Inference Based on Progressive-Stress Accelerated Life-Testing for Extended Distribution via the Marshall-Olkin Family Under Progressive Type-II Censoring with Optimality Techniques

Abstract

This paper explores a progressive-stress accelerated life test under progressive type-II censoring with binomial random removal. It assumes a cumulative exposure model in which the lifetimes of test units follow a Marshall–Olkin length-biased exponential distribution. The study derives maximum likelihood and Bayes estimates of the model parameters and constructs Bayes estimates of the unknown parameters under various loss functions. In addition, this study provides approximate, credible, and bootstrapping confidence intervals for the estimators. Moreover, it evaluates three optimal test methods to determine the most effective censoring approach based on various optimality criteria. A real-life dataset is analyzed to demonstrate the proposed procedures and simulation studies used to compare two different designs of the progressive-stress test.

1. Introduction

Advancements in science and technology have led to incredibly durable and complex products such as silicone seals, computers, and LEDs. Reliability is a cornerstone of product quality, so manufacturers invest heavily in testing, design, and other efforts to ensure dependable performance. Ideally, this would involve a wealth of life-testing data. However, a challenge arises with highly reliable products, as their extended lifespans often mean that very few or even zero failures occur within a reasonable testing period under normal operating conditions. The traditional maximum likelihood (ML) method faces challenges in this scenario.

Accelerated life testing (ALT) is employed to expedite failure detection. ALT applies higher-than-usual stress conditions (such as temperature and voltage) to products to induce earlier failures and assess their reliability. These data are then analyzed to estimate a product’s lifespan under normal use. There are different ways to apply stress in ALT: Constant stress maintains a fixed level, step-stress increases it gradually in discrete stages, and progressive stress continuously increases it over time. For a more comprehensive understanding of these accelerated models, refer to Nelson’s [1] study. Using ALT, manufacturers can gather critical failure data more quickly, even for highly reliable products. This approach improves product design, improves testing efficiency, and reduces costs. See Figure 1 to view the difference between the possible types of ALT, where ($S_1$, $S_2$, and $S_3$) or (A, B, and C) are stress levels. Additionally, NOC (normal operating conditions) are highlighted to indicate baseline performance under standard conditions. For more information, see [2].

Progressive-stress ALT adopts a different approach compared to constant and step-stress methods. Here, the stress level in each unit constantly increases over time. A specific example is a ramp stress test, where stress increases linearly. Various researchers have explored this approach, with studies focusing on different aspects. For instance, Yin and Sheng [3] investigated finding the maximum likelihood estimates (MLEs) for an exponential progressive-stress model, while Bai and Cha [4] explored optimal test designs and failure rate models under progressive stress conditions. AL-Hussaini et al. [5] discussed one-sample Bayesian prediction intervals based on progressively type-II censored data from the half-logistic distribution under the progressive stress model.

Figure 1 discussed (1) constant stress levels ($S_1,S_2,S_3$), (2) step-stress ($S_1,S_2,S_3$), and (3) progressive-stress (A, B, C). Types of accelerated life testing, illustrating constant stress, step-stress, and progressive-stress methods with different stress levels. For more information see [2].

In life testing, experiments often end before all units fail. These “censored data” help to reduce testing time and costs. Two common censoring methods are type-I and type-II censoring. Recently, progressive type-II censoring has gained popularity for analyzing data from highly reliable products. The process works as follows: Suppose n identical items are being tested. The experimenter predefines several failures (m, where m is less than n) that are expected to occur. The experimenter also predetermines how many surviving units to remove $(R_1,R_2,\cdots ,R_m)$ at each failure point. These removal numbers add up to the total number of surviving units that the experimenter will keep at the end $(n-m)$. The test ends after the m^th failure, with any remaining units being withdrawn. For more information on progressive type-II censoring, refer to Balakrishnan and Aggarwala [6] or Balakrishnan and Cramer [7].

The research on progressive-stress ALT extends beyond basic models based on censored samples. Rong-hua and Heliang [8] explored its application with the Weibull distribution and a tampered failure rate model. Other studies used progressive stress with various distribution models and statistical techniques. For instance, Abdel-Hamid and Al-Hussaini [9] examined progressive stress combined with finite mixture distributions, progressive censoring, and Bayesian analysis for different underlying lifetime distributions. This research demonstrated the versatility of progressive-stress ALT and its potential for analyzing complex failure data. Abdel-Hamid and Al-Hussaini [10] also studied inference for a progressive stress model based on Weibull distribution under progressive type-II censoring. Recently, studies have used progressive-stress ALT based on censoring schemes [11,12,13,14,15,16].

This research presents a novel methodology for testing the reliability of highly dependable products by integrating progressive-stress ALT with the Marshall–Olkin length-biased exponential (MOLBE) distribution under progressive type-II censoring. The MOLBE distribution offers greater flexibility in modeling hazard rates compared to other distributions, making it more accurate for analyzing censored data, especially for products with low failure rates. This study explores statistical estimation methods and optimal confidence interval analysis while providing simulation studies to assess the effectiveness of the proposed approach.

This study is structured to guide you through a new life testing approach for highly reliable products. Section 2 describes the model and testing assumptions. Section 3 and Section 4 estimate the model’s parameters using both MLEs and a modern Bayesian approach with Markov Chain Monte Carlo (MCMC). Section 5 constructs reliable confidence intervals for the parameters. Optimal censoring schemes for the progressive type-II censoring model are obtained in Section 6. To demonstrate the method’s application, Section 7 analyzes a real-world dataset. Section 8 reinforces the approach’s effectiveness through simulations, and Section 9 concludes this study by providing key takeaways.

2. Model Description and Test Assumption

In this section, an overview of the model under consideration is provided. Also, the ramp stress ALT model is constructed according to the assumptions outlined in the second subsection.

2.1. Marshall–Olkin Length-Biased Exponential Distribution

This study discusses a recently proposed lifetime distribution called the MOLBE distribution, which builds upon the length-biased exponential distribution and Marshall–Olkin family. It generalizes the concept of the length-biased exponential distribution. The mathematical details of this distribution are provided by ul Haq et al. [17], including its cumulative distribution function (CDF), probability density function (PDF), and hazard rate function (HRF). These functions describe the probability of failure within a certain time, the likelihood of failure at a specific time, and the instantaneous risk of failure over time, respectively.

$$\left\{\begin{array}{ccc}\hfill G(x;\alpha ,\beta )& ={\displaystyle \frac{1-\left(1+x\beta \right)e^{-\beta x}}{1-(1-\alpha )\left(1+x\beta \right)e^{-x\beta }}},\hfill & \\ \hfill g(x;\alpha ,\beta )& ={\displaystyle \frac{\alpha {\beta }^{2}x{e}^{-\beta x}}{{\left[1-(1-\alpha )\left(1+x\beta \right)e^{-x\beta }\right]}^2}},\hfill & x,\alpha ,\beta >0\\ \hfill h(x;\alpha ,\beta )& ={\displaystyle \frac{{\beta }^{2}x}{\left(1+x\beta \right)\left[1-(1-\alpha )\left(1+x\beta \right)e^{-x\beta }\right]}}.\hfill \end{array}\right.$$

(1)

A recent study by ul Haq et al. [17] explored the MOLBE distribution as a potential alternative to several existing models for analyzing lifetime data. These existing models included the standard exponential distribution and extensions of exponential, Marshall–Olkin extended exponential, moment exponential, and exponentiated moment exponential distributions. The implication is that MOLBE might offer advantages over these established models for accurately capturing the patterns observed in real-world lifetime data.

2.2. Progressive Stress ALT (PSALT) Structure and Assumptions

This subsection focuses on the underlying assumptions and the mathematical model for designing PSALT experiments. The following provides a breakdown of the key assumptions:

• Lifetime Distribution: The lifetime of each object being tested is assumed to follow a MOLBE distribution with specific parameters $(\alpha ,\beta )$. This distribution describes the probability of a unit failing at a certain time.
• Progressive Stress Levels: The stress applied to the units increases gradually over time according to a predefined law. This law relates the stress level (denoted by ${\mathsf{\Phi }}_i\left(t\right)$) at a specific stage (i) to the corresponding time (t). The time stages $(u_1,u_2,\cdots ,u_k)$ are ordered with increasing values, and each stage represents a specific stress level (which could be voltage, temperature, or another relevant factor).
• Stress and Lifetime Relationship: The relationship between the applied stress $\left({\mathsf{\Phi }}_i\left(t\right)\right)$ and the characteristic lifetime $(h_i={\displaystyle \frac{u_i}{u_{i-1}}})$ of the units at each stress level follows an inverse power law. This law describes how the lifetime of the units is expected to decrease as the stress level increases as follows:

  $${\mathsf{\Phi }}_i\left(t\right)={\displaystyle \frac{1}{{\delta }_0{\left[V_i\left(t\right)\right]}^{\eta }}};{\delta }_0>0,\eta >0,$$

  where ${\delta }_0$ and $\eta$ represent unidentified parameters; please refer to the experimental section regarding acceleration for further information on these methods (additional details on these acceleration techniques can be found in Chapter 2 of [1]).
• Typically, the lifespan of a unit follows a MOLBE distribution when operating under normal conditions. Additionally, the progressive stress, denoted as $V\left(t\right)$, increases linearly with time at a constant rate h, expressed as $V_i\left(t\right)=h_{i}t$, where V is greater than zero; $0<h_1<h_2<\cdots <h_k$.
• The k items tested are divided into $(k\ge 2)$ groups for the testing procedure.
• The effect of stress variation across different levels is modeled using a linear cumulative exposure model. For further elaboration, please refer to Nelson [1].

