.5270	0.0586	0.0496
D[31]	0.5319	0.1888	0.1248	0.5812	0.0750	0.0699	0.5314	0.0590	0.0483		0.5290	0.1810	0.1187	0.5819	0.0701	0.0690	0.5310	0.0497	0.0459
D[32]	0.5215	0.1596	0.1068	0.5802	0.0692	0.0634	0.5247	0.0495	0.0453		0.5248	0.1582	0.1067	0.5773	0.0667	0.0586	0.5254	0.0485	0.0419
D[33]	0.5372	0.1849	0.1202	0.5787	0.0699	0.0677	0.5243	0.0549	0.0478		0.5339	0.1734	0.1176	0.5798	0.0692	0.0588	0.5221	0.0493	0.0459
D[34]	0.5060	0.1519	0.1015	0.3828	0.0691	0.0489	0.4324	0.0459	0.0396		0.5060	0.1512	0.1014	0.3828	0.0599	0.0480	0.4324	0.0441	0.0386
D[35]	0.5080	0.1463	0.0992	0.3913	0.0678	0.0446	0.4324	0.0431	0.0327		0.5085	0.1462	0.0992	0.3913	0.0500	0.0432	0.4324	0.0380	0.0326
D[36]	0.5086	0.1502	0.1009	0.3884	0.0684	0.0459	0.4320	0.0436	0.0365		0.5080	0.1491	0.1000	0.3884	0.0553	0.0451	0.4320	0.0386	0.0364
Set-2→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.5, 1.0)										(1.5, 2.5)
D[11]	2.1264	0.5035	0.3145	1.5554	0.1372	0.0851	1.4420	0.0821	0.0480		1.7356	0.4824	0.3013	1.5578	0.1369	0.0823	1.4375	0.0819	0.0451
D[12]	1.9914	0.4491	0.3001	1.5565	0.1357	0.0851	1.4428	0.0809	0.0480		1.7106	0.4241	0.2861	1.5568	0.1341	0.0814	1.4375	0.0805	0.0446
D[13]	2.2097	0.7059	0.3931	1.5565	0.1387	0.0870	1.4361	0.0831	0.0492		1.6838	0.5936	0.3271	1.5556	0.1386	0.0864	1.4358	0.0831	0.0489
D[14]	1.5716	0.3996	0.1942	1.4795	0.0898	0.0535	1.4821	0.0504	0.0264		1.2768	0.3217	0.1657	1.4792	0.0877	0.0519	1.4811	0.0503	0.0263
D[15]	1.6161	0.3589	0.1490	1.4815	0.0835	0.0497	1.4828	0.0484	0.0261		1.7703	0.2843	0.1401	1.4815	0.0824	0.0496	1.4828	0.0484	0.0260
D[16]	1.9645	0.3685	0.1702	1.4799	0.0846	0.0503	1.4825	0.0499	0.0263		1.8031	0.3156	0.1491	1.4815	0.0824	0.0499	1.4827	0.0484	0.0261
D[21]	1.6255	0.2441	0.0940	1.4804	0.0824	0.0467	1.4816	0.0435	0.0253		1.4671	0.2277	0.0921	1.4807	0.0823	0.0451	1.4816	0.0431	0.0252
D[22]	1.8118	0.2346	0.0920	1.4823	0.0815	0.0451	1.4849	0.0432	0.0253		1.6156	0.2268	0.0902	1.4823	0.0813	0.0451	1.4849	0.0431	0.0251
D[23]	1.8957	0.2845	0.0948	1.4756	0.0828	0.0468	1.4773	0.0439	0.0255		1.8088	0.2291	0.0945	1.4749	0.0823	0.0463	1.4783	0.0432	0.0253
D[24]	1.7786	0.2326	0.0882	1.5695	0.0803	0.0431	1.5088	0.0394	0.0248		1.4047	0.2205	0.0882	1.5701	0.0795	0.0430	1.5088	0.0386	0.0247
D[25]	1.8883	0.2251	0.0777	1.5676	0.0682	0.0417	1.5094	0.0381	0.0246		2.2427	0.1980	0.0777	1.5676	0.0675	0.0417	1.5094	0.0380	0.0224
D[26]	1.8827	0.2292	0.0825	1.5676	0.0790	0.0428	1.5094	0.0388	0.0247		1.6123	0.2128	0.0801	1.5702	0.0777	0.0428	1.5088	0.0386	0.0246
D[31]	1.4258	0.1428	0.0491	1.5726	0.0398	0.0397	1.5104	0.0209	0.0202		1.6808	0.1190	0.0468	1.5748	0.0398	0.0394	1.5101	0.0208	0.0200
D[32]	1.7281	0.1277	0.0466	1.5726	0.0398	0.0393	1.5101	0.0208	0.0200		1.7281	0.1171	0.0444	1.5726	0.0398	0.0387	1.5101	0.0208	0.0200
D[33]	1.4398	0.1577	0.0495	1.5802	0.0420	0.0405	1.5130	0.0212	0.0209		2.0235	0.1577	0.0491	1.5779	0.0418	0.0405	1.5130	0.0211	0.0211
D[34]	1.7333	0.0907	0.0393	1.3758	0.0388	0.0297	1.4695	0.0208	0.0200		1.6871	0.0903	0.0393	1.3759	0.0386	0.0289	1.4694	0.0208	0.0181
D[35]	1.7334	0.0794	0.0392	1.3784	0.0386	0.0295	1.4701	0.0207	0.0198		1.7209	0.0756	0.0392	1.3798	0.0386	0.0270	1.4701	0.0207	0.0167
D[36]	1.7472	0.0833	0.0393	1.3771	0.0387	0.0295	1.4699	0.0208	0.0198		1.6962	0.0796	0.0392	1.3773	0.0386	0.0273	1.4695	0.0208	0.0175

,

Table A3. The APE (1st Col.), RMSE (2nd Col.), and MRAB (3rd Col.) results of $\theta$.

Design	MLE			Bayes							MLE			Bayes
Prior →				A			B							A			B
Set-1→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.2, 0.5)										(0.5, 0.8)
D[11]	0.3262	1.6181	0.8669	0.0986	0.5548	0.4847	0.1443	0.1855	0.1000		0.2841	1.5711	0.5851	0.1022	0.5244	0.4368	0.1482	0.1176	0.0923
D[12]	0.2888	1.7609	0.9168	0.0709	0.6413	0.4973	0.1253	0.1875	0.1034		0.2599	1.7289	0.6946	0.0816	0.5283	0.4795	0.1312	0.1459	0.0989
D[13]	0.3306	1.5795	0.7683	0.1007	0.5373	0.4462	0.1431	0.1836	0.0988		0.3543	1.5211	0.5449	0.0926	0.5119	0.4022	0.1304	0.1157	0.0920
D[14]	0.2449	1.5230	0.5226	0.2107	0.4222	0.2137	0.1108	0.1236	0.0840		1.1867	1.3552	0.4351	1.0389	0.3867	0.2049	1.1479	0.0928	0.0520
D[15]	0.4035	1.5781	0.5288	0.2238	0.4460	0.2196	0.1216	0.1352	0.0912		1.1759	1.3606	0.4714	1.0423	0.3916	0.2141	1.1573	0.0933	0.0528
D[16]	0.2036	1.3694	0.4388	0.2026	0.3688	0.2081	0.1005	0.1142	0.0784		1.3169	1.1399	0.4161	1.0407	0.3156	0.2038	1.1484	0.0884	0.0516
D[21]	0.2802	1.3118	0.3980	0.2195	0.3665	0.2067	0.1156	0.0824	0.0518		0.2262	1.1371	0.3922	0.2145	0.3135	0.2035	0.1126	0.0778	0.0507
D[22]	0.2219	1.3032	0.3937	0.2089	0.3295	0.2035	0.1108	0.0794	0.0507		0.2426	1.1242	0.3500	0.2146	0.3060	0.1988	0.1196	0.0751	0.0506
D[23]	0.3510	1.2100	0.3270	0.1971	0.3142	0.2030	0.1031	0.0786	0.0503		0.3924	1.0978	0.3240	0.1985	0.2918	0.1841	0.1041	0.0710	0.0500
D[24]	0.2248	1.2026	0.3135	0.2012	0.2691	0.1438	0.1893	0.0708	0.0498		0.2115	1.0554	0.3090	0.1971	0.1988	0.1335	0.1843	0.0682	0.0498
D[25]	0.3114	1.2093	0.3171	0.1928	0.2735	0.1520	0.1779	0.0710	0.0499		0.2334	1.0930	0.3156	0.1999	0.2029	0.1452	0.1879	0.0708	0.0498
D[26]	0.2255	1.1941	0.3048	0.2009	0.2543	0.1428	0.1881	0.0665	0.0466		0.2194	1.0196	0.2918	0.1983	0.1841	0.1329	0.1854	0.0665	0.0466
D[31]	0.2109	1.1803	0.2878	0.2047	0.1471	0.1318	0.1848	0.0348	0.0323		0.2441	0.9191	0.2484	0.1952	0.1401	0.1285	0.1747	0.0339	0.0319
D[32]	0.2752	1.1824	0.3021	0.2014	0.1492	0.1425	0.1879	0.0355	0.0353		0.1868	1.0117	0.2509	0.2032	0.1474	0.1309	0.1891	0.0354	0.0340
D[33]	0.2315	1.1656	0.2768	0.2074	0.1378	0.1314	0.1889	0.0338	0.0319		0.2352	0.9150	0.2407	0.2076	0.1366	0.1194	0.1936	0.0337	0.0319
D[34]	0.2544	1.0910	0.2534	0.1658	0.1375	0.1300	0.1417	0.0337	0.0317		0.2544	0.6060	0.2261	0.1658	0.1330	0.0518	0.1417	0.0329	0.0316
D[35]	0.2787	1.0886	0.2511	0.1613	0.1375	0.1298	0.1373	0.0337	0.0316		0.2771	0.5759	0.2147	0.1613	0.1330	0.0514	0.1373	0.0329	0.0313
D[36]	0.2626	0.9196	0.2149	0.1625	0.1319	0.1239	0.1369	0.0329	0.0302		0.2183	0.5143	0.2128	0.1625	0.1301	0.0438	0.1369	0.0327	0.0291
Set-2→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.5, 1.0)										(1.5, 2.5)
D[11]	1.1867	1.1055	0.7854	1.0389	0.1855	0.1438	1.1479	0.1128	0.0876		1.0904	0.9168	0.6060	1.0445	0.1827	0.1419	1.1582	0.1113	0.0864
D[12]	1.1759	1.1454	0.7924	1.0423	0.1881	0.1456	1.1573	0.1133	0.0878		0.9971	0.9173	0.6287	1.0385	0.1875	0.1452	1.1554	0.1115	0.0867
D[13]	1.3169	1.0630	0.7589	1.0407	0.1836	0.1425	1.1484	0.1115	0.0867		1.0374	0.8669	0.5759	1.0430	0.1815	0.1407	1.1582	0.1109	0.0863
D[14]	1.1898	0.7874	0.5258	1.2214	0.1102	0.0855	1.0946	0.0517	0.0346		1.1148	0.7799	0.5195	1.2254	0.1102	0.0855	1.0964	0.0515	0.0345
D[15]	1.1936	0.9173	0.6287	1.2227	0.1109	0.0864	1.0949	0.0531	0.0353		1.1092	0.8377	0.5682	1.2293	0.1109	0.0861	1.1002	0.0531	0.0353
D[16]	1.1839	0.7684	0.5254	1.2254	0.1068	0.0831	1.0973	0.0515	0.0345		1.1768	0.7683	0.5143	1.2254	0.1060	0.0824	1.0973	0.0510	0.0342
D[21]	1.1491	0.7651	0.4897	1.2286	0.0784	0.0518	1.1011	0.0510	0.0345		1.1926	0.7131	0.4834	1.2249	0.0708	0.0515	1.0967	0.0454	0.0342
D[22]	1.1408	0.6983	0.4834	1.2233	0.0778	0.0514	1.0960	0.0509	0.0343		1.2554	0.6983	0.4770	1.2225	0.0675	0.0510	1.0960	0.0432	0.0341
D[23]	1.1927	0.6969	0.4791	1.2240	0.0682	0.0509	1.0964	0.0438	0.0341		1.1927	0.6923	0.4615	1.2240	0.0675	0.0509	1.0964	0.0432	0.0341
D[24]	1.0633	0.6304	0.4116	1.2045	0.0574	0.0393	1.1825	0.0375	0.0262		1.1515	0.4871	0.3319	1.2016	0.0547	0.0388	1.1793	0.0363	0.0243
D[25]	1.0721	0.6438	0.4439	1.2025	0.0574	0.0400	1.1793	0.0393	0.0269		1.1484	0.6368	0.4180	1.2031	0.0569	0.0399	1.1793	0.0390	0.0269
D[26]	1.0893	0.5558	0.3803	1.2045	0.0560	0.0383	1.1825	0.0373	0.0253		1.1248	0.4759	0.3225	1.2045	0.0544	0.0376	1.1825	0.0357	0.0239
D[31]	1.0550	0.5363	0.3644	1.2044	0.0397	0.0327	1.1830	0.0274	0.0221		1.1672	0.4246	0.2947	1.2020	0.0396	0.0327	1.1825	0.0273	0.0221
D[32]	1.0891	0.5408	0.3719	1.2018	0.0397	0.0327	1.1792	0.0274	0.0221		1.1561	0.4423	0.3005	1.2012	0.0397	0.0327	1.1792	0.0274	0.0221
D[33]	1.0309	0.5326	0.3611	1.2044	0.0396	0.0327	1.1825	0.0273	0.0221		1.0309	0.4213	0.2892	1.2044	0.0383	0.0326	1.1825	0.0267	0.0220
D[34]	1.1617	0.4376	0.3057	1.2276	0.0383	0.0325	1.1549	0.0267	0.0219		1.2272	0.3671	0.2435	1.2306	0.0383	0.0325	1.1580	0.0267	0.0218
D[35]	1.1754	0.4844	0.3193	1.2316	0.0383	0.0327	1.1564	0.0267	0.0221		1.2257	0.4151	0.2888	1.2315	0.0383	0.0326	1.1584	0.0267	0.0219
D[36]	1.2236	0.4333	0.2849	1.2268	0.0383	0.0324	1.1550	0.0267	0.0218		1.2349	0.3467	0.2251	1.2291	0.0382	0.0324	1.1556	0.0266	0.0218

,

Table A4. The APE (1st Col.), RMSE (2nd Col.), and MRAB (3rd Col.) results of $R\left(x\right)$.

Design	MLE			Bayes							MLE			Bayes
Prior →				A			B							A			B
Set-1→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.2, 0.5)										(0.5, 0.8)
D[11]	0.7923	0.8863	0.7707	0.8553	0.6936	0.6839	0.7769	0.4639	0.4548		0.7923	0.8821	0.7707	0.8613	0.6910	0.6717	0.7875	0.4548	0.3969
D[12]	0.7877	1.2768	1.0989	0.8820	0.7757	0.7621	0.7886	0.5976	0.5253		0.7879	1.1320	0.9731	0.8723	0.7636	0.7502	0.7831	0.5439	0.4765
D[13]	0.7876	0.9727	0.8339	0.8564	0.7372	0.7156	0.7769	0.5038	0.4885		0.7880	0.9027	0.7774	0.8541	0.7251	0.7043	0.7749	0.4903	0.4037
D[14]	0.7903	0.7540	0.6689	0.7759	0.6049	0.5301	0.8305	0.4514	0.3945		0.9392	0.7540	0.6689	0.9500	0.6022	0.5223	0.9290	0.4264	0.3784
D[15]	0.7866	0.7786	0.6890	0.7693	0.6399	0.5331	0.8217	0.4519	0.4281		0.9380	0.7786	0.6890	0.9500	0.6230	0.5325	0.9289	0.4407	0.3787
D[16]	0.7838	0.8821	0.7637	0.7579	0.6792	0.5809	0.8120	0.4529	0.4422		0.9356	0.8690	0.7494	0.9499	0.6781	0.5769	0.9274	0.4521	0.3930
D[21]	0.7868	0.5352	0.5070	0.7854	0.4824	0.4564	0.8281	0.4225	0.3653		0.7867	0.5308	0.4969	0.7841	0.4815	0.4459	0.8272	0.3797	0.3607
D[22]	0.7789	0.5840	0.5322	0.7596	0.5185	0.4717	0.8143	0.4299	0.3930		0.7812	0.5802	0.5222	0.7624	0.5115	0.4608	0.8153	0.4225	0.3735
D[23]	0.7832	0.5393	0.5090	0.7753	0.4879	0.4675	0.8225	0.4264	0.3763		0.7843	0.5381	0.4999	0.7717	0.4878	0.4484	0.8140	0.3844	0.3689
D[24]	0.7913	0.4974	0.4401	0.8296	0.4309	0.4164	0.8114	0.3689	0.3520		0.7908	0.4764	0.4334	0.8308	0.4184	0.4127	0.8123	0.3675	0.3489
D[25]	0.7902	0.5126	0.4779	0.8293	0.4418	0.4324	0.8101	0.3759	0.3551		0.7906	0.4894	0.4482	0.8310	0.4358	0.4188	0.8135	0.3713	0.3490
D[26]	0.7895	0.5219	0.4912	0.8336	0.4603	0.4454	0.8164	0.4199	0.3626		0.7902	0.5167	0.4794	0.8303	0.4462	0.4388	0.8111	0.3790	0.3530
D[31]	0.7903	0.4148	0.4015	0.8263	0.3737	0.3593	0.8097	0.3488	0.3213		0.7899	0.4056	0.3915	0.8246	0.3730	0.3499	0.8078	0.3421	0.3187
D[32]	0.7862	0.4627	0.4365	0.8300	0.4085	0.4054	0.8145	0.3653	0.3503		0.7868	0.4538	0.4123	0.8292	0.4074	0.3891	0.8133	0.3592	0.3460
D[33]	0.7871	0.4312	0.4125	0.8269	0.4017	0.3769	0.8117	0.3557	0.3498		0.7872	0.4223	0.4112	0.8325	0.3852	0.3570	0.8154	0.3536	0.3434
D[34]	0.7892	0.3753	0.3687	0.7320	0.3637	0.3290	0.7705	0.1900	0.0556		0.7892	0.3700	0.3667	0.7320	0.3374	0.3273	0.7705	0.1897	0.0505
D[35]	0.7892	0.3903	0.3727	0.7300	0.3654	0.3399	0.7700	0.1913	0.0561		0.7893	0.3784	0.3708	0.7300	0.3422	0.3366	0.7700	0.1908	0.0509
D[36]	0.7881	0.4075	0.3910	0.7217	0.3717	0.3436	0.7669	0.1916	0.0671		0.7881	0.4018	0.3844	0.7217	0.3571	0.3372	0.7669	0.1910	0.0605
Set-2→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.5, 1.0)										(1.5, 2.5)
D[11]	0.9392	0.3953	0.3436	0.9500	0.2050	0.1959	0.9290	0.0894	0.0784		0.9377	0.3915	0.3422	0.9499	0.2048	0.1959	0.9275	0.0884	0.0775
D[12]	0.9356	0.4203	0.3631	0.9499	0.2155	0.2055	0.9274	0.0905	0.0791		0.9345	0.4164	0.3571	0.9501	0.2153	0.2054	0.9275	0.0894	0.0784
D[13]	0.9380	0.4174	0.3591	0.9500	0.2112	0.2017	0.9289	0.0899	0.0786		0.9370	0.4091	0.3528	0.9499	0.2086	0.1992	0.9275	0.0893	0.0782
D[14]	0.9344	0.3788	0.3332	0.9285	0.1900	0.1897	0.9382	0.0841	0.0739		0.9344	0.3779	0.3324	0.9285	0.1897	0.1894	0.9382	0.0803	0.0709
D[15]	0.9338	0.3849	0.3356	0.9286	0.1913	0.1899	0.9384	0.0877	0.0770		0.9339	0.3803	0.3345	0.9285	0.1908	0.1896	0.9382	0.0844	0.0740
D[16]	0.9311	0.3856	0.3386	0.9285	0.1921	0.1914	0.9383	0.0884	0.0775		0.9306	0.3844	0.3374	0.9279	0.1916	0.1910	0.9378	0.0877	0.0770
D[21]	0.9334	0.3016	0.2662	0.9287	0.0757	0.0701	0.9385	0.0673	0.0595		0.9334	0.2995	0.2636	0.9287	0.0747	0.0701	0.9385	0.0671	0.0523
D[22]	0.9298	0.3436	0.2889	0.9275	0.0835	0.0795	0.9373	0.0703	0.0605		0.9299	0.3358	0.2821	0.9277	0.0819	0.0778	0.9378	0.0682	0.0600
D[23]	0.9325	0.3111	0.2730	0.9286	0.0786	0.0742	0.9382	0.0696	0.0600		0.9330	0.3087	0.2730	0.9286	0.0770	0.0701	0.9382	0.0671	0.0528
D[24]	0.9358	0.2960	0.2556	0.9384	0.0743	0.0661	0.9347	0.0651	0.0523		0.9360	0.2951	0.2556	0.9384	0.0713	0.0651	0.9347	0.0584	0.0521
D[25]	0.9360	0.2989	0.2614	0.9384	0.0746	0.0689	0.9347	0.0651	0.0523		0.9360	0.2960	0.2610	0.9389	0.0734	0.0665	0.9349	0.0594	0.0521
D[26]	0.9349	0.2993	0.2634	0.9388	0.0747	0.0701	0.9349	0.0668	0.0528		0.9351	0.2977	0.2612	0.9388	0.0747	0.0688	0.9349	0.0665	0.0521
D[31]	0.9340	0.2408	0.2083	0.9390	0.0712	0.0584	0.9348	0.0534	0.0521		0.9340	0.2381	0.2082	0.9390	0.0709	0.0578	0.9348	0.0534	0.0505
D[32]	0.9321	0.2591	0.2261	0.9399	0.0735	0.0594	0.9354	0.0580	0.0521		0.9323	0.2580	0.2257	0.9398	0.0712	0.0585	0.9354	0.0573	0.0521
D[33]	0.9339	0.2441	0.2130	0.9390	0.0712	0.0584	0.9348	0.0559	0.0521		0.9341	0.2408	0.2130	0.9394	0.0710	0.0580	0.9348	0.0534	0.0509
D[34]	0.9316	0.2278	0.1941	0.9145	0.0700	0.0578	0.9300	0.0531	0.0472		0.9315	0.2259	0.1928	0.9145	0.0700	0.0556	0.9299	0.0530	0.0469
D[35]	0.9314	0.2296	0.1961	0.9142	0.0706	0.0580	0.9299	0.0534	0.0485		0.9314	0.2269	0.1942	0.9140	0.0700	0.0561	0.9296	0.0533	0.0470
D[36]	0.9304	0.2365	0.2077	0.9136	0.0706	0.0580	0.9298	0.0534	0.0490		0.9304	0.2354	0.2028	0.9136	0.0706	0.0578	0.9296	0.0534	0.0474

,

Table A5. The APE (1st Col.), RMSE (2nd Col.), and MRAB (3rd Col.) results of $h\left(x\right)$.

Design	MLE			Bayes							MLE			Bayes
Prior →				A			B							A			B
Set-1→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.2, 0.5)										(0.5, 0.8)
D[11]	1.4126	0.5899	0.5670	0.8554	0.4437	0.3953	1.2361	0.3604	0.2376		1.4134	0.5787	0.5481	0.8178	0.4222	0.3882	1.1747	0.3024	0.2053
D[12]	1.4542	0.8396	0.6775	0.7015	0.6253	0.5494	1.1667	0.4241	0.2690		1.4349	0.7429	0.6313	0.7574	0.5506	0.4923	1.1946	0.3772	0.2526
D[13]	1.4857	0.6574	0.6189	0.8473	0.5178	0.4313	1.2389	0.3807	0.2571		1.4709	0.6436	0.6014	0.8625	0.4442	0.4030	1.2531	0.3531	0.2073
D[14]	1.3230	0.4700	0.3290	1.3272	0.2784	0.2504	0.9304	0.2228	0.1499		0.9566	0.4643	0.2881	0.7351	0.2771	0.2441	0.9796	0.2003	0.1445
D[15]	1.4002	0.4817	0.3554	1.3666	0.3018	0.2523	0.9844	0.2378	0.1911		0.9925	0.4686	0.3302	0.7346	0.2875	0.2457	0.9801	0.2275	0.1453
D[16]	1.4238	0.5408	0.3812	1.4334	0.3475	0.2532	1.0424	0.2450	0.2047		1.0354	0.4775	0.3491	0.7365	0.2892	0.2463	0.9977	0.2288	0.1491
D[21]	1.3815	0.4453	0.2739	1.2833	0.2335	0.2153	0.9454	0.1591	0.1340		1.3846	0.3968	0.2606	1.2907	0.2206	0.1785	0.9511	0.1345	0.1178
D[22]	1.4452	0.4641	0.3257	1.4196	0.2731	0.2213	1.0205	0.1845	0.1378		1.4014	0.4537	0.2774	1.3999	0.2339	0.1912	1.0109	0.1416	0.1263
D[23]	1.4042	0.4506	0.2928	1.3420	0.2630	0.2181	0.9818	0.1600	0.1357		1.3921	0.4471	0.2700	1.3667	0.2207	0.1821	1.0311	0.1396	0.1200
D[24]	1.3223	0.3122	0.1979	1.1440	0.1888	0.1764	1.1772	0.1203	0.1161		1.3205	0.2968	0.1942	1.1348	0.1865	0.1706	1.1696	0.1163	0.1059
D[25]	1.3397	0.3304	0.2062	1.1456	0.2030	0.1793	1.1849	0.1310	0.1178		1.3296	0.2978	0.2048	1.1309	0.1903	0.1760	1.1624	0.1256	0.1159
D[26]	1.3551	0.3309	0.2102	1.1122	0.2062	0.1927	1.1401	0.1340	0.1306		1.3270	0.3034	0.2067	1.1391	0.1970	0.1781	1.1781	0.1307	0.1175
D[31]	1.3060	0.2657	0.1823	1.1697	0.1706	0.1584	1.1851	0.1159	0.1086		1.3142	0.2540	0.1823	1.1778	0.1655	0.1543	1.2000	0.1145	0.1022
D[32]	1.3591	0.3091	0.1909	1.1427	0.1877	0.1736	1.1642	0.1176	0.1143		1.3402	0.2905	0.1905	1.1498	0.1861	0.1671	1.1716	0.1148	0.1057
D[33]	1.3598	0.2861	0.1880	1.1626	0.1849	0.1727	1.1695	0.1161	0.1140		1.3547	0.2803	0.1868	1.1215	0.1813	0.1663	1.1383	0.1146	0.1040
D[34]	1.2840	0.2015	0.1671	1.3869	0.1598	0.1249	1.2437	0.1008	0.0570		1.2822	0.1997	0.1652	1.3869	0.1274	0.1022	1.2437	0.0957	0.0479
D[35]	1.2871	0.2061	0.1795	1.3906	0.1663	0.1269	1.2464	0.1070	0.0571		1.2844	0.2024	0.1762	1.3906	0.1289	0.1112	1.2464	0.1057	0.0489
D[36]	1.2863	0.2068	0.1813	1.4262	0.1671	0.1288	1.2682	0.1138	0.0622		1.2862	0.2066	0.1795	1.4262	0.1290	0.1145	1.2682	0.1059	0.0537
Set-2→
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)\to$	(0.5, 1.0)										(1.5, 2.5)
D[11]	0.9566	0.4568	0.3673	0.7351	0.2504	0.2441	0.9796	0.1192	0.1012		0.9714	0.4506	0.3604	0.7368	0.2497	0.2435	0.9970	0.1176	0.0996
D[12]	1.0354	0.5520	0.4317	0.7365	0.2540	0.2469	0.9977	0.1208	0.1019		1.0331	0.5408	0.4241	0.7343	0.2532	0.2463	0.9960	0.1192	0.1013
D[13]	0.9925	0.4932	0.3876	0.7346	0.2523	0.2457	0.9801	0.1199	0.1013		0.9944	0.4817	0.3807	0.7369	0.2497	0.2436	0.9970	0.1189	0.1005
D[14]	0.9677	0.3878	0.3138	1.0065	0.2000	0.1868	0.8721	0.1122	0.0949		0.9685	0.3850	0.3126	1.0065	0.1991	0.1857	0.8721	0.1073	0.0910
D[15]	0.9836	0.4061	0.3285	1.0043	0.2062	0.1923	0.8694	0.1173	0.0994		0.9898	0.3985	0.3243	1.0065	0.2027	0.1889	0.8712	0.1132	0.0957
D[16]	1.0118	0.4526	0.3612	1.0061	0.2105	0.1961	0.8703	0.1176	0.0996		1.0228	0.4497	0.3548	1.0138	0.2103	0.1959	0.8769	0.1173	0.0994
D[21]	0.9884	0.3309	0.2671	1.0039	0.0821	0.0693	0.8684	0.0664	0.0592		0.9884	0.3275	0.2641	1.0039	0.0821	0.0690	0.8684	0.0656	0.0584
D[22]	1.0513	0.3822	0.2987	1.0177	0.0881	0.0756	0.8823	0.0723	0.0600		1.0482	0.3816	0.2982	1.0144	0.0862	0.0741	0.8759	0.0699	0.0600
D[23]	1.0174	0.3323	0.2786	1.0055	0.0856	0.0715	0.8718	0.0691	0.0592		1.0085	0.3279	0.2780	1.0045	0.0835	0.0703	0.8718	0.0671	0.0592
D[24]	0.9859	0.3069	0.2493	0.9064	0.0816	0.0680	0.9329	0.0651	0.0584		0.9804	0.3046	0.2483	0.9064	0.0813	0.0671	0.9329	0.0624	0.0584
D[25]	0.9910	0.3080	0.2630	0.9064	0.0820	0.0680	0.9329	0.0656	0.0584		0.9800	0.3080	0.2624	0.9003	0.0816	0.0680	0.9302	0.0651	0.0584
D[26]	1.0049	0.3103	0.2664	0.9021	0.0821	0.0691	0.9302	0.0656	0.0584		0.9929	0.3093	0.2640	0.9022	0.0818	0.0688	0.9302	0.0656	0.0584
D[31]	0.9981	0.2540	0.2116	0.9006	0.0748	0.0670	0.9321	0.0624	0.0533		0.9981	0.2507	0.2116	0.9006	0.0741	0.0670	0.9321	0.0620	0.0489
D[32]	1.0376	0.3064	0.2401	0.8899	0.0813	0.0671	0.9255	0.0649	0.0539		1.0260	0.3007	0.2401	0.8917	0.0813	0.0671	0.9255	0.0622	0.0537
D[33]	1.0092	0.2550	0.2158	0.9006	0.0785	0.0671	0.9320	0.0624	0.0539		0.9972	0.2549	0.2122	0.8956	0.0741	0.0670	0.9321	0.0621	0.0489
D[34]	0.9749	0.2199	0.1820	1.1356	0.0695	0.0615	0.9789	0.0608	0.0484		0.9731	0.2161	0.1783	1.1366	0.0695	0.0608	0.9796	0.0570	0.0478
D[35]	0.9843	0.2249	0.1871	1.1387	0.0733	0.0623	0.9790	0.0608	0.0494		0.9737	0.2193	0.1809	1.1419	0.0695	0.0619	0.9831	0.0571	0.0482
D[36]	0.9959	0.2507	0.2070	1.1456	0.0741	0.0623	0.9813	0.0620	0.0499		0.9858	0.2420	0.2010	1.1454	0.0734	0.0622	0.9838	0.0620	0.0483

,

Table A6. The AIL (1st Col.) and CP (2nd Col.) results of $\alpha$.