Based on the assumption of the linear cumulative exposure model, a test unit’s CDF under progressive stress $V_u\left(t\right)$ can be expressed as

$$F_u(t;\alpha ,\beta )=G_u(\Delta t;\alpha ,\beta ),u=1,\cdots ,k,$$

(2)

where

$$\Delta t={\int }_0^t{\displaystyle \frac{1}{{\mathsf{\Phi }}_i\left(x\right)}}dx={\displaystyle \frac{{\delta }_0{h}_i^{\eta }{t}^{(\eta +1)}}{(\eta +1)}},$$

(3)

and $G_u(.)$ is the CDF of the MOLBE distribution under progressive stress $V_u\left(t\right)$ with the scale parameter $\beta$ taken as 1. Therefore, the CDF, PDF, and HRF of the MOLBE distribution are based on PSALT as follows:

$$\left\{\begin{array}{ccc}\hfill F_i(t;\alpha ,{\delta }_0,\eta )& ={\displaystyle \frac{1-\left(1+{\delta }_i{t}^{(\eta +1)}\right)e^{-{\delta }_i{t}^{(\eta +1)}}}{1-(1-\alpha )\left(1+{\delta }_i{t}^{(\eta +1)}\right)e^{-{\delta }_i{t}^{(\eta +1)}}}},\hfill & \\ \hfill f_i(t;\alpha ,{\delta }_0,\eta )& ={\displaystyle \frac{\alpha {\delta }_i^2{t}^{(\eta +1)}{e}^{-{\delta }_i{t}^{(\eta +1)}}}{{\left[1-(1-\alpha )\left(1+{\delta }_i{t}^{(\eta +1)}\right)e^{-{\delta }_i{t}^{(\eta +1)}}\right]}^2}},\hfill & t,\alpha ,{\delta }_0,\eta >0\\ \hfill h_i(t;\alpha ,{\delta }_0,\eta )& ={\displaystyle \frac{{\delta }_i^2{t}^{(\eta +1)}}{\left(1+{\delta }_i{t}^{(\eta +1)}\right)\left[1-(1-\alpha )\left(1+{\delta }_i{t}^{(\eta +1)}\right)e^{-{\delta }_i{t}^{(\eta +1)}}\right]}},\hfill \end{array}\right.$$

(4)

where ${\delta }_i={\displaystyle \frac{{\delta }_0{h}_i^{\eta }}{\eta +1}}$.

2.3. Progressive Type-II Censored

Various censoring schemes, including common type-I and type-II censoring, have been extensively discussed in the literature; see Kundu and Pradhan [18]. Recently, progressive censoring schemes have gained attention for their efficient resource utilization. Progressive type-II censoring extends type-II censoring by placing n units on a life test and observing only m failures. When a failure occurs, a predetermined number of surviving units are randomly removed. This process continues until the m-th failure, at which point all remaining units are removed. The censoring numbers $R_i$ are before the experiment. The sample is denoted by $t_{1:m:n},t_{2:m:n},\dots ,t_{m:m:n}$. For more information, see Balakrishnan and Aggarwala [6]. The likelihood function for progressive type-II is as follows:

$$L\left(\mathsf{\Omega }\right)=C\prod _{i=1}^{m}f(t_{(i:m:n)};\mathsf{\Omega }){\left[1-F(t_{(i:m:n)};\mathsf{\Omega })\right]}^{R_i},$$

(5)

where $\mathsf{\Omega }$ is a vector of parameters, and C is a fixed value and does not depend on $\mathsf{\Omega }$. For more information, see [19,20,21,22,23,24].

In some reliability studies, the number of patients dropping out is random. A progressive censoring scheme with random removals is needed, where each unit has the same removal probability p. The number of units withdrawn at each failure follows a binomial distribution: $R_1\sim \mathrm{binomial}(n-m,p)$, $R_j\sim binomial(n-m-{\sum }_{l=1}^{m-1}{R}_j,P);j=2,\cdots ,m-1$, and $R_m=n-m-{\sum }_{j=1}^{m-1}{R}_j$. For more details, see Tse et al. [25].

3. Maximum Likelihood Estimation

Consider n to be the total number of units under test, and let $V_1\left(t\right)<V_2\left(t\right)<\dots <V_k\left(t\right)$ represent the different stress levels applied during the test, with $V_1\left(t\right)$ being the use stress. At each increasing stress level $V_i\left(t\right)=h_{i}t$ for $i=1,2,\dots ,k$, $n_i$ identical units are tested. The progressive type-II censoring scheme is conducted as follows:

At the time of the first failure $t_{i1:m_i:n_i}$, $R_{i1}$ units are randomly withdrawn from the remaining $n_i-1$ surviving units. At the time of the second failure $t_{i2:m_i:n_i}$, $R_{i2}$ units are randomly withdrawn from the remaining $n_i-2-R_{i1}$ units. The test concludes at the time of the $m_i$-th failure $t_{i{m}_i:m_i:n_i}$; at this point, all remaining $R_{i{m}_i}=n_i-m_i-{\sum }_{j=1}^{m_i-1}{R}_{ij}$ units are withdrawn. Evidently, complete samples and type-II censored samples are specific cases of this scheme. Using this notation, the observed progressively censored data under the progressive-stress $V_i\left(t\right)$ are $t_{i1:m_i:n_i}<t_{i2:m_i:n_i}<\dots <t_{i{m}_i:m_i:n_i}$ for $i=1,2,\dots ,k$.

This section discusses the MLEs of the parameters $\alpha ,\eta$, and ${\delta }_0$ under progressive-stress $V_i\left(t\right)$ ALT when the data are progressive type-II censoring. The likelihood function of $\mathsf{\Omega }=(\alpha ,\eta ,{\delta }_0)$ can be expressed as follows:

$$\begin{array}{cc}\hfill L\left(\mathsf{\Omega }\right)& =\prod _{i=1}^k{C}_i\prod _{j=1}^{m_i}{\displaystyle \frac{{\alpha }^{R_{ij}+1}{\delta }_i^2{t}_{ij}^{(\eta +1)}{\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)}^{R_{ij}}{e}^{-{\delta }_i(R_{ij}+1)t_{ij}^{(\eta +1)}}}{{\left[1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}\right]}^{Rij+2}}}.\hfill \end{array}$$

(6)

The log-likelihood function can be written as

$$\begin{array}{cc}\hfill \mathsf{\ell }\left(\mathsf{\Omega }\right)& \propto 2\sum _{i=1}^k{m}_{i}ln{\delta }_i+ln\alpha \sum _{i=1}^k\sum _{j=1}^{m_i}(R_{ij}+1)+(\eta +1)\sum _{i=1}^k\sum _{j=1}^{m_i}ln{t}_{ij}-\sum _{i=1}^k{\delta }_i\sum _{j=1}^{m_i}(1+R_{ij})t_{ij}^{\eta +1}+\hfill \\ & \sum _{i=1}^k\sum _{j=1}^{m_i}{R}_{ij}ln\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)-\sum _{i=1}^k\sum _{j=1}^{m_i}(R_{ij}+2)ln\left[1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}\right].\hfill \end{array}$$

(7)

The log-likelihood equations for the parameters $\alpha$, $\eta$, and ${\delta }_0$ are, respectively, given by