Design	ACI[Norm]		ACI[Log-Norm]		BCI				HPD
Prior →					A		B		A		B
Set-1
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.2,0.5)$
D[11]	1.083	0.934	0.396	0.972	0.312	0.977	0.278	0.979	0.242	0.981	0.218	0.982
D[12]	1.166	0.929	0.480	0.968	0.313	0.977	0.280	0.979	0.242	0.981	0.221	0.982
D[13]	0.943	0.942	0.331	0.976	0.310	0.977	0.273	0.979	0.242	0.981	0.217	0.982
D[14]	0.786	0.950	0.321	0.976	0.273	0.979	0.227	0.982	0.218	0.982	0.146	0.986
D[15]	0.883	0.945	0.329	0.976	0.276	0.979	0.227	0.982	0.218	0.982	0.147	0.986
D[16]	0.673	0.957	0.291	0.978	0.267	0.980	0.222	0.982	0.218	0.982	0.145	0.986
D[21]	0.637	0.959	0.271	0.979	0.228	0.982	0.220	0.982	0.147	0.986	0.143	0.987
D[22]	0.633	0.959	0.265	0.980	0.226	0.982	0.217	0.982	0.146	0.986	0.142	0.987
D[23]	0.548	0.964	0.253	0.980	0.223	0.982	0.216	0.982	0.143	0.987	0.140	0.987
D[24]	0.485	0.967	0.246	0.981	0.215	0.983	0.155	0.986	0.142	0.987	0.129	0.987
D[25]	0.500	0.966	0.246	0.981	0.216	0.982	0.156	0.986	0.143	0.987	0.131	0.987
D[26]	0.468	0.968	0.242	0.981	0.209	0.983	0.154	0.986	0.142	0.987	0.129	0.987
D[31]	0.363	0.974	0.188	0.984	0.155	0.986	0.129	0.987	0.121	0.988	0.111	0.988
D[32]	0.454	0.969	0.225	0.982	0.156	0.986	0.131	0.987	0.121	0.988	0.115	0.988
D[33]	0.283	0.979	0.175	0.985	0.154	0.986	0.129	0.987	0.120	0.988	0.110	0.988
D[34]	0.276	0.979	0.154	0.986	0.122	0.988	0.120	0.988	0.111	0.988	0.100	0.989
D[35]	0.270	0.979	0.142	0.987	0.121	0.988	0.119	0.988	0.111	0.988	0.099	0.989
D[36]	0.185	0.984	0.136	0.987	0.121	0.988	0.118	0.988	0.110	0.988	0.099	0.989
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.5,0.8)$
D[11]	0.946	0.941	0.369	0.974	0.304	0.977	0.272	0.979	0.241	0.981	0.216	0.982
D[12]	1.041	0.936	0.390	0.973	0.310	0.977	0.278	0.979	0.242	0.981	0.220	0.982
D[13]	0.899	0.944	0.311	0.977	0.282	0.979	0.272	0.979	0.224	0.982	0.216	0.982
D[14]	0.610	0.960	0.273	0.979	0.270	0.979	0.222	0.982	0.213	0.983	0.143	0.987
D[15]	0.826	0.948	0.277	0.979	0.273	0.979	0.225	0.982	0.215	0.982	0.143	0.987
D[16]	0.591	0.961	0.269	0.979	0.266	0.980	0.220	0.982	0.213	0.983	0.142	0.987
D[21]	0.496	0.967	0.263	0.980	0.221	0.982	0.213	0.983	0.144	0.986	0.141	0.987
D[22]	0.474	0.968	0.253	0.980	0.221	0.982	0.211	0.983	0.143	0.987	0.140	0.987
D[23]	0.454	0.969	0.244	0.981	0.216	0.982	0.211	0.983	0.143	0.987	0.140	0.987
D[24]	0.348	0.975	0.219	0.982	0.205	0.983	0.155	0.986	0.139	0.987	0.127	0.987
D[25]	0.412	0.971	0.223	0.982	0.215	0.982	0.155	0.986	0.139	0.987	0.129	0.987
D[26]	0.336	0.976	0.219	0.982	0.184	0.984	0.154	0.986	0.139	0.987	0.127	0.987
D[31]	0.229	0.982	0.158	0.986	0.155	0.986	0.127	0.987	0.119	0.988	0.101	0.989
D[32]	0.239	0.981	0.167	0.985	0.155	0.986	0.129	0.987	0.119	0.988	0.105	0.989
D[33]	0.205	0.983	0.157	0.986	0.154	0.986	0.127	0.987	0.118	0.988	0.101	0.989
D[34]	0.168	0.985	0.150	0.986	0.121	0.988	0.119	0.988	0.111	0.988	0.100	0.989
D[35]	0.155	0.986	0.137	0.987	0.121	0.988	0.118	0.988	0.110	0.988	0.099	0.989
D[36]	0.149	0.986	0.123	0.988	0.120	0.988	0.118	0.988	0.108	0.989	0.099	0.989
Set-2
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.0,1.5)$
D[11]	0.946	0.940	0.400	0.973	0.296	0.979	0.271	0.980	0.242	0.982	0.220	0.983
D[12]	1.094	0.931	0.424	0.971	0.311	0.978	0.278	0.980	0.241	0.982	0.220	0.983
D[13]	0.899	0.943	0.390	0.973	0.288	0.979	0.271	0.980	0.242	0.982	0.216	0.984
D[14]	0.527	0.965	0.263	0.981	0.208	0.984	0.204	0.984	0.142	0.988	0.139	0.988
D[15]	0.864	0.945	0.301	0.978	0.208	0.984	0.204	0.984	0.143	0.988	0.140	0.988
D[16]	0.454	0.969	0.207	0.984	0.204	0.984	0.197	0.985	0.142	0.988	0.139	0.988
D[21]	0.452	0.969	0.207	0.984	0.204	0.984	0.192	0.985	0.141	0.988	0.138	0.988
D[22]	0.323	0.977	0.207	0.984	0.204	0.984	0.188	0.985	0.141	0.988	0.138	0.988
D[23]	0.308	0.978	0.207	0.984	0.204	0.984	0.187	0.985	0.141	0.988	0.138	0.988
D[24]	0.283	0.980	0.185	0.985	0.160	0.987	0.156	0.987	0.132	0.989	0.131	0.989
D[25]	0.301	0.978	0.186	0.985	0.161	0.987	0.158	0.987	0.134	0.988	0.131	0.989
D[26]	0.269	0.980	0.181	0.986	0.160	0.987	0.155	0.987	0.132	0.989	0.130	0.989
D[31]	0.245	0.982	0.170	0.986	0.122	0.989	0.119	0.989	0.113	0.990	0.102	0.990
D[32]	0.246	0.982	0.181	0.986	0.122	0.989	0.120	0.989	0.113	0.990	0.102	0.990
D[33]	0.237	0.982	0.166	0.987	0.122	0.989	0.119	0.989	0.113	0.990	0.101	0.990
D[34]	0.160	0.987	0.127	0.989	0.120	0.989	0.118	0.989	0.106	0.990	0.094	0.991
D[35]	0.187	0.985	0.156	0.987	0.121	0.989	0.119	0.989	0.112	0.990	0.100	0.990
D[36]	0.151	0.987	0.119	0.989	0.117	0.989	0.115	0.990	0.105	0.990	0.094	0.991
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.5,2.5)$
D[11]	0.915	0.942	0.312	0.978	0.278	0.980	0.270	0.980	0.235	0.982	0.215	0.984
D[12]	0.965	0.939	0.380	0.974	0.303	0.978	0.276	0.980	0.241	0.982	0.219	0.983
D[13]	0.841	0.946	0.311	0.978	0.273	0.980	0.269	0.980	0.220	0.983	0.215	0.984
D[14]	0.527	0.965	0.247	0.982	0.207	0.984	0.204	0.984	0.142	0.988	0.139	0.988
D[15]	0.780	0.950	0.265	0.981	0.208	0.984	0.204	0.984	0.143	0.988	0.139	0.988
D[16]	0.452	0.969	0.207	0.984	0.204	0.984	0.197	0.985	0.141	0.988	0.138	0.988
D[21]	0.345	0.976	0.207	0.984	0.204	0.984	0.190	0.985	0.141	0.988	0.138	0.988
D[22]	0.322	0.977	0.207	0.984	0.204	0.984	0.186	0.985	0.141	0.988	0.138	0.988
D[23]	0.304	0.978	0.206	0.984	0.203	0.984	0.182	0.986	0.141	0.988	0.138	0.988
D[24]	0.273	0.980	0.177	0.986	0.160	0.987	0.156	0.987	0.131	0.989	0.130	0.989
D[25]	0.292	0.979	0.181	0.986	0.160	0.987	0.156	0.987	0.132	0.989	0.131	0.989
D[26]	0.258	0.981	0.175	0.986	0.159	0.987	0.154	0.987	0.131	0.989	0.130	0.989
D[31]	0.218	0.983	0.169	0.986	0.122	0.989	0.119	0.989	0.113	0.990	0.101	0.990
D[32]	0.237	0.982	0.171	0.986	0.122	0.989	0.119	0.989	0.113	0.990	0.102	0.990
D[33]	0.203	0.984	0.165	0.987	0.121	0.989	0.119	0.989	0.112	0.990	0.100	0.990
D[34]	0.157	0.987	0.122	0.989	0.120	0.989	0.115	0.990	0.105	0.990	0.094	0.991
D[35]	0.181	0.986	0.153	0.987	0.121	0.989	0.119	0.989	0.111	0.990	0.097	0.991
D[36]	0.142	0.988	0.119	0.989	0.117	0.989	0.111	0.990	0.102	0.990	0.091	0.991

,

Table A7. The AIL (1st Col.) and CP (2nd Col.) results of $\gamma$.

Design	ACI[Norm]		ACI[Log-Norm]		BCI				HPD
Prior →					A		B		A		B
Set-1
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.2,0.5)$
D[11]	1.190	0.927	0.854	0.944	0.214	0.977	0.209	0.977	0.207	0.977	0.185	0.978
D[12]	1.015	0.936	0.845	0.945	0.213	0.977	0.208	0.977	0.204	0.977	0.184	0.978
D[13]	1.377	0.918	1.147	0.930	0.215	0.977	0.211	0.977	0.209	0.977	0.188	0.978
D[14]	0.981	0.938	0.753	0.950	0.212	0.977	0.183	0.978	0.167	0.979	0.159	0.980
D[15]	0.967	0.939	0.601	0.957	0.205	0.977	0.180	0.979	0.163	0.979	0.156	0.980
D[16]	0.969	0.939	0.745	0.950	0.209	0.977	0.183	0.978	0.165	0.979	0.156	0.980
D[21]	0.737	0.950	0.562	0.959	0.168	0.979	0.163	0.979	0.158	0.980	0.150	0.980
D[22]	0.604	0.957	0.526	0.961	0.166	0.979	0.160	0.980	0.156	0.980	0.148	0.980
D[23]	0.796	0.947	0.583	0.958	0.171	0.979	0.163	0.979	0.158	0.980	0.151	0.980
D[24]	0.510	0.962	0.481	0.963	0.165	0.979	0.159	0.980	0.155	0.980	0.141	0.981
D[25]	0.479	0.963	0.462	0.964	0.162	0.979	0.154	0.980	0.149	0.980	0.137	0.981
D[26]	0.499	0.962	0.472	0.964	0.165	0.979	0.159	0.980	0.152	0.980	0.137	0.981
D[31]	0.466	0.964	0.447	0.965	0.159	0.980	0.141	0.981	0.125	0.981	0.122	0.982
D[32]	0.423	0.966	0.382	0.968	0.156	0.980	0.134	0.981	0.123	0.981	0.121	0.982
D[33]	0.458	0.964	0.426	0.966	0.159	0.980	0.136	0.981	0.124	0.981	0.121	0.982
D[34]	0.393	0.968	0.359	0.970	0.127	0.981	0.126	0.981	0.123	0.981	0.120	0.982
D[35]	0.334	0.971	0.325	0.971	0.122	0.981	0.119	0.982	0.119	0.982	0.093	0.983
D[36]	0.353	0.970	0.327	0.971	0.124	0.981	0.124	0.981	0.122	0.981	0.120	0.982
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.5,0.8)$
D[11]	1.009	0.937	0.803	0.947	0.213	0.977	0.207	0.977	0.206	0.977	0.173	0.979
D[12]	0.937	0.940	0.693	0.953	0.212	0.977	0.205	0.977	0.201	0.978	0.172	0.979
D[13]	1.218	0.926	0.970	0.939	0.214	0.977	0.210	0.977	0.208	0.977	0.175	0.979
D[14]	0.865	0.944	0.594	0.958	0.211	0.977	0.175	0.979	0.165	0.979	0.151	0.980
D[15]	0.796	0.947	0.569	0.959	0.203	0.977	0.171	0.979	0.161	0.980	0.149	0.980
D[16]	0.821	0.946	0.581	0.958	0.209	0.977	0.173	0.979	0.163	0.979	0.149	0.980
D[21]	0.629	0.956	0.533	0.961	0.166	0.979	0.160	0.980	0.153	0.980	0.146	0.980
D[22]	0.553	0.960	0.502	0.962	0.166	0.979	0.160	0.980	0.153	0.980	0.145	0.980
D[23]	0.666	0.954	0.553	0.960	0.171	0.979	0.160	0.980	0.155	0.980	0.147	0.980
D[24]	0.481	0.963	0.479	0.963	0.164	0.979	0.157	0.980	0.148	0.980	0.140	0.981
D[25]	0.466	0.964	0.445	0.965	0.160	0.980	0.153	0.980	0.142	0.980	0.137	0.981
D[26]	0.479	0.963	0.464	0.964	0.162	0.979	0.157	0.980	0.144	0.980	0.137	0.981
D[31]	0.461	0.964	0.435	0.966	0.157	0.980	0.140	0.981	0.125	0.981	0.121	0.982
D[32]	0.393	0.968	0.382	0.968	0.154	0.980	0.133	0.981	0.121	0.982	0.120	0.982
D[33]	0.433	0.966	0.414	0.967	0.157	0.980	0.135	0.981	0.122	0.981	0.121	0.982
D[34]	0.361	0.969	0.350	0.970	0.127	0.981	0.126	0.981	0.120	0.982	0.119	0.982
D[35]	0.327	0.971	0.319	0.972	0.121	0.982	0.119	0.982	0.117	0.982	0.092	0.983
D[36]	0.345	0.970	0.320	0.971	0.124	0.981	0.123	0.981	0.120	0.982	0.117	0.982
Set-2
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.0,1.5)$
D[11]	1.021	0.929	0.323	0.971	0.213	0.977	0.207	0.978	0.191	0.979	0.183	0.979
D[12]	0.899	0.936	0.321	0.971	0.212	0.977	0.207	0.978	0.189	0.979	0.183	0.979
D[13]	1.139	0.922	0.387	0.967	0.214	0.977	0.210	0.978	0.191	0.979	0.185	0.979
D[14]	0.885	0.937	0.313	0.971	0.208	0.978	0.192	0.979	0.149	0.981	0.146	0.981
D[15]	0.648	0.951	0.294	0.973	0.205	0.978	0.190	0.979	0.147	0.981	0.144	0.982
D[16]	0.833	0.940	0.295	0.973	0.207	0.978	0.192	0.979	0.149	0.981	0.146	0.981
D[21]	0.639	0.952	0.247	0.975	0.165	0.980	0.159	0.981	0.140	0.982	0.139	0.982
D[22]	0.482	0.961	0.236	0.976	0.163	0.980	0.157	0.981	0.140	0.982	0.139	0.982
D[23]	0.647	0.951	0.265	0.974	0.168	0.980	0.162	0.980	0.140	0.982	0.139	0.982
D[24]	0.409	0.966	0.231	0.976	0.163	0.980	0.156	0.981	0.138	0.982	0.137	0.982
D[25]	0.360	0.969	0.203	0.978	0.162	0.981	0.153	0.981	0.138	0.982	0.134	0.982
D[26]	0.382	0.967	0.205	0.978	0.163	0.980	0.156	0.981	0.138	0.982	0.136	0.982
D[31]	0.322	0.971	0.193	0.979	0.129	0.982	0.128	0.983	0.126	0.983	0.095	0.984
D[32]	0.271	0.974	0.191	0.979	0.129	0.982	0.127	0.983	0.126	0.983	0.094	0.985
D[33]	0.339	0.970	0.194	0.979	0.129	0.982	0.128	0.983	0.127	0.983	0.098	0.984
D[34]	0.265	0.974	0.188	0.979	0.127	0.983	0.127	0.983	0.125	0.983	0.091	0.985
D[35]	0.245	0.976	0.151	0.981	0.112	0.984	0.108	0.984	0.087	0.985	0.087	0.985
D[36]	0.259	0.975	0.165	0.980	0.115	0.983	0.111	0.984	0.090	0.985	0.090	0.985
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.5,2.5)$
D[11]	0.967	0.932	0.318	0.971	0.210	0.978	0.202	0.978	0.187	0.979	0.183	0.979
D[12]	0.887	0.937	0.305	0.972	0.208	0.978	0.201	0.978	0.185	0.979	0.180	0.979
D[13]	1.024	0.929	0.323	0.971	0.212	0.978	0.204	0.978	0.188	0.979	0.183	0.979
D[14]	0.852	0.939	0.294	0.973	0.204	0.978	0.191	0.979	0.149	0.981	0.146	0.981
D[15]	0.629	0.953	0.291	0.973	0.202	0.978	0.189	0.979	0.147	0.981	0.144	0.982
D[16]	0.788	0.943	0.291	0.973	0.202	0.978	0.191	0.979	0.149	0.981	0.145	0.981
D[21]	0.604	0.954	0.233	0.976	0.163	0.980	0.157	0.981	0.140	0.982	0.139	0.982
D[22]	0.413	0.965	0.225	0.977	0.162	0.980	0.157	0.981	0.140	0.982	0.139	0.982
D[23]	0.610	0.954	0.259	0.975	0.166	0.980	0.162	0.980	0.140	0.982	0.139	0.982
D[24]	0.376	0.968	0.219	0.977	0.162	0.981	0.155	0.981	0.138	0.982	0.137	0.982
D[25]	0.339	0.970	0.194	0.979	0.160	0.981	0.153	0.981	0.138	0.982	0.133	0.982
D[26]	0.371	0.968	0.200	0.978	0.160	0.981	0.153	0.981	0.138	0.982	0.135	0.982
D[31]	0.285	0.973	0.192	0.979	0.129	0.982	0.128	0.983	0.126	0.983	0.094	0.985
D[32]	0.270	0.974	0.188	0.979	0.128	0.983	0.127	0.983	0.125	0.983	0.094	0.985
D[33]	0.337	0.970	0.193	0.979	0.129	0.982	0.128	0.983	0.126	0.983	0.097	0.984
D[34]	0.262	0.975	0.168	0.980	0.127	0.983	0.126	0.983	0.125	0.983	0.091	0.985
D[35]	0.155	0.981	0.129	0.982	0.111	0.984	0.107	0.984	0.087	0.985	0.087	0.985
D[36]	0.250	0.975	0.155	0.981	0.114	0.983	0.110	0.984	0.090	0.985	0.090	0.985

,

Table A8. The AIL (1st Col.) and CP (2nd Col.) results of $\theta$.

Design	ACI[Norm]		ACI[Log-Norm]		BCI				HPD
Prior →					A		B		A		B
Set-1
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.2,0.5)$
D[11]	0.720	0.941	0.307	0.970	0.269	0.973	0.243	0.975	0.193	0.978	0.184	0.979
D[12]	0.815	0.934	0.315	0.969	0.277	0.972	0.245	0.974	0.211	0.977	0.196	0.978
D[13]	0.676	0.944	0.303	0.970	0.265	0.973	0.233	0.975	0.188	0.978	0.183	0.979
D[14]	0.366	0.966	0.200	0.978	0.195	0.978	0.184	0.979	0.176	0.979	0.138	0.982
D[15]	0.430	0.961	0.212	0.977	0.201	0.977	0.187	0.978	0.183	0.979	0.140	0.982
D[16]	0.347	0.967	0.191	0.978	0.187	0.979	0.178	0.979	0.172	0.980	0.138	0.982
D[21]	0.345	0.967	0.190	0.978	0.178	0.979	0.154	0.981	0.138	0.982	0.137	0.982
Set-1
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.2,0.5)$
D[22]	0.319	0.969	0.182	0.979	0.173	0.980	0.153	0.981	0.135	0.982	0.135	0.982
D[23]	0.315	0.969	0.182	0.979	0.164	0.980	0.152	0.981	0.132	0.982	0.132	0.982
D[24]	0.279	0.972	0.124	0.983	0.122	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[25]	0.304	0.970	0.129	0.983	0.124	0.983	0.117	0.983	0.116	0.984	0.114	0.984
D[26]	0.223	0.976	0.123	0.983	0.121	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[31]	0.201	0.978	0.123	0.983	0.121	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[32]	0.218	0.976	0.123	0.983	0.121	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[33]	0.188	0.978	0.122	0.983	0.120	0.983	0.117	0.983	0.114	0.984	0.113	0.984
D[34]	0.149	0.981	0.122	0.983	0.120	0.983	0.106	0.984	0.103	0.984	0.098	0.985
D[35]	0.122	0.983	0.118	0.983	0.116	0.984	0.105	0.984	0.101	0.985	0.096	0.985
D[36]	0.118	0.983	0.116	0.984	0.106	0.984	0.105	0.984	0.101	0.985	0.095	0.985
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.5,0.8)$
D[11]	0.453	0.960	0.301	0.970	0.263	0.973	0.194	0.978	0.189	0.978	0.179	0.979
D[12]	0.565	0.952	0.312	0.970	0.274	0.972	0.214	0.977	0.207	0.977	0.192	0.978
D[13]	0.393	0.964	0.301	0.970	0.262	0.973	0.192	0.978	0.187	0.979	0.179	0.979
D[14]	0.360	0.966	0.195	0.978	0.190	0.978	0.180	0.979	0.142	0.982	0.138	0.982
D[15]	0.382	0.965	0.207	0.977	0.192	0.978	0.187	0.979	0.146	0.981	0.140	0.982
D[16]	0.342	0.968	0.190	0.978	0.181	0.979	0.173	0.980	0.141	0.982	0.138	0.982
D[21]	0.302	0.970	0.183	0.979	0.173	0.980	0.154	0.981	0.138	0.982	0.137	0.982
D[22]	0.285	0.972	0.177	0.979	0.166	0.980	0.153	0.981	0.135	0.982	0.135	0.982
D[23]	0.280	0.972	0.176	0.979	0.163	0.980	0.151	0.981	0.132	0.982	0.132	0.982
D[24]	0.221	0.976	0.123	0.983	0.121	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[25]	0.277	0.972	0.126	0.983	0.121	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[26]	0.216	0.976	0.122	0.983	0.120	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[31]	0.176	0.979	0.121	0.983	0.118	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[32]	0.201	0.978	0.122	0.983	0.120	0.983	0.117	0.983	0.115	0.984	0.113	0.984
D[33]	0.156	0.981	0.120	0.983	0.117	0.983	0.117	0.983	0.114	0.984	0.112	0.984
D[34]	0.122	0.983	0.120	0.983	0.117	0.983	0.106	0.984	0.102	0.985	0.097	0.985
D[35]	0.120	0.983	0.117	0.984	0.111	0.984	0.105	0.984	0.101	0.985	0.096	0.985
D[36]	0.118	0.983	0.114	0.984	0.105	0.984	0.105	0.984	0.101	0.985	0.095	0.985
Set-2
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.0,1.5)$
D[11]	0.720	0.942	0.307	0.971	0.243	0.976	0.213	0.978	0.198	0.979	0.187	0.980
D[12]	0.799	0.936	0.312	0.971	0.245	0.976	0.225	0.977	0.208	0.978	0.188	0.980
D[13]	0.631	0.948	0.303	0.971	0.233	0.977	0.208	0.978	0.190	0.980	0.172	0.981
D[14]	0.337	0.969	0.176	0.981	0.172	0.981	0.157	0.982	0.141	0.983	0.108	0.986
D[15]	0.467	0.960	0.177	0.981	0.173	0.981	0.159	0.982	0.143	0.983	0.108	0.985
D[16]	0.314	0.971	0.175	0.981	0.171	0.981	0.156	0.982	0.138	0.983	0.105	0.986
D[21]	0.309	0.971	0.173	0.981	0.170	0.981	0.141	0.983	0.108	0.985	0.104	0.986
D[22]	0.295	0.972	0.173	0.981	0.169	0.981	0.139	0.983	0.108	0.986	0.104	0.986
D[23]	0.271	0.974	0.173	0.981	0.169	0.981	0.136	0.983	0.105	0.986	0.103	0.986
D[24]	0.232	0.977	0.123	0.984	0.118	0.985	0.114	0.985	0.103	0.986	0.101	0.986
D[25]	0.249	0.975	0.123	0.984	0.118	0.985	0.115	0.985	0.105	0.986	0.102	0.986
D[26]	0.218	0.978	0.123	0.984	0.118	0.985	0.113	0.985	0.100	0.986	0.099	0.986
D[31]	0.217	0.978	0.122	0.984	0.117	0.985	0.112	0.985	0.098	0.986	0.097	0.986
D[32]	0.217	0.978	0.122	0.984	0.118	0.985	0.112	0.985	0.099	0.986	0.098	0.986
D[33]	0.202	0.979	0.122	0.985	0.117	0.985	0.111	0.985	0.098	0.986	0.097	0.986
D[34]	0.169	0.981	0.113	0.985	0.112	0.985	0.109	0.985	0.095	0.986	0.093	0.987
D[35]	0.174	0.981	0.113	0.985	0.112	0.985	0.111	0.985	0.096	0.986	0.094	0.987
D[36]	0.167	0.981	0.112	0.985	0.111	0.985	0.095	0.986	0.094	0.986	0.088	0.987
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.5,2.5)$
D[11]	0.453	0.961	0.269	0.974	0.231	0.977	0.207	0.978	0.189	0.980	0.171	0.981
D[12]	0.467	0.960	0.274	0.974	0.241	0.976	0.221	0.977	0.196	0.979	0.177	0.981
D[13]	0.393	0.965	0.265	0.974	0.231	0.977	0.198	0.979	0.189	0.980	0.171	0.981
D[14]	0.314	0.971	0.175	0.981	0.171	0.981	0.155	0.982	0.141	0.983	0.107	0.986
D[15]	0.360	0.967	0.177	0.981	0.172	0.981	0.156	0.982	0.142	0.983	0.108	0.986
D[16]	0.309	0.971	0.173	0.981	0.169	0.981	0.155	0.982	0.138	0.983	0.105	0.986
D[21]	0.309	0.971	0.173	0.981	0.169	0.981	0.141	0.983	0.108	0.986	0.104	0.986
D[22]	0.293	0.972	0.173	0.981	0.169	0.981	0.136	0.984	0.107	0.986	0.102	0.986
D[23]	0.246	0.976	0.172	0.981	0.168	0.981	0.135	0.984	0.105	0.986	0.101	0.986
D[24]	0.196	0.979	0.122	0.984	0.118	0.985	0.114	0.985	0.101	0.986	0.100	0.986
D[25]	0.236	0.976	0.122	0.984	0.118	0.985	0.115	0.985	0.103	0.986	0.100	0.986
D[26]	0.187	0.980	0.122	0.984	0.117	0.985	0.112	0.985	0.099	0.986	0.098	0.986
D[31]	0.166	0.981	0.122	0.985	0.117	0.985	0.111	0.985	0.097	0.986	0.097	0.986
D[32]	0.169	0.981	0.122	0.984	0.117	0.985	0.111	0.985	0.098	0.986	0.097	0.986
D[33]	0.163	0.982	0.122	0.985	0.117	0.985	0.110	0.985	0.097	0.986	0.095	0.986
D[34]	0.140	0.983	0.112	0.985	0.111	0.985	0.108	0.986	0.094	0.986	0.092	0.987
D[35]	0.161	0.982	0.112	0.985	0.112	0.985	0.108	0.985	0.095	0.986	0.093	0.987
D[36]	0.132	0.984	0.111	0.985	0.111	0.985	0.094	0.986	0.094	0.987	0.087	0.987

,

Table A9. The AIL (1st Col.) and CP (2nd Col.) results of $R\left(x\right)$.