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathsf{\partial }\mathsf{\ell }\left(\mathsf{\Omega }\right)}{\mathsf{\partial }\alpha }}& ={\displaystyle \frac{1}{\alpha }}\sum _{i=1}^k\sum _{j=1}^{m_i}(R_{ij}+1)-\sum _{i=1}^k\sum _{j=1}^{m_i}(R_{ij}+2){\displaystyle \frac{\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}{1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathsf{\partial }\mathsf{\ell }\left(\mathsf{\Omega }\right)}{\mathsf{\partial }\eta }}& ={\delta }_0\sum _{i=1}^k{\displaystyle \frac{2m_i}{{\delta }_i}}{\delta }_{i\left(\eta \right)}^{\prime }+\sum _{i=1}^k\sum _{j=1}^{m_i}ln{t}_{ij}-{\delta }_0\sum _{i=1}^k{\delta }_{i\left(\eta \right)}^{\prime }\sum _{j=1}^{m_i}(1+R_{ij})t_{ij}^{\eta +1}-\sum _{i=1}^k{\delta }_i\sum _{j=1}^{m_i}(1+R_{ij})t_{ij}^{\eta +1}ln{t}_{ij}+\hfill \\ & \sum _{i=1}^k\sum _{j=1}^{m_i}{R}_{ij}{\displaystyle \frac{{\delta }_i{t}_{ij}^{(\eta +1)}ln{t}_{ij}+{\delta }_0{\delta }_{i\left(\eta \right)}^{\prime }{t}_{ij}^{(\eta +1)}}{1+{\delta }_i{t}_{ij}^{(\eta +1)}}}-(1-\alpha )\sum _{i=1}^k\sum _{j=1}^{m_i}(R_{ij}+2){\displaystyle \frac{\left({\delta }_0{\delta }_{i\left(\eta \right)}^{\prime }+{\delta }_{i}ln{t}_{ij}\right){\delta }_i{t}_{ij}^{2(\eta +1)}{e}^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}{1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathsf{\partial }\mathsf{\ell }\left(\mathsf{\Omega }\right)}{\mathsf{\partial }{\delta }_0}}& =\sum _{i=1}^k{\displaystyle \frac{2m_i}{{\delta }_0}}-\sum _{i=1}^k{\displaystyle \frac{h_i^{\eta }}{\eta +1}}\sum _{j=1}^{m_i}(1+R_{ij})t_{ij}^{\eta +1}+{\displaystyle \frac{1}{\eta +1}}\sum _{i=1}^k\sum _{j=1}^{m_i}{R}_{ij}{\displaystyle \frac{h_i^{\eta }{t}_{ij}^{(\eta +1)}}{1+{\delta }_i{t}_{ij}^{(\eta +1)}}}-\hfill \\ & (1-\alpha )\sum _{i=1}^k{\displaystyle \frac{h_i^{\eta }}{\eta +1}}\sum _{j=1}^{m_i}(R_{ij}+2){\displaystyle \frac{{\delta }_i{t}_{ij}^{2(\eta +1)}{e}^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}{1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}},\hfill \end{array}$$

(10)

where ${\delta }_{i\left(\eta \right)}^{\prime }={\displaystyle \frac{\mathsf{\partial }{\delta }_i}{\mathsf{\partial }\eta }}={\displaystyle \frac{h_i^{\eta }(\eta +1)ln{h}_i-h_i^{\eta }}{{(\eta +1)}^2}}$.

As the likelihood equations for $\alpha$, $\eta$, and ${\delta }_0$ are nonlinear and challenging to solve analytically, numerical methods such as the Newton–Raphson method can be employed to find the MLEs of the distribution parameters $\alpha$, $\eta$, and ${\delta }_0$, denoted as $\widehat{\alpha }$, $\widehat{\eta }$, and $\widehat{{\delta }_0}$, respectively. In this study, the Newton–Raphson method was implemented using the maxlike function from the Maxlike library in R.

4. Bayesian Analysis

In this section, Bayesian analysis is explored using various prior and posterior distributions of the MOLBE distribution with PSALT based on the PTII censored sample.

4.1. Prior and Posterior Distribution

To carry out the Bayesian analysis, prior distributions for each unknown parameter need to be considered. To ensure that the Bayesian analysis is comparable with the likelihood-based analysis from Section 4, we assume that the three parameters $\alpha$, $\eta$, and ${\delta }_0$ follow gamma prior distributions. Thus, the following is true:

$$\begin{array}{cc}\hfill \pi \left(\alpha \right)& \propto {\alpha }^{a_1-1}{e}^{-{\displaystyle \frac{\alpha }{b1}}},\alpha >0,a_1,b_1>0,\hfill \\ \hfill \pi \left(\eta \right)& \propto {\eta }^{a_2-1}{e}^{-{\displaystyle \frac{\eta }{b_2}}},\eta >0,a_2,b_2>0,\hfill \\ \hfill \pi \left({\delta }_0\right)& \propto {\delta }_0^{a_3-1}{e}^{-{\displaystyle \frac{{\delta }_0}{b_3}}},{\delta }_0>0,a_3,b_3>0.\hfill \end{array}$$

Assuming that the parameters $\alpha ,\eta$, and ${\delta }_0$ are independent, the joint prior PDF is given by

$$\begin{array}{cc}\hfill \pi (\alpha ,\eta ,{\delta }_0)& \propto {\alpha }^{a_1-1}{\eta }^{a_2-1}{\delta }_0^{a_3-1}{e}^{-({\displaystyle \frac{\alpha }{b_1}}+{\displaystyle \frac{\eta }{b_2}}+{\displaystyle \frac{{\delta }_0}{b_3}})},\alpha >0,\eta >0,{\delta }_0>0.\hfill \end{array}$$

(11)

Together with the likelihood function in (11), by using the Bayesian theorem, the joint posterior density of $\alpha ,\eta$, and ${\delta }_0$ can be written as follows:

$$\begin{array}{cc}\hfill \pi \left(\mathsf{\Omega }\right|\mathit{t})& \propto L\left(\mathsf{\Omega }\right)\pi (\alpha ,\eta ,{\delta }_0)\hfill \\ \hfill & \propto \prod _{i=1}^k\prod _{j=1}^{m_i}{\displaystyle \frac{\alpha {\delta }_i^2{t}_{ij}^{(\eta +1)}{e}^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}{{\left[1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}\right]}^2}}{\left[{\displaystyle \frac{\alpha \left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}{1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}}\right]}^{R_{ij}}\times \hfill \\ \hfill & {\alpha }^{a_1-1}{\eta }^{a_2-1}{\delta }_0^{a_3-1}{e}^{-({\displaystyle \frac{\alpha }{b_1}}+{\displaystyle \frac{\eta }{b_2}}+{\displaystyle \frac{{\delta }_0}{b_3}})}.\hfill \end{array}$$

(12)

4.2. Symmetric and Asymmetric Loss Functions

This study investigates two distinct loss functions used in Bayesian estimation. The first one is the squared error loss function (SELF), which treats overestimation and underestimation equally, making it symmetric in nature when estimating parameters. The second option is the linear exponential loss function (LLF), which is asymmetric and assigns different weights to overestimation and underestimation.

The Bayes estimate of the function of parameters $D=D\left(\mathsf{\Omega }\right)$ based on SELF is expressed as follows:

$$\begin{array}{c}\hfill {\tilde{D}}_{\mathrm{SELF}}={\int }_{\mathsf{\Omega }}D\pi \left(\mathsf{\Omega }\right)d\mathsf{\Omega }.\end{array}$$

(13)

The Bayes estimate under LLF of D is given by

$$\begin{array}{c}\hfill {\tilde{D}}_{\mathrm{LLF}}={\displaystyle \frac{-1}{\nu }}ln\left[{\int }_{\mathsf{\Omega }}{e}^{-\nu D}\pi \left(\mathsf{\Omega }\right)d\mathsf{\Omega }\right],\end{array}$$

(14)

where $\nu \ne 0$ is the shape parameter of the LLF.

It is important to highlight that the Bayes estimates of D in Equations (13) and (14) are similar to a ratio of three multiple integrals, which cannot be simplified analytically. Therefore, it is advisable to use an approximation technique to compute these estimates, as outlined in the following subsection.

4.3. MCMC Method

In this context, the MCMC method is used to generate samples from the posterior distribution and subsequently calculate the Bayes estimates of D under ramp stress ALT. The conditional posterior distributions of $\alpha$, $\eta$, and ${\delta }_0$ are obtained from the joint posterior density function given in Equation (12) as follows:

$$\begin{array}{cc}\hfill \pi \left(\alpha \right|\eta ,{\delta }_0,t)& \propto {\alpha }^{a_1-1}{e}^{-{\displaystyle \frac{\alpha }{b_1}}}\prod _{i=1}^k\prod _{j=1}^{m_i}{\displaystyle \frac{{\alpha }^{R_{ij}+1}}{{\left[1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}\right]}^{R_{ij}+2}}},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill \pi \left(\eta \right|\alpha ,{\delta }_0,\mathit{t})& \propto {\eta }^{a_2-1}{e}^{-{\displaystyle \frac{\eta }{b_2}}}\prod _{i=1}^k\prod _{j=1}^{m_i}{\displaystyle \frac{{\left(\frac{h_i^{\eta }}{\eta +1}\right)}^2{t}_{ij}^{(\eta +1)}{e}^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}{{\left[1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}\right]}^2}}{\left[{\displaystyle \frac{\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}{1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}}}\right]}^{R_{ij}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \pi \left({\delta }_0\right|\alpha ,\eta ,\mathit{t})& \propto {\delta }_0^{a_3-1}{e}^{-{\displaystyle \frac{{\delta }_0}{b_3}}}\prod _{i=1}^k\prod _{j=1}^{m_i}{\displaystyle \frac{{\delta }_0^2{\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)}^{R_{ij}}{e}^{-{\delta }_i(1+R_{ij})t_{ij}^{(\eta +1)}}}{{\left[1-(1-\alpha )\left(1+{\delta }_i{t}_{ij}^{(\eta +1)}\right)e^{-{\delta }_i{t}_{ij}^{(\eta +1)}}\right]}^{R_{ij}+2}}}.\hfill \end{array}$$