Design	ACI[Norm]		ACI[Log-Norm]		BCI				HPD
Prior →					A		B		A		B
Set-1
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.2,0.5)$
D[11]	0.251	0.949	0.235	0.953	0.148	0.971	0.140	0.972	0.137	0.973	0.123	0.976
D[12]	0.297	0.940	0.279	0.944	0.206	0.959	0.177	0.965	0.175	0.965	0.152	0.970
D[13]	0.274	0.945	0.247	0.950	0.168	0.966	0.148	0.970	0.142	0.972	0.131	0.974
D[14]	0.237	0.952	0.198	0.960	0.135	0.973	0.132	0.974	0.116	0.977	0.114	0.977
D[15]	0.240	0.952	0.201	0.960	0.135	0.973	0.133	0.974	0.118	0.977	0.116	0.977
D[16]	0.246	0.950	0.230	0.954	0.138	0.973	0.134	0.973	0.121	0.976	0.121	0.976
D[21]	0.200	0.960	0.182	0.964	0.115	0.977	0.110	0.978	0.097	0.981	0.090	0.982
D[22]	0.236	0.952	0.188	0.962	0.116	0.977	0.113	0.978	0.106	0.979	0.097	0.981
D[23]	0.215	0.957	0.184	0.963	0.116	0.977	0.113	0.978	0.102	0.980	0.091	0.982
D[24]	0.184	0.963	0.153	0.970	0.110	0.978	0.091	0.982	0.090	0.983	0.079	0.985
D[25]	0.184	0.963	0.158	0.969	0.111	0.978	0.096	0.981	0.090	0.982	0.081	0.984
D[26]	0.188	0.962	0.176	0.965	0.113	0.978	0.096	0.981	0.090	0.982	0.081	0.984
D[31]	0.154	0.969	0.145	0.971	0.090	0.983	0.088	0.983	0.080	0.985	0.071	0.986
D[32]	0.177	0.965	0.146	0.971	0.090	0.982	0.090	0.983	0.082	0.984	0.078	0.985
D[33]	0.158	0.969	0.146	0.971	0.090	0.982	0.089	0.983	0.080	0.984	0.078	0.985
D[34]	0.145	0.971	0.129	0.975	0.087	0.983	0.072	0.986	0.028	0.995	0.014	0.998
D[35]	0.145	0.971	0.129	0.974	0.088	0.983	0.077	0.985	0.028	0.995	0.015	0.998
D[36]	0.146	0.971	0.141	0.972	0.089	0.983	0.079	0.985	0.028	0.995	0.015	0.998
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.5,0.8)$
D[11]	0.249	0.950	0.234	0.953	0.143	0.972	0.137	0.973	0.134	0.973	0.122	0.976
D[12]	0.281	0.943	0.277	0.944	0.199	0.960	0.176	0.965	0.171	0.966	0.151	0.970
D[13]	0.272	0.945	0.246	0.950	0.149	0.970	0.145	0.971	0.137	0.973	0.126	0.975
D[14]	0.236	0.952	0.197	0.960	0.135	0.973	0.128	0.975	0.116	0.977	0.114	0.978
D[15]	0.239	0.952	0.201	0.960	0.135	0.973	0.132	0.974	0.116	0.977	0.115	0.977
D[16]	0.245	0.951	0.229	0.954	0.138	0.973	0.133	0.974	0.121	0.976	0.121	0.976
D[21]	0.199	0.960	0.181	0.964	0.113	0.978	0.110	0.978	0.096	0.981	0.090	0.982
D[22]	0.235	0.953	0.188	0.962	0.115	0.977	0.112	0.978	0.106	0.979	0.097	0.981
D[23]	0.213	0.957	0.183	0.963	0.115	0.977	0.112	0.978	0.101	0.980	0.091	0.982
D[24]	0.183	0.963	0.153	0.970	0.107	0.979	0.091	0.982	0.089	0.983	0.079	0.985
D[25]	0.184	0.963	0.158	0.969	0.108	0.979	0.095	0.981	0.090	0.983	0.080	0.984
D[26]	0.188	0.962	0.175	0.965	0.112	0.978	0.096	0.981	0.090	0.982	0.080	0.984
D[31]	0.154	0.969	0.145	0.971	0.089	0.983	0.088	0.983	0.079	0.985	0.071	0.986
D[32]	0.177	0.965	0.145	0.971	0.090	0.982	0.089	0.983	0.082	0.984	0.078	0.985
D[33]	0.158	0.969	0.145	0.971	0.090	0.983	0.089	0.983	0.080	0.985	0.078	0.985
D[34]	0.145	0.971	0.128	0.975	0.087	0.983	0.072	0.986	0.026	0.996	0.014	0.998
D[35]	0.145	0.971	0.129	0.974	0.088	0.983	0.077	0.985	0.026	0.996	0.015	0.998
D[36]	0.146	0.971	0.141	0.972	0.089	0.983	0.079	0.985	0.027	0.995	0.015	0.998
Set-2
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.0,1.5)$
D[11]	0.136	0.951	0.136	0.951	0.035	0.990	0.034	0.990	0.022	0.995	0.021	0.995
D[12]	0.149	0.946	0.148	0.946	0.037	0.989	0.036	0.990	0.023	0.995	0.021	0.995
D[13]	0.144	0.948	0.141	0.949	0.036	0.990	0.035	0.990	0.023	0.995	0.021	0.995
D[14]	0.130	0.953	0.129	0.954	0.028	0.993	0.026	0.993	0.020	0.996	0.020	0.996
D[15]	0.130	0.953	0.129	0.954	0.028	0.993	0.026	0.993	0.020	0.996	0.020	0.996
D[16]	0.134	0.952	0.134	0.952	0.029	0.993	0.027	0.993	0.020	0.996	0.020	0.996
D[21]	0.110	0.961	0.109	0.961	0.019	0.996	0.019	0.996	0.019	0.996	0.019	0.996
D[22]	0.123	0.956	0.122	0.956	0.020	0.996	0.019	0.996	0.019	0.996	0.019	0.996
D[23]	0.112	0.960	0.112	0.960	0.019	0.996	0.019	0.996	0.019	0.996	0.019	0.996
D[24]	0.106	0.963	0.105	0.963	0.019	0.996	0.018	0.996	0.018	0.996	0.018	0.997
D[25]	0.106	0.963	0.106	0.963	0.019	0.996	0.019	0.996	0.018	0.996	0.018	0.997
D[26]	0.109	0.961	0.109	0.961	0.019	0.996	0.019	0.996	0.018	0.996	0.018	0.997
D[31]	0.088	0.969	0.088	0.970	0.018	0.997	0.017	0.997	0.017	0.997	0.016	0.997
D[32]	0.095	0.967	0.094	0.967	0.018	0.997	0.018	0.997	0.017	0.997	0.016	0.997
D[33]	0.094	0.967	0.094	0.967	0.018	0.997	0.018	0.997	0.017	0.997	0.016	0.997
D[34]	0.086	0.970	0.086	0.970	0.017	0.997	0.016	0.997	0.015	0.998	0.014	0.998
D[35]	0.087	0.970	0.086	0.970	0.017	0.997	0.016	0.997	0.015	0.998	0.014	0.998
D[36]	0.088	0.970	0.088	0.970	0.017	0.997	0.016	0.997	0.015	0.998	0.015	0.998
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.5,2.5)$
D[11]	0.136	0.951	0.136	0.951	0.035	0.990	0.034	0.990	0.022	0.995	0.021	0.995
D[12]	0.149	0.946	0.148	0.946	0.037	0.989	0.036	0.990	0.022	0.995	0.021	0.995
D[13]	0.144	0.948	0.141	0.949	0.036	0.990	0.035	0.990	0.022	0.995	0.021	0.995
D[14]	0.130	0.953	0.128	0.954	0.028	0.993	0.026	0.993	0.020	0.996	0.020	0.996
D[15]	0.130	0.953	0.129	0.954	0.028	0.993	0.026	0.993	0.020	0.996	0.020	0.996
D[16]	0.134	0.952	0.134	0.952	0.028	0.993	0.027	0.993	0.020	0.996	0.020	0.996
D[21]	0.110	0.961	0.109	0.961	0.019	0.996	0.019	0.996	0.019	0.996	0.019	0.996
D[22]	0.123	0.956	0.122	0.957	0.020	0.996	0.019	0.996	0.019	0.996	0.019	0.996
D[23]	0.112	0.960	0.111	0.961	0.019	0.996	0.019	0.996	0.019	0.996	0.019	0.996
D[24]	0.106	0.963	0.105	0.963	0.019	0.996	0.018	0.996	0.018	0.996	0.018	0.997
D[25]	0.106	0.963	0.106	0.963	0.019	0.996	0.019	0.996	0.018	0.996	0.018	0.997
D[26]	0.109	0.961	0.109	0.961	0.019	0.996	0.019	0.996	0.018	0.996	0.018	0.997
D[31]	0.088	0.969	0.088	0.970	0.018	0.997	0.017	0.997	0.017	0.997	0.016	0.997
D[32]	0.095	0.967	0.094	0.967	0.018	0.997	0.018	0.997	0.017	0.997	0.016	0.997
D[33]	0.094	0.967	0.094	0.967	0.018	0.997	0.018	0.997	0.017	0.997	0.016	0.997
D[34]	0.086	0.970	0.086	0.970	0.017	0.997	0.016	0.997	0.015	0.998	0.014	0.998
D[35]	0.087	0.970	0.086	0.970	0.017	0.997	0.016	0.997	0.015	0.998	0.014	0.998
D[36]	0.088	0.970	0.088	0.970	0.017	0.997	0.016	0.997	0.015	0.998	0.015	0.998

and

Table A10. The AIL (1st Col.) and CP (2nd Col.) results of $h\left(x\right)$.

Design	ACI[Norm]		ACI[Log-Norm]		BCI				HPD
Prior →					A		B		A		B
Set-1
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.2,0.5)$
D[11]	2.145	0.920	1.960	0.926	1.336	0.944	1.175	0.949	1.093	0.951	0.963	0.955
D[12]	2.415	0.912	2.216	0.918	1.860	0.929	1.627	0.935	1.502	0.939	1.324	0.944
D[13]	2.232	0.917	2.133	0.920	1.541	0.938	1.287	0.946	1.199	0.948	1.046	0.953
D[14]	1.968	0.925	1.552	0.938	0.723	0.962	0.682	0.964	0.631	0.965	0.625	0.965
D[15]	2.024	0.924	1.615	0.936	0.730	0.962	0.703	0.963	0.635	0.965	0.630	0.965
D[16]	2.040	0.923	1.794	0.930	0.760	0.961	0.750	0.962	0.650	0.965	0.641	0.965
D[21]	1.418	0.942	1.325	0.944	0.646	0.965	0.539	0.968	0.534	0.968	0.513	0.969
D[22]	1.656	0.935	1.528	0.938	0.690	0.963	0.580	0.967	0.555	0.967	0.540	0.968
D[23]	1.585	0.937	1.500	0.939	0.654	0.964	0.549	0.968	0.544	0.968	0.524	0.968
D[24]	1.190	0.948	1.125	0.950	0.539	0.968	0.516	0.969	0.499	0.969	0.490	0.969
D[25]	1.303	0.945	1.244	0.947	0.549	0.968	0.518	0.968	0.509	0.969	0.501	0.969
D[26]	1.353	0.944	1.296	0.945	0.555	0.967	0.529	0.968	0.515	0.969	0.510	0.969
D[31]	1.053	0.953	0.896	0.957	0.510	0.969	0.505	0.969	0.485	0.969	0.473	0.970
D[32]	1.142	0.950	1.097	0.951	0.525	0.968	0.513	0.969	0.496	0.969	0.485	0.969
D[33]	1.101	0.951	1.007	0.954	0.513	0.969	0.505	0.969	0.492	0.969	0.479	0.970
D[34]	0.779	0.961	0.754	0.961	0.496	0.969	0.488	0.969	0.400	0.972	0.170	0.979
D[35]	0.853	0.958	0.765	0.961	0.498	0.969	0.491	0.969	0.405	0.972	0.187	0.978
D[36]	0.918	0.957	0.827	0.959	0.504	0.969	0.494	0.969	0.408	0.972	0.188	0.978
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(0.5,0.8)$
D[11]	2.066	0.922	1.819	0.930	1.276	0.946	1.151	0.950	1.046	0.953	0.945	0.956
D[12]	2.378	0.913	2.193	0.919	1.713	0.933	1.504	0.939	1.379	0.943	1.208	0.948
D[13]	2.166	0.919	1.873	0.928	1.336	0.944	1.263	0.946	1.094	0.951	0.983	0.955
D[14]	1.771	0.931	1.519	0.939	0.722	0.962	0.679	0.964	0.631	0.965	0.625	0.965
D[15]	1.838	0.929	1.589	0.937	0.724	0.962	0.688	0.963	0.635	0.965	0.630	0.965
D[16]	1.846	0.929	1.711	0.933	0.755	0.961	0.741	0.962	0.650	0.965	0.641	0.965
D[21]	1.349	0.944	1.269	0.946	0.643	0.965	0.534	0.968	0.533	0.968	0.512	0.969
D[22]	1.614	0.936	1.515	0.939	0.689	0.963	0.578	0.967	0.550	0.968	0.539	0.968
D[23]	1.583	0.937	1.499	0.939	0.650	0.965	0.547	0.968	0.543	0.968	0.520	0.968
D[24]	1.172	0.949	1.113	0.951	0.533	0.968	0.513	0.969	0.492	0.969	0.484	0.969
D[25]	1.269	0.946	1.222	0.948	0.544	0.968	0.515	0.969	0.508	0.969	0.491	0.969
D[26]	1.318	0.945	1.266	0.946	0.550	0.968	0.528	0.968	0.511	0.969	0.504	0.969
D[31]	0.918	0.957	0.896	0.957	0.509	0.969	0.495	0.969	0.478	0.970	0.472	0.970
D[32]	1.109	0.951	1.044	0.953	0.517	0.969	0.513	0.969	0.487	0.969	0.483	0.970
D[33]	1.021	0.954	0.969	0.955	0.509	0.969	0.495	0.969	0.480	0.970	0.478	0.970
D[34]	0.766	0.961	0.742	0.962	0.488	0.969	0.479	0.970	0.364	0.973	0.164	0.979
D[35]	0.831	0.959	0.753	0.961	0.491	0.969	0.489	0.969	0.368	0.973	0.179	0.979
D[36]	0.837	0.959	0.812	0.960	0.495	0.969	0.492	0.969	0.370	0.973	0.180	0.979
Set-2
$({\mathsf{ȷ}}_1,{\mathsf{ȷ}}_2)=(1.0,1.5)$
D[11]	2.130	0.919	1.595	0.936	0.400	0.974	0.364	0.975	0.240	0.979	0.236	0.980
D[12]	2.225	0.916	1.942	0.925	0.413	0.974	0.375	0.975	0.245	0.979	0.240	0.979
D[13]	2.140	0.919	1.731	0.932	0.405	0.974	0.368	0.975	0.243	0.979	0.239	0.979
D[14]	1.753	0.931	1.501	0.939	0.347	0.976	0.340	0.976	0.234	0.980	0.231	0.980
D[15]	1.893	0.927	1.551	0.938	0.360	0.976	0.352	0.976	0.238	0.979	0.234	0.980
D[16]	2.085	0.920	1.593	0.936	0.370	0.975	0.361	0.976	0.239	0.979	0.236	0.980
D[21]	1.354	0.944	1.261	0.947	0.226	0.980	0.222	0.980	0.217	0.980	0.217	0.980
D[22]	1.649	0.934	1.447	0.941	0.231	0.980	0.229	0.980	0.219	0.980	0.219	0.980
D[23]	1.389	0.943	1.288	0.946	0.229	0.980	0.227	0.980	0.219	0.980	0.218	0.980
D[24]	1.247	0.947	1.168	0.950	0.219	0.980	0.216	0.980	0.213	0.980	0.212	0.980
D[25]	1.250	0.947	1.180	0.949	0.220	0.980	0.217	0.980	0.216	0.980	0.214	0.980
D[26]	1.293	0.946	1.202	0.949	0.220	0.980	0.217	0.980	0.217	0.980	0.216	0.980
D[31]	1.002	0.955	0.963	0.956	0.215	0.980	0.212	0.980	0.183	0.981	0.180	0.981
D[32]	1.233	0.948	1.155	0.950	0.218	0.980	0.215	0.980	0.186	0.981	0.183	0.981
D[33]	1.033	0.954	0.987	0.956	0.218	0.980	0.215	0.980	0.183	0.981	0.180	0.981
D[34]	0.905	0.958	0.873	0.959	0.178	0.981	0.175	0.981	0.171	0.982	0.166	0.982
D[35]	0.914	0.958	0.882	0.959	0.187	0.981	0.179	0.981	0.176	0.981	0.173	0.982
D[36]	0.984	0.956	0.944	0.957	0.188	0.981	0.180	0.981	0.180	0.981	0.173	0.982
$(j_1,j_2)=(1.5,2.5)$
D[11]	1.846	0.928	1.589	0.936	0.398	0.974	0.361	0.976	0.239	0.979	0.236	0.980
D[12]	2.149	0.918	1.873	0.927	0.408	0.974	0.370	0.975	0.241	0.979	0.237	0.979
D[13]	2.029	0.922	1.708	0.932	0.398	0.974	0.362	0.975	0.240	0.979	0.236	0.980
D[14]	1.680	0.933	1.462	0.940	0.347	0.976	0.340	0.976	0.232	0.980	0.229	0.980
D[15]	1.691	0.933	1.519	0.939	0.355	0.976	0.348	0.976	0.235	0.980	0.232	0.980
D[16]	1.771	0.930	1.525	0.938	0.369	0.975	0.360	0.976	0.236	0.979	0.234	0.980
D[21]	1.321	0.945	1.230	0.948	0.224	0.980	0.221	0.980	0.217	0.980	0.217	0.980
D[22]	1.646	0.934	1.444	0.941	0.229	0.980	0.228	0.980	0.219	0.980	0.219	0.980
D[23]	1.330	0.945	1.239	0.947	0.228	0.980	0.225	0.980	0.219	0.980	0.218	0.980
D[24]	1.205	0.949	1.132	0.951	0.219	0.980	0.216	0.980	0.213	0.980	0.212	0.980
D[25]	1.222	0.948	1.145	0.950	0.219	0.980	0.216	0.980	0.216	0.980	0.214	0.980
D[26]	1.293	0.946	1.202	0.949	0.220	0.980	0.217	0.980	0.217	0.980	0.216	0.980
D[31]	0.984	0.956	0.944	0.957	0.215	0.980	0.212	0.980	0.180	0.981	0.176	0.981
D[32]	1.180	0.949	1.120	0.951	0.218	0.980	0.215	0.980	0.186	0.981	0.182	0.981
D[33]	0.988	0.955	0.947	0.957	0.218	0.980	0.215	0.980	0.183	0.981	0.180	0.981
D[34]	0.885	0.959	0.855	0.960	0.178	0.981	0.175	0.981	0.170	0.982	0.158	0.982
D[35]	0.903	0.958	0.871	0.959	0.180	0.981	0.178	0.981	0.175	0.981	0.166	0.982
D[36]	0.976	0.956	0.939	0.957	0.183	0.981	0.180	0.981	0.176	0.981	0.166	0.982

summarize the outcomes of the proposed simulation experiments, highlighting the behavior of $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$. The main observations can be outlined as follows:

• Across all simulated scenarios, the proposed estimation procedures demonstrate stable performance. This is reflected in consistently low values of RMSE, MRAB, and AIL, alongside CPs that remain close to the nominal 95% level.
• Increasing n or $q_i$ ($i=1,2$) generally improves estimation accuracy. Similarly, reducing the number of censored observations contributes to more precise estimates. Additionally, longer censoring times ${\mathsf{ȷ}}_i$ ($i=1,2$) tend to enhance the reliability and precision of estimates for all coefficients.
• Under Prior-B, the Bayesian estimates outperform those based on Prior-A. Overall, both Bayesian approaches produce more efficient and less biased estimates compared to their frequentist counterparts. This behavior can be explained by the fact that the hyperparameters of Prior-B provide prior distributions that are more compatible with the true parameter values used in the simulation design, leading to posterior distributions with reduced variability and improved estimation precision.
• Comparisons of the 95% interval methods reveal that:

  –

  Intervals constructed via the HPD (for either prior) generally provide superior coverage and shorter interval lengths than those obtained with the BCI approach. Both types of credible intervals tend to outperform the asymptotic intervals (ACI-NA and ACI-NL) in terms of reliability and precision.

  –

  For $\gamma$ and $R\left(x\right)$, the ACI[Log-Norm] results are slightly narrower and more accurate than the corresponding ACI-[Norm] intervals.

  –

  For $\alpha$, $\theta$, and $h\left(x\right)$, the ACI[Norm] approach shows marginally better performance than ACI[Log-Norm].

  –

  The CPs for both Bayesian (BCI/HPD) and classical (ACI-NA/ACI-NL) intervals generally achieve or exceed the nominal 95% across most scenarios.

  –

  The HPD intervals consistently outperform the two approaches of ACIs in terms of AIL and CP behaviors. This result is expected because HPD intervals are constructed directly from the posterior distribution and therefore fully incorporate the uncertainty arising from both the observed data and the prior information.

  –

  In contrast, both ACIs are based on asymptotic normal and log-normal approximations to the likelihood function, which may be less accurate for moderate sample sizes or in the presence of censoring. Consequently, HPD intervals tend to produce more efficient interval estimates in censored lifetime models.

• When the true values in NOT-Exp$(\vartheta ,\theta )$ are increased, it can be noted that all point and interval estimation findings of $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$ become satisfactory in terms of lowest RMSE, lowest MRAB, AIL results, and highest CP values.
• Evaluating the performance of the proposed censoring designs listed in Table 1:

  –

  The right schemes $\left[i3\right]$ and $\left[i6\right]$ provide the most efficient estimates for $\alpha$ and $\theta$.

  –

  The left schemes $\left[i1\right]$ and $\left[i4\right]$ provide the most efficient estimates for $R\left(x\right)$ and $h\left(x\right)$.

  –

  The middle schemes $\left[i2\right]$ and $\left[i5\right]$ provide the most efficient estimates for $\gamma$.

• Overall, once the UT2-PH dataset is generated, the Bayesian point and credible interval methodologies demonstrate clear advantages over other methods. We finally recommend the MCMC framework, which delivers more efficient, stable, and reliable inference for the NOT-Exp parameters and associated reliability functions.

6. Real-World Applications

This section examines two genuine datasets from the toxicology and banking sectors, aiming to: (i) assess the flexibility, accuracy, and practical relevance of the NOT-Exp model; (ii) demonstrate how the model’s inferential outcomes can support real-world decision-making; and (iii) benchmark its performance against twelve existing lifespan distributions. Table 2 provides a summary of the datasets employed in this comparative analysis, detailed as follows:

• Toxicological Application: Variations in airborne exposure and their influence on urinary metabolite concentrations are a central focus in toxicology and environmental health research. Airborne chemicals, once inhaled, undergo absorption, distribution, metabolism, and excretion, leading to the formation of measurable metabolites in biological matrices such as urine. Understanding how fluctuations in airborne exposure translate into changes in urinary metabolite concentrations is crucial for accurately assessing human exposure, validating toxicokinetic and physiologically based pharmacokinetic (PBPK) models, and identifying dose–response relationships (see Valavanidis et al. [15] for more details). This application (say, App.1) investigates the impact of varying airborne exposure on urinary metabolite concentrations for thirty individual human subjects exposed to acetone under controlled conditions. Airborne exposure was expressed in mg/m³ (milligrams per cubic meter of air), while urinary metabolite concentrations were expressed in mg/g creatinine. This dataset was first provided by Kumagai and Matsunaga [16] and reanalyzed by Peter et al. [17].
• Banking Management Application: Banking systems are designed to manage customer flow and provide financial services efficiently, but they often involve queues that lead to long waiting times before customers receive service. Bank waiting times refer to the duration customers spend from arrival until being attended by a teller or service agent, and they represent a fundamental performance indicator for evaluating service efficiency, improving customer satisfaction, and optimizing the allocation of human and operational resources; Cowdrey et al. [18]. This application (say, App.2) examines one hundred recorded waiting times (in minutes) representing the duration customers spend in a bank before being served. This dataset was presented by Ghitany et al. [19] and later reanalyzed by Alsubie [20].

Table 2. The datasets of airborne variations and banking service systems.

Airborne Variations
1.5	1.7	2.1	2.2	2.4	2.5	2.6	3.8	3.8	4.2	4.3
5.6	6.0	7.0	7.5	9.3	9.9	10.2	10.6	12.3	12.9	13.7
14.1	17.8	27.6	31.0	42.0	45.6	51.9	91.3	131.8
Bank Waiting Times
0.8	0.8	1.3	1.5	1.8	1.9	1.9	2.1	2.6	2.7	2.9
3.1	3.2	3.3	3.5	3.6	4.0	4.1	4.2	4.2	4.3	4.3
4.4	4.4	4.6	4.7	4.7	4.8	4.9	4.9	5.0	5.3	5.5
5.7	5.7	6.1	6.2	6.2	6.2	6.3	6.7	6.9	7.1	7.1
7.1	7.1	7.4	7.6	7.7	8.0	8.2	8.6	8.6	8.6	8.8
8.8	8.9	8.9	9.5	9.6	9.7	9.8	10.7	10.9	11.0	11.0
11.1	11.2	11.2	11.5	11.9	12.4	12.5	12.9	13.0	13.1	13.3
13.6	13.7	13.9		14.1	15.4	15.4	17.3	17.3	18.1	18.2
18.4	18.9	19.0	19.9	20.6	21.3	21.4	21.9	23.0	27.0	31.6
33.1	38.5

Table 2. The datasets of airborne variations and banking service systems.

Airborne Variations
1.5	1.7	2.1	2.2	2.4	2.5	2.6	3.8	3.8	4.2	4.3
5.6	6.0	7.0	7.5	9.3	9.9	10.2	10.6	12.3	12.9	13.7
14.1	17.8	27.6	31.0	42.0	45.6	51.9	91.3	131.8
Bank Waiting Times
0.8	0.8	1.3	1.5	1.8	1.9	1.9	2.1	2.6	2.7	2.9
3.1	3.2	3.3	3.5	3.6	4.0	4.1	4.2	4.2	4.3	4.3
4.4	4.4	4.6	4.7	4.7	4.8	4.9	4.9	5.0	5.3	5.5
5.7	5.7	6.1	6.2	6.2	6.2	6.3	6.7	6.9	7.1	7.1
7.1	7.1	7.4	7.6	7.7	8.0	8.2	8.6	8.6	8.6	8.8
8.8	8.9	8.9	9.5	9.6	9.7	9.8	10.7	10.9	11.0	11.0
11.1	11.2	11.2	11.5	11.9	12.4	12.5	12.9	13.0	13.1	13.3
13.6	13.7	13.9		14.1	15.4	15.4	17.3	17.3	18.1	18.2
18.4	18.9	19.0	19.9	20.6	21.3	21.4	21.9	23.0	27.0	31.6
33.1	38.5

Briefly,

Table 3. Summary for the datasets in App.i (for $i=1,2$).

App.	Mean	Mode	${\mathcal{Q}}_1$	${\mathcal{Q}}_2$	${\mathcal{Q}}_3$	St.D	Skew.
1	19.01	3.80	3.80	9.30	15.95	28.55	2.65
2	9.88	7.10	4.68	8.10	13.03	7.237	1.47

presents a comparative summary of the two datasets in App.i (for $i=1,2$) through key descriptive statistics including measures of central tendency (mean, mode), quartiles ($Q_i$ for $i=1,2,3$), variability (standard deviation (St.D)), and distribution shape (skewness). These statistics provide an initial understanding of the underlying data distributions, essential for guiding subsequent analyses or interpretations. Overall, App.1 demonstrates a higher average value and greater variability with a more pronounced right skew, suggesting a dataset with some extremely high values influencing the mean, while App.2 is still positively skewed, appears more concentrated around its mode and median with less dispersion.

Using total time on test (TTT) plots and violin within boxplot (VB) diagnostics, Figure 6 provides a graphical analysis of two datasets from App.i (for $i=1,2$). The subplots in Figure 6a reveal that the two datasets from App.i (for $i=1,2$) provide decreasing and increasing failure rates, respectively, which both align with the same NOT-Exp failure rates depicted in Figure 1b. In contrast, the violin plots support these observations, showing App.1 has a right-skewed distribution with several outliers, while App.2’s data are more symmetric and concentrated.

To assess the suitability and flexibility of the proposed NOT-Exp model, it was fitted to two complete real datasets and its performance was compared against twelve alternative lifetime models, each representing diverse HR behaviors (see Table 4). Model optimum was conducted using eight widely recognized goodness-of-fit and information criteria: (i) negative log-likelihood ($\mathsf{NLL}$), (ii) Akaike ($\mathsf{A}$), (iii) Bayesian ($\mathsf{B}$), (iv) consistent Akaike ($\mathsf{CA}$), (v) Hannan–Quinn criterion ($\mathsf{HQ}$), (vi) Anderson–Darling ($\mathsf{AD}$), (vii) Cramér–von Mises ($\mathsf{CvM}$), and (viii) the Kolmogorov–Smirnov statistic ($\mathsf{KS}$) along with its associated p-value.

All computations were performed using the $\mathsf{AdequacyModel}$ package (v2.0.0) (Marinho et al. [21]) (see Table 5). For each candidate model, parameters along with their standard errors (St.Es) were estimated via maximum likelihood. Additionally, various goodness-of-fit measures were computed and are reported in the same table. The preferred model is determined as the one minimizing the information-based criteria (i)–(v) while simultaneously maximizing the p-value associated with the $\mathsf{KS}$ statistic. From Table 5, the NOT-Exp model provides the best overall fit to the datasets in App.i (for $i=1,2$), outperforming all alternative lifetime models considered.

Table 4. Twelve competitive models of the NOT-Exp distribution.

Author(s)	Symbol	Model
Bagdonavicius and Nikulin [22]	PGW $(\alpha ,\gamma ,\theta )$	Power Generalized Weibull
Peng and Yan [23]	NEW $(\alpha ,\gamma ,\theta )$	New Extended Weibull
Mahdavi and Kundu [24]	APE $(\gamma ,\theta )$	Alpha-Power Exponential
Pinho et al. [25]	HEE $(\alpha ,\gamma ,\theta )$	Harris-Extended Exponential
Alotaibi et al. [26]	EP $(\alpha ,\gamma ,\theta )$	Exponentiated-Pham
Mudholkar and Srivastava [27]	EW $(\alpha ,\gamma ,\theta )$	Exponentiated Weibull
Oguntunde et al. [28]	WE $(\alpha ,\gamma ,\theta )$	Weibull Exponential
Nadarajah and Haghighi [29]	NH $(\gamma ,\theta )$	Nadarajah–Haghighi
Gupta and Kundu [30]	GE $(\gamma ,\theta )$	Generalized Exponential
Birnbaum and Saunders [31]	BS $(\gamma ,\theta )$	Birnbaum–Saunders
Weibull [32]	W $(\gamma ,\theta )$	Weibull
Johnson et al. [33]	G $(\gamma ,\theta )$	Gamma

Table 4. Twelve competitive models of the NOT-Exp distribution.

Author(s)	Symbol	Model
Bagdonavicius and Nikulin [22]	PGW $(\alpha ,\gamma ,\theta )$	Power Generalized Weibull
Peng and Yan [23]	NEW $(\alpha ,\gamma ,\theta )$	New Extended Weibull
Mahdavi and Kundu [24]	APE $(\gamma ,\theta )$	Alpha-Power Exponential
Pinho et al. [25]	HEE $(\alpha ,\gamma ,\theta )$	Harris-Extended Exponential
Alotaibi et al. [26]	EP $(\alpha ,\gamma ,\theta )$	Exponentiated-Pham
Mudholkar and Srivastava [27]	EW $(\alpha ,\gamma ,\theta )$	Exponentiated Weibull
Oguntunde et al. [28]	WE $(\alpha ,\gamma ,\theta )$	Weibull Exponential
Nadarajah and Haghighi [29]	NH $(\gamma ,\theta )$	Nadarajah–Haghighi
Gupta and Kundu [30]	GE $(\gamma ,\theta )$	Generalized Exponential
Birnbaum and Saunders [31]	BS $(\gamma ,\theta )$	Birnbaum–Saunders
Weibull [32]	W $(\gamma ,\theta )$	Weibull
Johnson et al. [33]	G $(\gamma ,\theta )$	Gamma

Table 5. The NOT-Exp fitting and its competitors from App.i (for $i=1,2$).

Model	$\mathit{\alpha }$		$\mathit{\gamma }$		$\mathit{\theta }$		$\mathsf{NLL}$	$\mathsf{A}$	$\mathsf{CA}$	$\mathsf{B}$	$\mathsf{HQ}$	$\mathsf{AD}$	$\mathsf{CvM}$	$\mathsf{KS}$	p-Value
	Est.	St.E	Est.	St.E	Est.	St.E
App.1
NOT-Exp	0.0121	0.0145	8.2571	8.3908	0.0162	0.0110	115.96	237.92	238.81	242.23	239.33	0.2152	0.0287	0.0935	0.9493
PGW	0.0030	0.0059	13.635	7.5739	0.0255	0.0155	116.50	239.09	240.00	243.46	240.52	0.2366	0.0313	0.0985	0.9244
NEW	4.0576	1.5384	0.4488	0.1520	0.4545	0.2743	116.90	239.86	240.76	244.20	241.27	0.2160	0.0294	0.0947	0.9451
APE	-	-	0.0440	0.0842	0.0242	0.0133	119.47	242.94	243.37	245.81	243.88	0.6876	0.1039	0.1203	0.7609
HEE	0.0136	0.0561	1.1475	4.5699	15.253	60.316	119.98	245.95	246.84	250.26	247.36	0.6748	0.1002	0.1255	0.7129
EP	0.1031	0.0313	3.8999	0.9714	62.604	67.090	117.08	240.16	241.05	244.46	241.56	0.3342	0.0446	0.0939	0.9483
EW	8.9994	41.509	21.260	42.862	0.2828	0.1620	117.02	240.05	240.94	244.35	241.45	0.3213	0.0425	0.0957	0.9373
WE	7.4202	3.9030	0.7653	0.0991	0.0040	0.0020	121.55	249.09	249.98	253.39	250.49	1.0355	0.1661	0.1717	0.3205
NH	-	-	0.5319	0.1268	0.1808	0.0958	119.51	243.02	243.45	245.89	243.96	0.6289	0.0938	0.1273	0.6968
GE	-	-	0.8110	0.1901	0.0454	0.0117	121.87	247.75	248.18	250.62	248.68	1.1182	0.1815	0.1970	0.1801
BS	-	-	1.3370	0.1684	10.301	1.9825	116.68	239.26	239.69	242.35	240.20	0.3533	0.0504	0.1488	0.4990
W	-	-	0.8202	0.1063	0.0995	0.0403	120.98	245.96	246.39	248.82	246.89	0.9432	0.1499	0.1603	0.4031
G	-	-	0.8012	0.1755	0.0422	0.0125	121.74	247.49	247.91	250.35	248.42	1.0969	0.1777	0.1904	0.2112
App.2
NOT-Exp	2.6162	6.9501	2.5701	1.0643	0.1497	0.0270	317.00	640.00	640.25	647.81	643.16	0.1270	0.0170	0.0368	0.9993
PGW	0.0322	0.0127	2.0169	0.3881	0.4571	0.1626	317.07	640.14	640.39	647.96	643.30	0.1275	0.0174	0.0379	0.9988
NEW	1.7880	1.1060	1.2248	0.1726	0.0662	0.0361	317.02	640.04	640.29	647.86	643.21	0.1761	0.0238	0.0465	0.9820
APE	-	-	20.982	13.9661	0.1829	0.0197	319.04	642.07	642.20	647.98	644.18	0.4175	0.0666	0.0526	0.9452
HEE	6.2972	3.3510	0.1368	0.0177	10.969	8.1686	317.52	641.05	641.30	648.86	644.21	0.1361	0.0191	0.0415	0.9953
EP	0.3405	0.1025	1.8353	0.4145	7.6234	5.0187	317.15	640.30	640.55	648.12	643.19	0.1272	0.0173	0.0376	0.9987
EW	0.1905	0.1065	2.6747	1.6430	0.9057	0.2570	317.03	640.07	640.32	647.88	643.23	0.1271	0.0175	0.0384	0.9984
WE	18.634	14.842	1.3636	0.1070	0.0100	0.0047	319.50	645.00	645.25	652.82	648.16	0.5010	0.0796	0.0629	0.8233
NH	-	-	3.3367	1.8424	0.0212	0.0139	323.45	650.90	651.02	656.11	653.01	0.6969	0.1113	0.1076	0.1976
GE	-	-	2.1834	0.3343	0.1592	0.0175	317.10	642.02	642.14	647.93	644.14	0.1428	0.0207	0.0402	0.9969
BS	-	-	0.8462	0.0597	7.2078	0.5562	320.33	644.66	644.78	649.87	646.77	0.5987	0.0808	0.0801	0.5426
W	-	-	1.4581	0.1089	0.0305	0.0095	318.73	641.46	641.59	647.97	643.57	0.3960	0.0629	0.0576	0.8941
G	-	-	2.0091	0.2639	0.2034	0.0303	317.30	642.43	642.56	647.94	644.55	0.1823	0.0276	0.0425	0.9935

Table 5. The NOT-Exp fitting and its competitors from App.i (for $i=1,2$).