(17)

Since the conditional posterior distributions of the parameters $\alpha ,\eta$, and ${\delta }_0$ cannot be simplified into well-known distribution forms, we employ the Metropolis–Hastings algorithm, as referenced in Upadhyay and Gupta [26]. To calculate the Bayes estimates of $D=D(\alpha ,\eta ,{\delta }_0)$ under SELF and LLF, the following procedure is used:

• Set initial values for $\alpha$, $\eta$, and ${\delta }_0$, for instance, ${\alpha }_0=\widehat{\alpha }$, ${\eta }_0=\widehat{\eta }$, and ${\delta }_{0_0}={\widehat{\delta }}_0$.
• Set $j=1$.
• Generate proposed values $\tilde{\alpha }\sim N(\widehat{\alpha },{\sigma }_{11})$, $\tilde{\eta }\sim N(\widehat{\eta },{\sigma }_{22})$, and ${\tilde{\delta }}_0\sim N({\widehat{\delta }}_0,{\sigma }_{33})$, where $\sigma$ represents the variance–covariance matrix.
• Calculate

  $$\begin{array}{c}\hfill T_j={\displaystyle \frac{\pi ({\alpha }_j,{\eta }_j,{\delta }_{0_j}|t)}{\pi ({\alpha }_{j-1},{\eta }_{j-1},{\delta }_{0_{j-1}}|t)}}.\end{array}$$
• Accept $({\alpha }_j,{\eta }_j,{\delta }_{0_j})$ with a probability of $min(1,T_j)$.
• Repeat steps (3) to (5) $\mathbb{I}$ times to obtain $\mathbb{I}$ samples for the parameters $(\alpha ,\eta ,{\delta }_0)$.
• Calculate the Bayes estimates of $\alpha ,\eta$, and ${\delta }_0$ under SELF and LLF using Equations (13) and (14) as follows:

$${\tilde{\alpha }}_{\mathrm{SELF}}={\displaystyle \frac{1}{\acute{\mathbb{I}}}}\sum _{j=1}^{\acute{\mathbb{I}}}{\alpha }_j,{\tilde{\eta }}_{\mathrm{SELF}}={\displaystyle \frac{1}{\acute{\mathbb{I}}}}\sum _{j=1}^{\acute{\mathbb{I}}}{\eta }_j,{\widetilde{{\delta }_0}}_{\mathrm{SELF}}={\displaystyle \frac{1}{\acute{\mathbb{I}}}}\sum _{j=1}^{\acute{\mathbb{I}}}{\delta }_{0_j}.$$

(18)

$${\tilde{\alpha }}_{\mathrm{LLF}}={\displaystyle \frac{-1}{\nu }}ln\left[{\displaystyle \frac{1}{\acute{\mathbb{I}}}}\sum _{j=1}^{\acute{\mathbb{I}}}{e}^{-\nu {\alpha }_j}\right],{\tilde{\eta }}_{\mathrm{LLF}}={\displaystyle \frac{-1}{\nu }}ln\left[{\displaystyle \frac{1}{\acute{\mathbb{I}}}}\sum _{j=1}^{\acute{\mathbb{I}}}{e}^{-\nu {\eta }_j}\right],{\widetilde{{\delta }_0}}_{\mathrm{LLF}}={\displaystyle \frac{-1}{\nu }}ln\left[{\displaystyle \frac{1}{\acute{\mathbb{I}}}}\sum _{j=1}^{\acute{\mathbb{I}}}{e}^{-\nu {\delta }_{0_j}}\right].$$

(19)

where $\acute{\mathbb{I}}=\mathbb{I}-I_0$, and $I_0$ is also known as a burn-in sample.

5. Interval Estimation

In this section, we construct approximate confidence intervals (ACIs) using the asymptotic normality of MLEs, as well as BCI- and HPD-credible intervals for the parameters $\alpha$, $\eta$, and ${\delta }_0$.

5.1. Approximate CI

As closed-form solutions for the MLEs of the unknown parameters are not available, it is challenging to determine their exact distributions. Consequently, exact confidence intervals (CIs) for the parameters cannot be computed. Therefore, ACIs for $\alpha$, $\eta$, and ${\delta }_0$ are developed using large sample approximations.

ACIs for $\mathsf{\Omega }=(\alpha ,\eta ,{\delta }_0)$ can be computed using the observed Fisher information matrix (FIM), leveraging the asymptotic normality properties of the MLEs. The observed information matrix, derived from the unknown parameters of the log-likelihood function, is used to estimate the asymptotic variance–covariance of the MLEs.

$$\begin{array}{c}\hfill I\left(\widehat{\mathsf{\Omega }}\right)=-{\left({\displaystyle \frac{{\mathsf{\partial }}^2\mathsf{\ell }\left(\mathsf{\Omega }\right)}{\mathsf{\partial }{\mathsf{\Omega }}_i\mathsf{\partial }{\mathsf{\Omega }}_j}}\right)}_{\langle \widehat{\mathsf{\Omega }}\rangle }.\end{array}$$

(20)

Next, the approximate asymptotic variance–covariance matrix is established as

$$\begin{array}{c}\hfill Cov\left(\widehat{\mathsf{\Omega }}\right)=I^{-1}\left(\widehat{\mathsf{\Omega }}\right).\end{array}$$

(21)

The MLEs of parameters approximately follow the multivariate normal distribution with mean $\mathsf{\Omega }$ and variance–covariance matrix $Cov\left(\widehat{\mathsf{\Omega }}\right)$, namely $\widehat{\mathsf{\Omega }}\sim N(\mathsf{\Omega },Cov\left(\widehat{\mathsf{\Omega }}\right))$. Therefore, for arbitrary $0<ϵ<1$, the $100(1-ϵ)\%$ ACIs of the unknown parameters can be expressed as follows:

$$\begin{array}{c}\hfill \left(\widehat{\mathsf{\Omega }}-z_{{\displaystyle \frac{ϵ}{2}}}\sqrt{Var\left(\widehat{\mathsf{\Omega }}\right)},\widehat{\mathsf{\Omega }}+z_{{\displaystyle \frac{ϵ}{2}}}\sqrt{Var\left(\widehat{\mathsf{\Omega }}\right)}\right),\end{array}$$

(22)

where $z_{{\displaystyle \frac{ϵ}{2}}}$ is the $(1-ϵ)$ quantile of the standard normal distribution $N(0,1)$.

5.2. Bootstrap Confidence Intervals

The bootstrap is a resampling technique used for statistical inference. It is the likelihood applied in the estimation of CIs; see Efron [27] for more details. In this section, we construct confidence intervals for the unknown parameters $\alpha ,\eta ,$ and ${\delta }_0$ using the parametric bootstrap approach. Specifically, we consider the percentile bootstrap (PB) and bootstrap-t (BT) methods for constructing confidence intervals. For further details on the bootstrap technique, refer to [28,29,30,31].

5.2.1. Percentile Bootstrap CI

• Compute the MLEs of ALT for parameters of TCPE distribution;
• Generate bootstrap samples using $\alpha ,\eta$, and ${\delta }_0$ to obtain the bootstrap estimate of $\alpha$ say ${\alpha }^b$, $\eta$ say ${\eta }^b$, and ${\delta }_0$ say ${\delta }_0^b$ using the bootstrap sample;
• Repeat step (2) $\mathcal{B}$ times to obtain $({\alpha }^{b\left(1\right)},{\alpha }^{b\left(2\right)},\dots {\alpha }^{b\left(\mathcal{B}\right)}),({\eta }^{b\left(1\right)},{\eta }^{b\left(2\right)},\dots ,{\eta }^{b\left(\mathcal{B}\right)})$ and $({\delta }_0^{b\left(1\right)},{\delta }_0^{b\left(2\right)},\dots ,{\delta }_0^{b\left(\mathcal{B}\right)})$;
• Arrange $({\alpha }^{b\left(1\right)},{\alpha }^{b\left(2\right)},\dots {\alpha }^{b\left(\mathcal{B}\right)}),({\eta }^{b\left(1\right)},{\eta }^{b\left(2\right)},\dots ,{\eta }^{b\left(\mathcal{B}\right)})$ and $({\delta }_0^{b\left(1\right)},{\delta }_0^{b\left(2\right)},\dots ,{\delta }_0^{b\left(\mathcal{B}\right)})$ in ascending order as $({\alpha }^{b\left[1\right]},{\alpha }^{b\left[2\right]},\dots {\alpha }^{b\left[\mathcal{B}\right]}),({\eta }^{b\left[1\right]},{\eta }^{b\left[2\right]},\dots ,{\eta }^{b\left(\right[\mathcal{B}\left]\right)})$, and $({\delta }_0^{b\left[1\right]},{\delta }_0^{b\left[2\right]},\dots ,{\delta }_0^{b\left[\mathcal{B}\right]})$;
• Two-sided $100(1-ϵ)\%$ BP confidence interval for the unknown parameters $\eta ,\alpha ,{\delta }_0$, where $k=1,2$ and $\mathsf{\Omega }$ are given by $[{\alpha }^{b\left(\left[\mathcal{B}{\displaystyle \frac{ϵ}{2}}\right]\right)},{\alpha }^{b\left(\left[\mathcal{B}(1-{\displaystyle \frac{ϵ}{2}})\right]\right)}],[{\eta }^{b\left(\left[\mathcal{B}{\displaystyle \frac{ϵ}{2}}\right]\right)},{\eta }^{b\left(\left[\mathcal{B}(1-{\displaystyle \frac{ϵ}{2}})\right]\right)}]$ and $[{\delta }_0^{b\left(\left[\mathcal{B}{\displaystyle \frac{ϵ}{2}}\right]\right)},{\delta }_0^{b\left(\left[\mathcal{B}(1-{\displaystyle \frac{ϵ}{2}})\right]\right)}]$.