Model	$\mathit{\alpha }$		$\mathit{\gamma }$		$\mathit{\theta }$		$\mathsf{NLL}$	$\mathsf{A}$	$\mathsf{CA}$	$\mathsf{B}$	$\mathsf{HQ}$	$\mathsf{AD}$	$\mathsf{CvM}$	$\mathsf{KS}$	p-Value
	Est.	St.E	Est.	St.E	Est.	St.E
App.1
NOT-Exp	0.0121	0.0145	8.2571	8.3908	0.0162	0.0110	115.96	237.92	238.81	242.23	239.33	0.2152	0.0287	0.0935	0.9493
PGW	0.0030	0.0059	13.635	7.5739	0.0255	0.0155	116.50	239.09	240.00	243.46	240.52	0.2366	0.0313	0.0985	0.9244
NEW	4.0576	1.5384	0.4488	0.1520	0.4545	0.2743	116.90	239.86	240.76	244.20	241.27	0.2160	0.0294	0.0947	0.9451
APE	-	-	0.0440	0.0842	0.0242	0.0133	119.47	242.94	243.37	245.81	243.88	0.6876	0.1039	0.1203	0.7609
HEE	0.0136	0.0561	1.1475	4.5699	15.253	60.316	119.98	245.95	246.84	250.26	247.36	0.6748	0.1002	0.1255	0.7129
EP	0.1031	0.0313	3.8999	0.9714	62.604	67.090	117.08	240.16	241.05	244.46	241.56	0.3342	0.0446	0.0939	0.9483
EW	8.9994	41.509	21.260	42.862	0.2828	0.1620	117.02	240.05	240.94	244.35	241.45	0.3213	0.0425	0.0957	0.9373
WE	7.4202	3.9030	0.7653	0.0991	0.0040	0.0020	121.55	249.09	249.98	253.39	250.49	1.0355	0.1661	0.1717	0.3205
NH	-	-	0.5319	0.1268	0.1808	0.0958	119.51	243.02	243.45	245.89	243.96	0.6289	0.0938	0.1273	0.6968
GE	-	-	0.8110	0.1901	0.0454	0.0117	121.87	247.75	248.18	250.62	248.68	1.1182	0.1815	0.1970	0.1801
BS	-	-	1.3370	0.1684	10.301	1.9825	116.68	239.26	239.69	242.35	240.20	0.3533	0.0504	0.1488	0.4990
W	-	-	0.8202	0.1063	0.0995	0.0403	120.98	245.96	246.39	248.82	246.89	0.9432	0.1499	0.1603	0.4031
G	-	-	0.8012	0.1755	0.0422	0.0125	121.74	247.49	247.91	250.35	248.42	1.0969	0.1777	0.1904	0.2112
App.2
NOT-Exp	2.6162	6.9501	2.5701	1.0643	0.1497	0.0270	317.00	640.00	640.25	647.81	643.16	0.1270	0.0170	0.0368	0.9993
PGW	0.0322	0.0127	2.0169	0.3881	0.4571	0.1626	317.07	640.14	640.39	647.96	643.30	0.1275	0.0174	0.0379	0.9988
NEW	1.7880	1.1060	1.2248	0.1726	0.0662	0.0361	317.02	640.04	640.29	647.86	643.21	0.1761	0.0238	0.0465	0.9820
APE	-	-	20.982	13.9661	0.1829	0.0197	319.04	642.07	642.20	647.98	644.18	0.4175	0.0666	0.0526	0.9452
HEE	6.2972	3.3510	0.1368	0.0177	10.969	8.1686	317.52	641.05	641.30	648.86	644.21	0.1361	0.0191	0.0415	0.9953
EP	0.3405	0.1025	1.8353	0.4145	7.6234	5.0187	317.15	640.30	640.55	648.12	643.19	0.1272	0.0173	0.0376	0.9987
EW	0.1905	0.1065	2.6747	1.6430	0.9057	0.2570	317.03	640.07	640.32	647.88	643.23	0.1271	0.0175	0.0384	0.9984
WE	18.634	14.842	1.3636	0.1070	0.0100	0.0047	319.50	645.00	645.25	652.82	648.16	0.5010	0.0796	0.0629	0.8233
NH	-	-	3.3367	1.8424	0.0212	0.0139	323.45	650.90	651.02	656.11	653.01	0.6969	0.1113	0.1076	0.1976
GE	-	-	2.1834	0.3343	0.1592	0.0175	317.10	642.02	642.14	647.93	644.14	0.1428	0.0207	0.0402	0.9969
BS	-	-	0.8462	0.0597	7.2078	0.5562	320.33	644.66	644.78	649.87	646.77	0.5987	0.0808	0.0801	0.5426
W	-	-	1.4581	0.1089	0.0305	0.0095	318.73	641.46	641.59	647.97	643.57	0.3960	0.0629	0.0576	0.8941
G	-	-	2.0091	0.2639	0.2034	0.0303	317.30	642.43	642.56	647.94	644.55	0.1823	0.0276	0.0425	0.9935

To facilitate interpretation of the comparative results and complement the numerical goodness-of-fit statistics reported in Table 5, a set of graphical diagnostics is provided in Figure 7. All candidate distributions were fitted to the datasets under identical sampling conditions to ensure a fair comparison. Figure 7 presents several graphical tools commonly used for model assessment, including the empirical and fitted PDF/RF curves, probability–probability (P–P) plots, and quantile–quantile (Q–Q) plots for the datasets in App.i $(i=1,2)$. These plots provide a visual comparison between the empirical distribution of the data and the fitted theoretical models. The subplots in Figure 7 consistently indicate that the NOT-Exp distribution provides the closest agreement with the empirical patterns of the data compared with the competing models. In particular, the fitted curves closely follow the empirical density and reliability behavior, while the P–P and Q–Q plots exhibit points lying near the reference line. These graphical diagnostics further support the superiority of the NOT-Exp model for the considered datasets arising from toxicological and banking management applications. Figure 8 indicates that the log-likelihood contours for $\alpha$, $\gamma$, and $\theta$ from App.i (for $i=1,2$), existed and are unique.

Based on the complete datasets for App.i ($i=1,2$) reported in Table 2, multiple UT2-PH censored samples are generated under different combinations of $(q_1,q_2)$, where $(q_1,q_2)=(10,19)$ for App.1 and $(20,40)$ for App.2. These samples are constructed using various configurations of ${\mathsf{ȷ}}_i$$(i=1,2)$ and progressive censoring schemes $\mathbf{S}$, as detailed in

Table 6. Different UT2-PH censored datasets from App.i (for $i=1,2$).

Sample	S	${\mathsf{ȷ}}_1\left({\mathit{d}}_1\right)$	${\mathsf{ȷ}}_2\left({\mathit{d}}_2\right)$	${\mathcal{S}}^{\mathit{\star }}$	${\mathcal{T}}^{\mathit{\star }}$	Data
App.1
$\mathcal{S}\left[11\right]$	($2^{\left[6\right]}$,$0^{\left[13\right]}$)	43(19)	45(19)	0	51.9	1.5, 2.1, 2.2, 2.5, 2.6, 3.8, 3.8, 4.3, 6, 7, 9.3, 10.2, 10.6, 12.3, 14.1, 17.8, 27.6, 42, 51.9
$\mathcal{S}\left[12\right]$	($0^{\left[6\right]}$,$2^{\left[6\right]}$,$0^{\left[7\right]}$)	46(15)	50(15)	4	46	1.5, 1.7, 2.1, 2.2, 2.4, 2.5, 2.6, 3.8, 6, 7.5, 10.6, 12.3, 14.1, 31, 45.6
$\mathcal{S}\left[13\right]$	($2^{\left[3\right]}$,$0^{\left[13\right]}$,$2^{\left[3\right]}$)	32(10)	35(10)	15	32	1.5, 2.6, 3.8, 4.2, 5.6, 7.5, 9.3, 12.9, 17.8, 31
$\mathcal{S}\left[14\right]$	($0^{\left[13\right]}$,$2^{\left[6\right]}$)	3(7)	5(8)	23	5	1.5, 1.7, 2.1, 2.2, 2.4, 2.5, 2.6, 3.8
App.2
$\mathcal{S}\left[21\right]$	($5^{\left[12\right]}$,$0^{\left[28\right]}$)	28(40)	30(40)	0	27	0.8, 1.3, 1.9, 2.9, 3.2, 3.3, 4, 4.1, 4.2, 4.7, 4.9, 5, 5.5, 5.7, 6.2, 6.9, 7.1, 7.4, 7.6, 7.7,
						8, 8.6, 8.8, 8.9, 9.5 10.7 11.1 11.5 12.4, 12.5, 13.3, 14.1, 15.4, 18.1, 18.4, 18.9, 19.9, 21.4, 23, 27
$\mathcal{S}\left[22\right]$	($0^{\left[14\right]}$,$5^{\left[12\right]}$,$0^{\left[14\right]}$)	25(30)	27(30)	10	25	0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2, 3.3, 3.5, 4.1, 4.4, 4.6, 4.9, 5.5,
						6.2, 8.6, 9.5, 12.4, 12.5, 13.7, 17.3, 18.2, 20.6, 23
$\mathcal{S}\left[23\right]$	($5^{\left[6\right]}$,$0^{\left[28\right]}$,$5^{\left[6\right]}$)	8(11)	22(20)	50	21.4	0.8, 1.5, 2.7, 3.2, 4.2, 4.3, 4.7, 5.3, 6.1, 6.3, 7.6, 8.9, 9.7, 11, 12.4, 12.9, 13.3, 15.4, 19.9, 21.4
$\mathcal{S}\left[24\right]$	($0^{\left[28\right]}$,$5^{\left[12\right]}$)	2(7)	4(15)	85	4	0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2, 3.3, 3.5

. Following Algorithm 1, and assuming the absence of prior information about the parameter vector $\mathit{\xi }$, Bayesian inference for the NOT-Exp model parameters $\alpha$, $\gamma$, and $\theta$, along with the associated reliability measures $R\left(x\right)$ and $h\left(x\right)$ evaluated at $x=0.1$, is carried out using non-informative gamma priors for all parameters. The MCMC procedure is implemented with $\wp =40,000$ iterations, of which ${\wp }^{\star }=30,000$ are retained for posterior inference.

Table 7. Estimates of $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$ from App.1 (1st Row) and App.2 (2nd Row).

Sample	Par.	MLE		MCMC		ACI[Norm]			BCI			ACI[Log-Norm]			HPD
		Est.	St.E	Est.	St.E	Low.	Upp.	IW	Low.	Upp.	IW	Low.	Upp.	IW	Low.	Upp.	IW
$\mathcal{S}\left[11\right]$	$\alpha$	0.0016	0.0013	0.0016	0.0002	0.0010	0.0042	0.0052	0.0011	0.0023	0.0012	0.0003	0.0080	0.0077	0.0011	0.0022	0.0012
$\mathcal{S}\left[21\right]$		325.27	5.9316	325.27	0.0058	313.64	336.89	23.252	325.25	325.29	0.0394	313.85	337.10	23.257	325.25	325.29	0.0392
	$\gamma$	164.83	8.6903	164.83	0.0006	147.80	181.86	34.065	164.83	164.83	0.0040	148.65	182.77	34.126	164.83	164.83	0.0040
		3.1379	0.6418	3.1379	0.0058	1.8799	4.3958	2.5159	3.1182	3.1579	0.0397	2.1015	4.6854	2.5838	3.1182	3.1579	0.0397
	$\theta$	0.0434	0.0319	0.0434	0.0006	0.0091	0.1058	0.1250	0.0414	0.0453	0.0039	0.0103	0.1832	0.1729	0.0413	0.0452	0.0039
		0.1711	0.0272	0.1707	0.0049	0.1177	0.2245	0.1068	0.1543	0.1873	0.0330	0.1252	0.2337	0.1085	0.1543	0.1873	0.0330
	$R\left(x\right)$	0.9410	0.0323	0.9412	0.0176	0.8776	0.9944	0.1267	0.8674	0.9836	0.1163	0.8797	1.0065	0.1268	0.8811	0.9892	0.1081
		0.9426	0.0200	0.9427	0.0039	0.9034	0.9819	0.0785	0.9288	0.9553	0.9979	0.9042	0.9827	0.0785	0.9918	0.9771	0.9342
	$h\left(x\right)$	0.7001	0.0359	0.0886	0.0187	0.6298	0.7705	0.1407	0.0353	0.1602	0.1250	0.6332	0.7741	0.1409	0.0310	0.1529	0.1219
		0.0487	0.0116	0.9173	0.0013	0.0259	0.0715	0.0456	0.8650	0.8579	0.8506	0.0305	0.0777	0.0473	0.8128	0.7971	0.7891
$\mathcal{S}\left[12\right]$	$\alpha$	0.0006	0.0002	0.0006	0.0000	0.0003	0.0009	0.0006	0.0005	0.0008	0.0003	0.0004	0.0010	0.0007	0.0005	0.0008	0.0003
$\mathcal{S}\left[21\right]$		52.061	8.3900	52.061	0.0058	35.617	68.505	32.888	52.042	52.081	0.0392	37.961	71.399	33.438	52.041	52.080	0.0391
	$\gamma$	266.57	0.0002	266.57	0.0006	266.57	266.57	0.0010	266.57	266.57	0.0039	266.57	266.57	0.0010	266.57	266.57	0.0039
		1.3621	0.2793	1.3619	0.0058	0.8147	1.9095	1.0947	1.3426	1.3816	0.0390	0.9114	2.0358	1.1244	1.3426	1.3815	0.0390
	$\theta$	0.0440	0.0002	0.0440	0.0001	0.0435	0.0444	0.0009	0.0438	0.0442	0.0004	0.0435	0.0444	0.0009	0.0438	0.0442	0.0004
		0.0667	0.0184	0.0664	0.0041	0.0307	0.1027	0.0720	0.0528	0.0805	0.0277	0.0389	0.1145	0.0756	0.0526	0.0802	0.0276
	$R\left(x\right)$	0.8393	0.0744	0.8311	0.0239	0.6935	0.9852	0.2917	0.7384	0.8991	0.1607	0.7054	0.9986	0.2932	0.7490	0.9061	0.1571
		0.9002	0.0263	0.9006	0.0075	0.8487	0.9517	0.1030	0.8745	0.9252	0.0507	0.8501	0.9532	0.1031	0.8738	0.9245	0.0507
	$h\left(x\right)$	0.1822	0.0546	0.1865	0.0168	0.0753	0.2891	0.2138	0.1338	0.2475	0.1136	0.1013	0.3276	0.2263	0.1322	0.2453	0.1131
		0.0453	0.0086	0.0451	0.0035	0.0284	0.0622	0.0338	0.0337	0.0573	0.0236	0.0312	0.0658	0.0346	0.0339	0.0575	0.0235
$\mathcal{S}\left[13\right]$	$\alpha$	0.0055	0.0098	0.0055	0.0001	0.0038	0.0248	0.0386	0.0053	0.0057	0.0004	0.0002	0.1830	0.1828	0.0053	0.0057	0.0004
$\mathcal{S}\left[23\right]$		160.16	8.4237	160.16	0.0058	143.65	176.67	33.020	160.14	160.17	0.0393	144.47	177.55	33.079	160.14	160.18	0.0392
	$\gamma$	37.847	15.718	37.847	0.0006	7.0399	68.655	61.615	37.845	37.849	0.0039	16.770	85.417	68.648	37.845	37.849	0.0039
		2.7551	0.7683	2.7550	0.0057	1.2493	4.2610	3.0116	2.7358	2.7747	0.0390	1.5951	4.7589	3.1638	2.7356	2.7746	0.0389
	$\theta$	0.0321	0.0508	0.0321	0.0001	0.0067	0.1316	0.1990	0.0319	0.0323	0.0004	0.0015	0.7106	0.7091	0.0319	0.0323	0.0004
		0.1788	0.0410	0.1784	0.0054	0.0983	0.2592	0.1608	0.1603	0.1968	0.0366	0.1140	0.2803	0.1663	0.1597	0.1962	0.0365
	$R\left(x\right)$	0.9503	0.0399	0.9501	0.0015	0.8721	0.9985	0.1564	0.9447	0.9551	0.0104	0.8752	1.0318	0.1566	0.9448	0.9552	0.0104
		0.9104	0.0364	0.9105	0.0056	0.8391	0.9817	0.1427	0.8909	0.9290	0.0381	0.8418	0.9846	0.1428	0.8915	0.9296	0.0381
	$h\left(x\right)$	0.0780	0.0450	0.0781	0.0017	0.0010	0.1662	0.1764	0.0723	0.0842	0.0119	0.0251	0.2417	0.2166	0.0723	0.0841	0.0119
		0.0681	0.0196	0.0680	0.0041	0.0297	0.1066	0.0769	0.0546	0.0823	0.0277	0.0387	0.1198	0.0810	0.0542	0.0818	0.0276
$\mathcal{S}\left[14\right]$	$\alpha$	0.0567	0.0617	0.0567	0.0001	0.0003	0.1778	0.2421	0.0566	0.0569	0.0004	0.0067	0.4788	0.4721	0.0566	0.0569	0.0004
$\mathcal{S}\left[24\right]$		206.90	11.864	206.90	0.0058	183.65	230.15	46.506	206.88	206.92	0.0391	184.91	231.51	46.604	206.88	206.92	0.0391
	$\gamma$	233.46	24.902	233.46	0.0006	184.65	282.26	97.614	233.46	233.46	0.0039	189.41	287.74	98.327	233.46	233.46	0.0039
		6.8015	3.3175	6.8013	0.0057	0.2993	13.304	13.004	6.7821	6.8211	0.0391	2.6147	17.692	15.078	6.7818	6.8208	0.0390
	$\theta$	1.1753	0.4381	1.1753	0.0001	0.3167	2.0339	1.7172	1.1751	1.1755	0.0004	0.5661	2.4402	1.8741	1.1751	1.1755	0.0004
		1.1349	0.2609	1.1349	0.0058	0.6235	1.6463	1.0228	1.1153	1.1545	0.0392	0.7232	1.7810	1.0578	1.1155	1.1546	0.0391
	$R\left(x\right)$	0.7321	0.1177	0.7321	0.0003	0.5015	0.9627	0.4613	0.7309	0.7333	0.0023	0.5343	1.0032	0.4690	0.7309	0.7332	0.0023
		0.2044	0.0840	0.2045	0.0032	0.0397	0.3691	0.3293	0.1938	0.2156	0.0218	0.0913	0.4575	0.3661	0.1935	0.2153	0.0217
	$h\left(x\right)$	0.6244	0.2425	0.6244	0.0005	0.1491	1.0997	0.9506	0.6227	0.6261	0.0033	0.2916	1.3368	1.0451	0.6227	0.6261	0.0033
		1.0274	0.2708	1.0274	0.0071	0.4967	1.5581	1.0614	1.0034	1.0514	0.0480	0.6130	1.7222	1.1092	1.0041	1.0519	0.0479