5.2.2. Bootstrap-T CI

For Bootstrap-T CI, follow these steps:

• Same as steps (1, 2) in BP;
• Compute the t-statistic of $\mathsf{\Omega }$ as

  $$T={\displaystyle \frac{{\widehat{\mathsf{\Omega }}}^b-\widehat{\mathsf{\Omega }}}{\sqrt{V\left({\widehat{\mathsf{\Omega }}}^b\right)}}}$$

  where $V\left({\widehat{\mathsf{\Omega }}}^b\right)$ is the asymptotic variance of ${\widehat{\mathsf{\Omega }}}^b$, and it can be obtained using the FIM;
• Repeat steps 2–3 $\mathcal{B}$ times and obtain $(T^{\left(1\right)},T^{\left(2\right)},\dots ,T^{\left(\mathcal{B}\right)})$;
• Arrange $(T^{\left(1\right)},T^{\left(2\right)},\dots ,T^{\left(\mathcal{B}\right)})$ in ascending order as $(T^{\left[1\right]},T^{\left[2\right]},\dots ,T^{\left[\mathcal{B}\right]})$;
• A two-sided $100(1-ϵ)\%$ BP confidence interval for the unknown parameters $\eta$, $\alpha$, and ${\delta }_0$ is given by:

  $$\begin{array}{cc}\hfill & \left[\alpha +T^{\left[\lfloor \mathcal{B}{\displaystyle \frac{ϵ}{2}}\rfloor \right]}\sqrt{V\left({\alpha }^b\right)},\alpha +T^{\left[\lfloor \mathcal{B}(1-{\displaystyle \frac{ϵ}{2}})\rfloor \right]}\sqrt{V\left({\alpha }^b\right)}\right],\hfill \\ \hfill & \left[\eta +T^{\left[\lfloor \mathcal{B}{\displaystyle \frac{ϵ}{2}}\rfloor \right]}\sqrt{V\left({\eta }^b\right)},\eta +T^{\left[\lfloor \mathcal{B}(1-{\displaystyle \frac{ϵ}{2}})\rfloor \right]}\sqrt{V\left({\eta }^b\right)}\right],\hfill \\ \hfill & \left[{\delta }_0+T^{\left[\lfloor \mathcal{B}{\displaystyle \frac{ϵ}{2}}\rfloor \right]}\sqrt{V\left({\delta }_0^b\right)},{\delta }_0+T^{\left[\lfloor \mathcal{B}(1-{\displaystyle \frac{ϵ}{2}})\rfloor \right]}\sqrt{V\left({\delta }_0^b\right)}\right].\hfill \end{array}$$

5.3. Highest Posterior Density (HPD) Credible Interval

To compute the Bayesian HPD credible CIs (CCIs) of any function of $\mu$, set the credible level to $100(1-ϵ)\%$ and apply the MCMC method described in Section 4.3 based on its point estimation. The following are the specific steps:

• To obtain $\mathbb{I}$ groups of samples, repeat Steps (1)–(6) of the MCMC method sampling in Section 5.3;
• The samples from the aforementioned $\mathbb{I}$ groups are sorted in turn to produce $W_{\left(1\right)},W_{\left(2\right)},\cdots ,$$W_{\left(\mathbb{I}\right)}$, where $W_i=W_{\left({\mu }_i\right)}$;
• The $100(1-ϵ)\%$ HPD credible interval of $W\left(\mu \right)$ is $(W_{\left(i^{*}\right)},W_{(i^{*}+(1-{\displaystyle \frac{ϵ}{2}})\mathbb{I})})$, where $i^{*}$ satisfies the equation $W_{(i^{*}+(1-{\displaystyle \frac{ϵ}{2}})\mathbb{I})}-W_{\left(i^{*}\right)}=min\{W_{(i^{*}+(1-{\displaystyle \frac{ϵ}{2}})\mathbb{I})}-W_{\left(i^{*}\right)}\}$, for $1\le i\le \mathbb{I}-(1-{\displaystyle \frac{ϵ}{2}})\mathbb{I}$.

The benefit of the Bayesian credible interval over the classical confidence interval is that it does not require creating a pivot in advance.

6. Optimal Censoring Plan

The ideal method for gathering data through censoring has been extensively studied recently by Burkschat [32] and Pradhan and Kundu [26]. Progressive-stress ALT based on progressive type-II censoring involves removing items at specific points during an experiment, allowing for various combinations $(R_{i1},R_{i2},\cdots ,R_{i{m}_i})$ of removal times based on the predetermined number of total items $\left(n_i\right)$ and observed failures $\left(m_i\right)$.

Before selecting a particular sampling plan, it is crucial to determine which progressive censoring method yields the most informative data about the unknown parameters that we aim to estimate. This involves two main challenges:

• How can we estimate the unknown parameters using a given progressive censoring scheme?
• How can we evaluate the value of different progressive censoring schemes?

To address these challenges for MOLBE under progressive-stress ALT based on progressive type-II censoring samples with various schemes via binomial removal, this study establishes a set of optimality criteria, as outlined in

Table 1. Optimization criteria and methods.

Criterion	Method	Target
$O1$	trace ${\left[I_{3\times 3}\right]}^{-1}$	Minimum
$O2$	det ${\left[I_{3\times 3}\right]}^{-1}$	Minimum
$O3$	trace $\left[I_{3\times 3}\right]$	Maximum

. The table also includes several popular information measures that can help to identify the best progressive-stress ALT based on progressive type-II censoring for a given experiment.

The three optimality criteria ($O1,O2$, and $O3$) defined to identify the most informative progressive-stress ALT are based on a progressive type-II censoring scheme for estimating multiple unknown parameters.

• $O3$: This criterion maximizes the observed FIM, represented as $\left(\left[I_{3\times 3}\right]\right)$. FIM quantifies the amount of information an experiment provides about the parameters estimated. In this context, a higher value signifies more informative data.
• $O1$ and $O2$: These criteria both aim to reduce the uncertainty in the estimates. They do so by minimizing the determinant and the trace (sum of the diagonal elements) of the inverse of the FIM $\left({\left[I_{3\times 3}\right]}^{-1}\right)$. Lower values indicate less uncertainty.

7. An Application

In this section, a real-world dataset from Nelson [1] is presented to demonstrate the practical application of MLEs and Bayesian estimation methods. The data are shown as follows:

Failure data with 30 kilovolts (kv): 7.74, 17.05, 20.46, 21.02, 22.66, 43.40, 47.30, 139.07, 144.12, 175.88, and 194.90. Failure data with 32 kv: 0.27, 0.40, 0.69, 0.79, 2.75, 3.91, 9.88, 13.95, 15.93, 27.80, 53.24, 82.85, 89.29, 100.58, and 215.10. Failure data with 34 kv: 0.19, 0.78, 0.96, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 12.06, 31.75, 32.52, 33.91, 36.71, and 72.89. Failure data with 36 kv: 0.35, 0.59, 0.96, 0.99, 1.69, 1.97, 2.07, 2.58, 2.71, 2.9, 3.67, 3.99, 5.35, 13.77, and 25.50.

This section analyzes the breakdown time of insulating oil under different stress levels (30 kv, 32 kv, 34 kv, and 36 kv). The data at 30 kv are considered the “normal stress” case. Before diving deeper, the researchers check if the chosen MOLBE distribution accurately reflects the breakdown time data at each stress level. To achieve this, we use standard error (SE), AIC, BIC, the Kolmogorov–Smirnov (K-S) test statistic (KSD), and its corresponding p-value (PVKS) for each stress level. The K-S test and its p-value help us to assess how well the MOLBE distribution fits the observed data patterns.

The results in

Table 2. The MLEs, SE, AIC, BIC, and K-S statistics with the associated PVKS for the oil breakdown for MOLBE distribution.

Failure Data With KV		Estimate	SE	AIC	BIC	KSD	PVKS
30	$\alpha$	0.0149	0.0099	121.5858	122.3816	0.2667	0.3503
	$\beta$	0.1705	0.2453
32	$\alpha$	0.0193	0.0108	163.9172	165.3333	0.2950	0.1187
	$\beta$	0.0231	0.0289
34	$\alpha$	0.0509	0.0315	148.4514	150.3402	0.2190	0.2788
	$\beta$	0.0426	0.0536
36	$\alpha$	0.1013	0.1104	75.7273	77.1434	0.1078	0.9870
	$\beta$	0.0250	0.0527

indicate a good fit between the MOLBE distribution and the data for each stress level. This is because the p-values from the K-S test (included in Table 2) are greater than 0.05, which suggests a high probability that the data come from the MOLBE distribution. Figure 2 visually confirms this by plotting the TTT, estimated HF, and empirical CDFs of the data alongside the theoretical CDF, histogram of the data, and the theoretical PDF, QQ, and PP plots of the MOLBE distribution for each stress level. The close alignment between the data and the theoretical function plots of Figure 2 reinforces the conclusion that the MOLBE distribution is a suitable model for these data.

Figure 2 showcases various statistical plots of oil breakdown datasets for the MOLBE distribution, with each row representing a different dataset or parameter setting. The columns feature distinct types of plots to assess the distribution’s fit to the data. The first column displays TTT (total time on test) plots, which reveal failure patterns. Concave shapes indicate early failures, while convex shapes suggest wear-out failures. The second column presents the estimated hazard rate, reflecting how the failure rate changes over time, with fluctuating trends indicating a non-constant hazard function. The third column compares the empirical CDF with the fitted distribution to demonstrate how well the model matches the observed data.

Further, the fourth column features the estimated PDF, displaying the distribution of failure times with a histogram for empirical data and a red curve for the fitted distribution. The fifth column includes a QQ plot, assessing the goodness of fit by comparing theoretical and empirical quantiles. Any deviations from the straight line suggest discrepancies in the model. Lastly, the PP plot in the final column evaluates how well the empirical distribution matches the theoretical one, with points on the diagonal indicating a good fit. Together, these plots provide valuable insights into the MOLBE distribution’s suitability for modeling oil breakdown data, highlighting areas where the model aligns with or deviates from empirical data.

This study utilizes a real-world dataset to demonstrate a specific life testing approach. The data are subjected to progressive-stress ALT combined with progressive type-II censoring that removes units based on a binomial probability.

Table 3. Data under progressive-stress ALT based on progressive type-II censoring.

p	j	${\mathit{t}}_1$	${\mathit{R}}_1$	${\mathit{t}}_2$	${\mathit{R}}_2$	${\mathit{t}}_3$	${\mathit{R}}_3$	${\mathit{t}}_4$	${\mathit{R}}_4$
0.25	1	7.74	1	0.27	2	0.19	0	0.35	2
	2	17.05	0	0.4	0	0.78	2	0.59	0
	3	20.46	0	0.69	1	0.96	4	0.96	0
	4	22.66	0	0.79	1	1.31	2	0.99	0
	5	43.4	0	2.75	0	3.16	0	1.69	0
	6	47.3	0	13.95	0	4.15	0	1.97	0
	7	139.07	0	15.93	1	7.35	1	2.07	0
	8	144.12	0	27.8	0	8.27	0	2.58	0
	9	175.88	0	53.24	0	33.91	0	2.71	0
	10	194.9	0	100.58	0	36.71	0	3.67	3
0.85	1	7.74	1	0.27	2	0.19	0	0.35	2
	2	17.05	0	0.4	0	0.78	2	0.59	0
	3	20.46	0	0.69	1	0.96	4	0.96	0
	4	22.66	0	0.79	1	1.31	2	0.99	0
	5	43.4	0	2.75	0	3.16	0	1.69	0
	6	47.3	0	13.95	0	4.15	0	1.97	0
	7	139.07	0	15.93	1	7.35	1	2.07	0
	8	144.12	0	27.8	0	8.27	0	2.58	0
	9	175.88	0	53.24	0	33.91	0	2.71	0
	10	194.9	0	100.58	0	36.71	0	3.67	3

presents the results of applying this testing method under two scenarios:

• Binomial removal probability (p) of 0.25;
• Binomial removal probability (p) of 0.85.

Essentially, the table showcases the outcomes of the life-testing approach under different probabilities for removing units during the testing process.

We analyze the data that used progressive-stress testing with progressive type-II censoring. For each test, we calculate the MLEs and Bayesian estimates of the parameters ${\delta }_0,\eta ,$ and $\alpha$. These estimates are presented in

Table 4. MLEs and Bayesian estimation under progressive-stress ALT based on progressive type-II censoring.

			MLEs					Bayesian
${\mathit{m}}_{\mathit{i}}$	p		Estimates	SE	Lower	Upper	LACI	Estimates	SE	Lower	Upper	LCCI
$n_i$	0	${\delta }_0$	0.0165	0.0061	0.0045	0.0284	0.0239	0.0181	0.0039	0.0106	0.0257	0.0151
		$\eta$	0.1076	0.0389	0.0315	0.1838	0.1523	0.1103	0.0500	0.0269	0.2051	0.1782
		$\alpha$	0.0101	0.0021	0.0060	0.0143	0.0083	0.0103	0.0028	0.0052	0.0158	0.0105
10	0.25	${\delta }_0$	0.0157	0.0066	0.0027	0.0287	0.0260	0.0181	0.0047	0.0097	0.0271	0.0175
		$\eta$	0.3389	0.0477	0.0691	0.2562	0.1872	0.1559	0.0574	0.0526	0.2679	0.2153
		$\alpha$	0.0079	0.0019	0.0042	0.0117	0.0075	0.0084	0.0027	0.0034	0.0135	0.0101
	0.85	${\delta }_0$	0.0138	0.0062	0.0017	0.0259	0.0242	0.0155	0.0037	0.0089	0.0231	0.0141
		$\eta$	0.3086	0.0524	0.0454	0.2508	0.2054	0.1560	0.0702	0.0368	0.2955	0.2588
		$\alpha$	0.0060	0.0013	0.0035	0.0085	0.0049	0.0063	0.0018	0.0031	0.0099	0.0068

. Additionally, Table 4 presents 95% confidence intervals (CIs) along with the lengths of these intervals. For the MLEs, the length is represented as ACIs (LACI), while for the Bayesian method, the length is represented as CCIs (LCCI). To check that the MLE values have maximum points of likelihood, the likelihood profile is depicted in Figure 3 for the parameter in the first case. Based on this figure (Figure 3), the values of MLEs have a maximum point and uniqueness solution. Additionally, MCMC plots are shown in Figure 4 to determine the convergence and normality of the outputs of the MH algorithm via MCMC for Bayesian estimation and each stress level.

8. Simulation Study

This section compares the performance of maximum likelihood and Bayesian estimation methods through simulations. The simulations involve various scenarios using progressive-stress ALT based on progressive type-II censoring schemes with random removal. To achieve this, the analysis leverages the R 4.3.0 software package for extensive calculations. The simulations generate data samples under progressive-stress ALT based on progressive type-II censoring schemes with random removal configurations by specifying the values of models. The specific steps involved in defining these parameter values are detailed later in the study.

• The parameter values $({\delta }_0,\alpha ,\eta )$ in this simulation are as follows:
  Case 1 is (${\delta }_0=0.6,\eta =0.5,\alpha =0.5$), Case 2 is (${\delta }_0=1.5,\eta =0.5,\alpha =0.5$),
  Case 3 is (${\delta }_0=1.5,\eta =2,\alpha =0.5$), Case 4 is (${\delta }_0=1.5,\eta =2,\alpha =2.5$), and
  Case 5 is (${\delta }_0=1.5,\eta =1.3,\alpha =1.2$).
  The sample sizes for each stress level are defined as follows: $(n_1=10,n_2=10,n_3=8,$$n_4=8)$, $(n_1=30,n_2=30,n_3=25,n_4=20)$, and $(n_1=50,n_2=50,n_3=40,n_4=30)$; the censored samples $m_i=n_{i}r$ and relative censoring size $r_i$ are determined to be 0.75 and 0.9 for each stress level, and (they must be whole numbers and not fractions). The process used is as follows:
• Four samples from a uniform distribution with a range of $(0,1)$ are generated.
• MOLBE samples are generated using the inverse of the CDF. However, this inverse does not have a closed mathematical form. Therefore, the ’uniroot’ function in R, which is a root-finding method rather than an iterative algorithm, is used to obtain the required values. As a result, four samples following the MOLBE distribution are generated.
• In addition, the probability of binomial removal p is considered to be $0.25$ and $0.85$ for each stress level l, $l=1,2,\cdots ,k$.
• The algorithm technique of a progressive-stress ALT based on progressive type-II censoring has produced a MOLBE distribution with a size of $n_i$, a censoring size of $m_i$, and ratios of time stages of stress $h_i$ of 100%, 71.43%, 55.55%, and 41.47%, respectively, for each stress level.