reports the resulting interval estimates, including the 95% ACIs (normal and log-normal) and the corresponding 95% credible intervals (BCI and HPD), together with their associated interval widths (IWs), for $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$. The results indicate that the Bayesian MCMC-based estimators consistently outperform their frequentist counterparts, as reflected by smaller St.Es and narrower IWs. Moreover, when comparing classical asymptotic intervals with credible intervals in terms of minimum IW, both 95% credible intervals (BCI and HPD) generally provide more precise inference, further highlighting their practical advantage.

To assess the existence and uniqueness of the frequentist estimation results using the gathered samples $\mathsf{S}\left[i1\right]$ (for $i=1,2$) listed in Table 6, contour diagrams for the parameters $\alpha$, $\gamma$, and $\theta$ are displayed in Figure 9. The graphical evidence in Figure 9 is consistent with the numerical MLEs reported in Table 7, thereby supporting their validity.

Ensuring convergence of the Markov chains is a critical step in the implementation of the MCMC procedure. To this end, based on the same $\mathsf{S}\left[11\right]$ and $\mathsf{S}\left[21\right]$ samples, Figure 10 illustrates the corresponding trace diagnostics and posterior density estimates for all model parameters. For each parameter, including $\alpha$, $\gamma$, $\theta$, $R\left(x\right)$, and $h\left(x\right)$, the posterior mean is indicated by a solid blue line, while the limits of the 95% BCI are marked by dashed blue lines. The traces exhibit good mixing behavior and stable trajectories, providing clear evidence that the NH sampler has achieved satisfactory convergence. From App.2, the posterior densities of all NOT-Exp parameters are approximately symmetric. In contrast, from App.1, those of ($\alpha$,$h\left(x\right)$) and $R\left(x\right)$ are highly positive and negative skewed, respectively, and those of $\gamma$ and $\theta$ are approximately symmetric.

The visualization of interval limits for the NOT-Exp lifespan metrics $R\left(x\right)$ and $h\left(x\right)$ is crucial, as it plays a fundamental role in lifetime data analysis by providing time-dependent quantification of uncertainty beyond point estimation. In this context, comparing asymptotic and credible intervals enables a comprehensive assessment of inferential robustness across different methodological frameworks. Such comparisons are particularly informative for moderate sample sizes, where departures from asymptotic assumptions may affect inferential accuracy. Using $\mathcal{S}\left[i1\right]$ (for $i=1,2$), Figure 11 displays the $95\%$ interval bounds for $R\left(x\right)$ and $h\left(x\right)$ obtained using the ACI[Norm], ACI[Log-Norm], BCI, and HPD methods. The results show that the interval estimates for $R\left(x\right)$ exhibit smooth monotonic decay over time, while the corresponding bounds for $h\left(x\right)$ follow the characteristic pattern implied by the NOT-Exp model, confirming the internal coherence of the fitted model.

It is observed that the ACI[Log-Norm] intervals are generally tighter and more stable than their ACI[Norm] counterparts, highlighting the advantage of logarithmic transformation in accommodating skewness and enforcing positivity. Moreover, the Bayesian HPD intervals tend to be narrower than the BCI intervals, indicating more efficient uncertainty quantification, while remaining in close agreement with the frequentist intervals across the time domain.

7. Optimal Plans

Choosing an optimal progressive censoring pattern (OPCP) from a set of competing designs is an important issue in reliability and lifetime analysis. For fixed values of the total sample size n, the observed failure sizes $q_i,i=1,2$, and the two pre-specified thresholds ${\mathsf{ȷ}}_i,i=1,2$, a progressive censoring scheme $\mathbf{S}$ is said to be preferable to another scheme if it provides greater information about the unknown model parameters contained in the lifespan distribution under examination. This section introduces four accuracy-based criteria, denoted collectively by ${\psi }_i,i=1,2,3,4$, which are used to assess and compare alternative progressive censoring designs and to identify the most suitable pattern for application to the real data sets analyzed in the previous section. For comprehensive discussions on optimal schemes, the reader is referred to Ng et al. [34], Pradhan and Kundu [35], Sen et al. [36,37], Lin et al. [38], Nassar and Elshahhat [39], and the references cited therein. In the present study, four optimality criteria are employed and summarized in

Table 8. Four criteria for defining the OPCP.

Criterion	Goal
${\psi }_1$	$\mathrm{Max}trace\left(\mathbf{I}\left(\widehat{\xi }\right)\right)$
${\psi }_2$	$\mathrm{Min}trace\left({\mathbf{I}}^{-1}\left(\widehat{\xi }\right)\right)$
${\psi }_3$	$\mathrm{Min}det\left({\mathbf{I}}^{-1}\left(\widehat{\xi }\right)\right)$
${\psi }_4$	$\mathrm{Min}\left[\widehat{Var}(log\left({\widehat{Q}}_u\right))\right],0<u<1$

.

The criteria listed in Table 8 operate as follows: ${\psi }_1$ maximizes the trace of the observed FI matrix $\mathbf{I}\left(\widehat{\mathit{\xi }}\right)$, thereby favoring designs with higher overall information content; ${\psi }_2$ minimizes the trace of the IFI matrix ${\mathbf{I}}^{-1}\left(\widehat{\mathit{\xi }}\right)$, which is associated with minimizing the total variance of the parameter estimates; ${\psi }_3$ minimizes the determinant of the same IFI matrix, and ${\psi }_4$ minimizes the estimated variance of the log-MLE of the uth NOT-Exp quantile (say, $Q_u$), emphasizing precision in quantile estimation. We re-recommend utilizing the delta approach here to develop the variance of $Q_u$. All fitted results of ${\psi }_i,i=1,2,3,4$, using all UT2-PH censored datasets $\mathcal{S}\left[ij\right]$ from App.i (for $i=1,2$ and $j=1,2,3,4$) listed in Table 6, are provided in

Table 9. Fitted four O-T2-PC plans from App.i (for $i=1,2$).

Sample	${\mathit{\psi }}_1$	${\mathit{\psi }}_2$	${\mathit{\psi }}_3$	${\mathit{\psi }}_4$
$\mathit{u}\to$				0.3	0.6	0.9
App.1
$\mathcal{S}\left[11\right]$	1.03 × 107	7.55 × 101	7.49 × 10−9	6.50 × 10−1	2.91 × 100	7.40 × 101
$\mathcal{S}\left[12\right]$	3.21 × 108	1.38 × 10−7	1.06 × 10−23	5.03 × 10−1	2.06 × 100	1.27 × 101
$\mathcal{S}\left[13\right]$	4.63 × 105	2.47 × 102	1.42 × 10−6	1.48 × 100	8.00 × 100	3.70 × 102
$\mathcal{S}\left[14\right]$	2.53 × 103	6.20 × 102	4.79 × 10−2	6.78 × 10−2	1.31 × 10−1	5.73 × 10−1
App.2
$\mathcal{S}\left[21\right]$	4.11 × 103	7.05 × 101	3.53 × 10−3	3.83 × 10−1	1.14 × 100	5.33 × 100
$\mathcal{S}\left[22\right]$	1.01 × 104	3.56 × 101	5.48 × 10−4	1.40 × 100	8.82 × 100	6.96 × 101
$\mathcal{S}\left[23\right]$	1.67 × 103	7.16 × 101	2.49 × 10−2	6.58 × 10−1	2.01 × 100	9.71 × 100
$\mathcal{S}\left[24\right]$	5.26 × 101	1.52 × 102	2.96 × 101	4.45 × 10−2	8.06 × 10−2	2.85 × 10−1

. In this table, for clarity, 5.73E-01 means 5.73 $\times {10}^{-1}$.

To provide more specific information, the four optimality criteria $({\psi }_i,i=1,2,\dots ,4)$ serve as complementary measures for evaluating candidate censoring plans in reliability experiments. In general, criteria such as ${\psi }_1$ and ${\psi }_2$ emphasize estimation efficiency by reducing estimator variability, thereby yielding more precise parameter and reliability estimates. In contrast, ${\psi }_3$ and ${\psi }_4$ offer more balanced alternatives when practical constraints such as testing time, cost, or early termination of the experiment are important considerations. Consequently, the choice of an optimal censoring plan should depend on the primary objective of the study: plans optimizing ${\psi }_1$ or ${\psi }_2$ are preferable when precise statistical inference is the main goal, whereas ${\psi }_3$ or ${\psi }_4$ may be more suitable when experimental resources are limited and a compromise between estimation accuracy and experimental efficiency is required.

Based on the real-world datasets arising from toxicological and banking management applications in Table A6, denoted as App.i $(i=1,2)$, the fitted OPCP results summarized in Table 9 reveal that the middle T2-P scheme implemented in $\mathcal{S}\left[i2\right]$ and the right T2-P scheme implemented in $\mathcal{S}\left[i4\right]$ consistently outperform the remaining censoring plans. These results demonstrate superior inferential efficiency under the proposed optimality criteria and are in full agreement with the conclusions drawn from the Monte Carlo study in Section 5.

8. Conclusions, Recommendations, and Future Research

This study has developed a comprehensive reliability and inferential framework for the NOT-Exp distribution under the UT2-PH plan, addressing both methodological and applied challenges commonly encountered in modern lifetime data analysis. By integrating the flexibility of the NOT-Exp model with a unified censoring structure, the proposed approach successfully accommodates complex hazard rate behaviors and realistic data incompleteness within a single coherent framework. The derived likelihood-based and Bayesian inferential procedures provide reliable estimation of model parameters as well as key reliability measures, including the survival and hazard rate functions, under diverse censoring scenarios. From an inferential perspective, the combination of likelihood estimation, asymptotic confidence interval constructions, and Bayesian MCMC-based inference offers a balanced toolkit for practitioners. The comparative results demonstrate that Bayesian estimators, particularly those based on HPD intervals, exhibit superior stability and efficiency in heavily censored settings, while the log-normal asymptotic intervals effectively address positivity constraints inherent to reliability parameters. The extensive Monte Carlo investigation further confirms that estimator accuracy improves with increasing effective sample size and reduced censoring intensity, and that the proposed methodology remains robust across a wide range of censoring designs and prior specifications. The real-data applications to airborne toxicological variations and bank customer waiting times highlight the practical relevance of the proposed framework. These two datasets represent contrasting application domains with distinct data-generating mechanisms, yet both exhibit non-constant hazard structures and partial observability. The consistent superiority of the NOT-Exp model over twelve competing lifetime distributions across goodness-of-fit measures underscores its adaptability and modeling strength. Importantly, the reliability and hazard-based interpretations obtained from these applications provide actionable insights for environmental risk assessment and service system management, reinforcing the applied value of the proposed methods.

8.1. Recommendations

From the theoretical findings and empirical evidence, the following recommendations can be made:

• Based on the theoretical findings and empirical evidence, the following recommendations can be made:
• The NOT-Exp distribution is strongly recommended for lifetime data exhibiting complex hazard shapes that cannot be adequately captured by classical exponential-type models.
• The UT2-PH censoring scheme should be favored in practical life-testing experiments where multiple termination criteria and progressive withdrawals are present, as it provides flexibility without sacrificing inferential tractability.
• Bayesian MCMC-based inference is recommended in moderate-to-small samples or heavily censored settings, where classical asymptotic approximations may be less reliable.
• Practitioners in toxicology, environmental monitoring, and service operations should consider reliability-based modeling to better quantify persistence, risk, and system performance.

8.2. Future Directions

Several promising extensions naturally arise from this work, as follows:

• First, the proposed framework may be extended to other members of the odd-family or exponential-generated distributions under UT2-PH or adaptive censoring schemes, for example, odd Weibull, odd gamma, odd log-logistic, and odd Lomax, among others.
• Second, incorporating covariates through accelerated failure time or proportional hazard-type regressions based on the NOT-Exp baseline would significantly broaden its applicability.
• Third, future studies may explore optimal censoring design problems that aim to minimize estimator variance or experimental costs under the NOT-Exp model, such as a cost-minimization or a meta-heuristic algorithm.
• Fourth, multivariate and dependent lifetime extensions—such as shared frailty or copula-based NOT-Exp models—represent an important avenue for analyzing correlated failure data. Finally, machine learning–assisted Bayesian computation and approximate inference techniques could be integrated to further enhance computational efficiency in large-scale or high-dimensional reliability studies.
• In summary, this study provides a unified, flexible, and practically relevant contribution to censored lifetime modeling, offering both theoretical advancement and applied insight. The proposed NOT-Exp using the UT2-PH-based framework lays a solid foundation for future developments in reliability theory and its multidisciplinary applications.