This subsection evaluates the accuracy of different parameter estimates through simulations. The analysis compares two approaches:

• MLEs: This method provides point estimates for the model parameters ($\alpha ,\eta$, and ${\delta }_0$). The researchers calculate the mean squared error (MSE) and estimated bias of these MLEs to evaluate their accuracy.
• Bayesian Estimation: This method incorporates prior information about the parameters. Here, the researchers use a gamma prior distribution with different settings (SELF, LLF with $\nu$ = −0.5, and LLF with $\nu$ = 0.5) to estimate the parameters.

The simulations involve 5000 runs and analyze various aspects of the point estimates:

• We estimate bias for MLEs and the Bayesian estimates of $\alpha ,\eta$ and ${\delta }_0$;
• We calculate the MSE for MLEs and the Bayesian estimates;
• We determine the optimality measures via FIM for MLEs to determine the best schemes that have a minimum or maximum value of binomial removal.

For interval estimates, the following definitions are used:

• Length of CIs: This refers to the width of the interval capturing the true parameter value with a certain confidence level (e.g., 95%). The analysis compares three types of CIs, ACIs, Bootstrap-P CIs, and Bootstrap-T CIs for each parameter.
• Coverage Probability (CP): This represents the proportion of times the constructed CIs contain the true parameter value when repeated sampling is conducted. For example, a 95% CI should ideally contain the true parameter 95% of the time.
• Bayesian Credible Ranges: Similar to CIs, these represent the range of likely values for the parameters based on the Bayesian approach.

Interestingly, to set up the Bayesian estimation effectively, the researchers leverage the MLEs results. They use the MLEs estimate and its variance–covariance matrix to determine suitable values for the hyper-parameters of the gamma prior distribution.

Since exact formulas are unavailable for the best MLEs, a numerical optimization method, BFGS (Broyden–Fletcher–Goldfarb–Shanno), implemented in the R 4.3.0 package ‘MaxLik’. This package calculates the MLEs for each parameter based on the data. For Bayesian estimates, the researchers ran simulations with (12,000 samples) using MCMC and discarded initial adjustments (burn-in) by removing the first 2000 iterations. Finally, the researchers used R’s “coda” tools to obtain the most likely Bayesian values and their credible ranges (HPD intervals) for the same parameters.

The findings from

Table 5. Point estimation via bias and MSE for parameters: Case 1.

${\mathit{\delta }}_0=0.6,\mathit{\eta }=0.5,\mathit{\alpha }=0.5$				MLEs		SELF		LLF 1		LLF 2
Sample Sizes	p	r		Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	0.0336	0.1189	−0.0109	0.0207	−0.0036	0.0213	−0.0180	0.0202
			$\eta$	0.0858	0.2237	0.0557	0.0711	0.0802	0.0812	0.0320	0.0629
			$\alpha$	0.4252	0.9883	0.2387	0.2119	0.2744	0.2597	0.2041	0.1705
		0.9	${\delta }_0$	0.0285	0.1028	0.0093	0.0202	0.0034	0.0203	0.0178	0.0192
			$\eta$	0.0604	0.2084	0.0419	0.0617	0.0646	0.0699	0.0199	0.0550
			$\alpha$	0.4113	0.8082	0.2156	0.1892	0.2482	0.2273	0.1836	0.1541
	0.85	0.75	${\delta }_0$	0.2292	0.2018	0.0481	0.0234	0.0572	0.0252	0.0392	0.0219
			$\eta$	0.0379	0.2336	0.0416	0.0602	0.0655	0.0694	0.0185	0.0527
			$\alpha$	0.7595	1.4761	0.2838	0.2405	0.3260	0.3005	0.2429	0.1901
		0.9	${\delta }_0$	0.1747	0.1451	0.0458	0.0225	0.0547	0.0247	0.0315	0.0203
			$\eta$	0.0145	0.1981	0.0404	0.0594	0.0607	0.0679	0.0162	0.0526
			$\alpha$	0.5044	0.9377	0.2189	0.1742	0.2537	0.2138	0.1854	0.1404
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	0.0726	0.0451	0.0285	0.0098	0.0327	0.0103	0.0243	0.0094
			$\eta$	0.0307	0.0936	0.0263	0.0268	0.0397	0.0292	0.0134	0.0249
			$\alpha$	0.3154	0.3588	0.2060	0.1099	0.2281	0.1289	0.1845	0.0930
		0.9	${\delta }_0$	0.0566	0.0313	0.0203	0.0074	0.0308	0.0078	0.0213	0.0070
			$\eta$	0.0041	0.0696	0.0187	0.0216	0.0309	0.0235	0.0068	0.0201
			$\alpha$	0.1779	0.1607	0.1401	0.0771	0.1586	0.1122	0.1228	0.0566
	0.85	0.75	${\delta }_0$	0.0872	0.0452	0.0434	0.0117	0.0479	0.0126	0.0389	0.0110
			$\eta$	0.0157	0.0823	0.0193	0.0252	0.0326	0.0275	0.0064	0.0233
			$\alpha$	0.2814	0.2877	0.1875	0.1789	0.2127	0.2984	0.1623	0.0951
		0.9	${\delta }_0$	0.0574	0.0334	0.0194	0.0079	0.0396	0.0084	0.0322	0.0075
			$\eta$	0.0131	0.0703	0.0182	0.0209	0.0319	0.0225	0.0059	0.0196
			$\alpha$	0.1612	0.1699	0.1215	0.0554	0.1379	0.0654	0.1058	0.0467
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.0414	0.0208	0.0195	0.0047	0.0223	0.0050	0.0167	0.0045
			$\eta$	0.0400	0.0515	0.0256	0.0151	0.0351	0.0162	0.0163	0.0141
			$\alpha$	0.1819	0.1240	0.1339	0.0473	0.1476	0.0548	0.1207	0.0407
		0.9	${\delta }_0$	0.0396	0.0176	0.0162	0.0046	0.0209	0.0048	0.0138	0.0044
			$\eta$	−0.0020	0.0454	0.0083	0.0120	0.0165	0.0127	0.0025	0.0115
			$\alpha$	0.1089	0.0795	0.0913	0.0312	0.1013	0.0354	0.0816	0.0275
	0.85	0.75	${\delta }_0$	0.0449	0.0211	0.0275	0.0052	0.0303	0.0055	0.0246	0.0050
			$\eta$	0.0086	0.0567	0.0104	0.0153	0.0197	0.0162	0.0012	0.0147
			$\alpha$	0.1433	0.1097	0.1129	0.0412	0.1252	0.0471	0.1011	0.0360
		0.9	${\delta }_0$	0.0372	0.0166	0.0241	0.0039	0.0265	0.0040	0.0216	0.0037
			$\eta$	0.0024	0.0426	0.0104	0.0123	0.0185	0.0130	0.0011	0.0118
			$\alpha$	0.1001	0.0701	0.0805	0.0244	0.0903	0.0279	0.0711	0.0214

,

Table 6. Point estimation via bias and MSE for parameters: Case 2.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=0.5,\mathit{\alpha }=0.5$				MLEs		SELE		LLF 1		LLF 2
Sample Sizes	p	r		Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	0.2899	0.6285	−0.0536	0.1332	−0.0791	0.1432	−0.0979	0.1289
			$\eta$	0.1059	0.2563	0.0670	0.0721	0.0928	0.0830	0.0422	0.0631
			$\alpha$	0.4790	1.3910	0.2276	0.2614	0.2699	0.4112	0.1877	0.1691
		0.9	${\delta }_0$	0.2204	0.6138	0.0392	0.1039	0.0687	0.1204	−0.0071	0.0936
			$\eta$	0.0590	0.2064	0.0511	0.0639	0.0744	0.0727	0.0286	0.0568
			$\alpha$	0.4306	0.9840	0.1852	0.1313	0.2167	0.1624	0.1547	0.1051
	0.85	0.75	${\delta }_0$	0.5558	1.0261	0.1018	0.1444	0.1585	0.1740	0.0468	0.1238
			$\eta$	0.0486	0.2367	0.0450	0.0642	0.0687	0.0733	0.0221	0.0567
			$\alpha$	0.8992	2.4661	0.2670	0.2102	0.3064	0.2581	0.2288	0.1687
		0.9	${\delta }_0$	0.4214	0.8263	0.1003	0.1390	0.1381	0.1667	0.0428	0.1189
			$\eta$	0.0324	0.1902	0.0418	0.0635	0.0572	0.0731	0.0216	0.0558
			$\alpha$	0.5813	1.4819	0.2192	0.1901	0.2539	0.2327	0.1858	0.1533
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	0.1641	0.2555	0.0573	0.0544	0.0829	0.0614	0.0319	0.0491
			$\eta$	0.0292	0.0866	0.0241	0.0256	0.0374	0.0280	0.0111	0.0237
			$\alpha$	0.3252	0.3876	0.2039	0.1061	0.2264	0.1255	0.1821	0.0889
		0.9	${\delta }_0$	0.1639	0.1901	0.0485	0.0468	0.0819	0.0539	0.0306	0.0411
			$\eta$	0.0111	0.0714	0.0209	0.0207	0.0328	0.0225	0.0092	0.0193
			$\alpha$	0.1990	0.1724	0.1434	0.0703	0.1607	0.0819	0.1267	0.0599
	0.85	0.75	${\delta }_0$	0.2323	0.3039	0.1023	0.0669	0.1297	0.0768	0.0752	0.0587
			$\eta$	0.0042	0.0825	0.0174	0.0238	0.0304	0.0258	0.0047	0.0222
			$\alpha$	0.2950	0.3270	0.1809	0.0978	0.2021	0.1146	0.1604	0.0829
		0.9	${\delta }_0$	0.1438	0.2013	0.0882	0.0452	0.1117	0.0526	0.0649	0.0391
			$\eta$	−0.0028	0.0722	0.0089	0.0202	0.0204	0.0216	−0.0024	0.0192
			$\alpha$	0.1478	0.1498	0.1098	0.0464	0.1253	0.0547	0.0949	0.0393
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.1242	0.1410	0.0602	0.0342	0.0778	0.0383	0.0427	0.0309
			$\eta$	0.0408	0.0538	0.0289	0.0165	0.0383	0.0177	0.0197	0.0154
			$\alpha$	0.2033	0.1417	0.1448	0.0524	0.1584	0.0598	0.1317	0.0459
		0.9	${\delta }_0$	0.1058	0.1122	0.0586	0.0244	0.0740	0.0276	0.0413	0.0218
			$\eta$	0.0034	0.0426	0.0116	0.0120	0.0199	0.0127	0.0035	0.0114
			$\alpha$	0.1115	0.0720	0.0831	0.0254	0.0930	0.0289	0.0735	0.0222
	0.85	0.75	${\delta }_0$	0.1375	0.1535	0.0780	0.0367	0.0962	0.0417	0.0599	0.0324
			$\eta$	0.0142	0.0540	0.0176	0.0156	0.0272	0.0168	0.0082	0.0147
			$\alpha$	0.1665	0.1228	0.1217	0.0442	0.1345	0.0507	0.1092	0.0384
		0.9	${\delta }_0$	0.0874	0.1060	0.0633	0.0266	0.0784	0.0299	0.0483	0.0238
			$\eta$	0.0058	0.0469	0.0132	0.0138	0.0217	0.0146	0.0049	0.0131
			$\alpha$	0.0895	0.0684	0.0802	0.0259	0.0900	0.0297	0.0707	0.0226

,

Table 7. Point estimation via bias and MSE for parameters: Case 3.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=2,\mathit{\alpha }=0.5$				MLEs		SELE		LLF 1		LLF 2
Sample Sizes	p	r		Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	0.0977	0.6453	−0.0579	0.1229	−0.1043	0.1316	−0.1005	0.1191
			$\eta$	0.1026	0.2612	0.0729	0.0796	0.1215	0.0919	0.0257	0.0728
			$\alpha$	0.5458	1.7351	0.2096	0.1757	0.2436	0.2150	0.1766	0.1414
		0.9	${\delta }_0$	0.0892	0.5859	0.0421	0.1003	0.0861	0.1141	−0.0076	0.0916
			$\eta$	0.0243	0.2389	0.0302	0.0632	0.0729	0.0701	−0.0115	0.0601
			$\alpha$	0.4958	1.2014	0.1905	0.1444	0.2212	0.1748	0.1611	0.1181
	0.85	0.75	${\delta }_0$	0.5095	0.8795	0.0889	0.1219	0.1394	0.1425	0.0395	0.1077
			$\eta$	0.0496	0.2232	0.0506	0.0555	0.0961	0.0652	0.0065	0.0504
			$\alpha$	0.9235	2.6703	0.2384	0.1901	0.2721	0.2271	0.2058	0.1577
		0.9	${\delta }_0$	0.4176	0.7344	0.0813	0.1224	0.1308	0.1410	0.0308	0.1049
			$\eta$	0.0287	0.2065	0.0332	0.0512	0.0759	0.0581	−0.0063	0.0482
			$\alpha$	0.6312	1.5220	0.2164	0.1689	0.2500	0.2068	0.1839	0.1365
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	0.1831	0.2453	0.0627	0.0558	0.0861	0.0620	0.0393	0.0507
			$\eta$	0.0366	0.0926	0.0266	0.0235	0.0446	0.0253	0.0087	0.0224
			$\alpha$	0.3556	0.4286	0.2059	0.1060	0.2262	0.1224	0.1862	0.0912
		0.9	${\delta }_0$	0.1347	0.1642	0.0618	0.0417	0.0809	0.0469	0.0352	0.0373
			$\eta$	0.0259	0.0695	0.0187	0.0176	0.0345	0.0188	0.0030	0.0171
			$\alpha$	0.1892	0.1705	0.1292	0.0553	0.1443	0.0641	0.1147	0.0476
	0.85	0.75	${\delta }_0$	0.1959	0.2392	0.0860	0.0574	0.1106	0.0653	0.0614	0.0510
			$\eta$	0.0039	0.0892	0.0104	0.0224	0.0288	0.0235	−0.0078	0.0219
			$\alpha$	0.2776	0.2903	0.1653	0.0838	0.1845	0.0979	0.1468	0.0714
		0.9	${\delta }_0$	0.1326	0.1619	0.0726	0.0420	0.0933	0.0477	0.0519	0.0373
			$\eta$	0.0032	0.0728	0.0101	0.0181	0.0253	0.0191	−0.0013	0.0176
			$\alpha$	0.1676	0.1580	0.1130	0.0480	0.1276	0.0561	0.0988	0.0409
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.1288	0.1272	0.0527	0.0287	0.0682	0.0316	0.0373	0.0262
			$\eta$	0.0373	0.0545	0.0233	0.0137	0.0349	0.0146	0.0117	0.0132
			$\alpha$	0.2052	0.1446	0.1321	0.0423	0.1440	0.0478	0.1205	0.0374
		0.9	${\delta }_0$	0.0702	0.0882	0.0479	0.0225	0.0612	0.0249	0.0346	0.0204
			$\eta$	0.0238	0.0416	0.0138	0.0107	0.0240	0.0112	0.0036	0.0104
			$\alpha$	0.0982	0.0735	0.0809	0.0259	0.0901	0.0295	0.0720	0.0227
	0.85	0.75	${\delta }_0$	0.1019	0.1227	0.0605	0.0314	0.0761	0.0349	0.0448	0.0283
			$\eta$	0.0268	0.0552	0.0175	0.0146	0.0295	0.0153	0.0056	0.0141
			$\alpha$	0.1423	0.1087	0.1081	0.0411	0.1194	0.0468	0.0971	0.0359
		0.9	${\delta }_0$	0.0643	0.0881	0.0468	0.0193	0.0598	0.0214	0.0339	0.0176
			$\eta$	0.0022	0.0418	0.0052	0.0108	0.0155	0.0111	−0.0051	0.0107
			$\alpha$	0.0836	0.0653	0.0679	0.0208	0.0762	0.0234	0.0598	0.0185

,

Table 8. Point estimation via bias and MSE for parameters: Case 4.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=2,\mathit{\alpha }=2.5$				MLEs		SELE		LLF 1		LLF 2
Sample Sizes	p	r		Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	−0.0818	0.1390	−0.0606	0.0440	−0.0427	0.0436	−0.0784	0.0450
			$\eta$	0.0763	0.1404	0.0607	0.0419	0.0887	0.0474	0.0331	0.0381
			$\alpha$	0.8172	4.5202	0.9918	3.3952	1.6173	7.5147	0.4530	1.3515
		0.9	${\delta }_0$	0.0363	0.1194	0.0090	0.0325	0.0268	0.0347	−0.0088	0.0310
			$\eta$	0.0342	0.1238	0.0310	0.0355	0.0558	0.0386	0.0066	0.0338
			$\alpha$	0.8092	4.0522	0.8062	2.2885	1.3667	4.7404	0.3319	0.9732
	0.85	0.75	${\delta }_0$	0.1753	0.1452	0.0619	0.0423	0.0828	0.0471	0.0411	0.0384
			$\eta$	0.0316	0.1176	0.0296	0.0299	0.0560	0.0330	0.0038	0.0283
			$\alpha$	1.8205	7.7573	1.2538	4.0872	1.9820	8.4912	0.6283	1.6431
		0.9	${\delta }_0$	0.1471	0.1236	0.0608	0.0352	0.0807	0.0397	0.0409	0.0316
			$\eta$	0.0181	0.1150	0.0215	0.0287	0.0458	0.0311	−0.0024	0.0276
			$\alpha$	1.2703	5.4607	0.9264	2.6470	1.5158	5.4016	0.4084	1.0915
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	0.0808	0.0481	0.0331	0.0142	0.0416	0.0152	0.0245	0.0134
			$\eta$	0.0276	0.0498	0.0183	0.0120	0.0283	0.0126	0.0084	0.0116
			$\alpha$	1.0854	3.2062	0.8600	1.6373	1.2069	2.8613	0.5603	0.8704
		0.9	${\delta }_0$	0.0758	0.0477	0.0314	0.0125	0.0401	0.0135	0.0204	0.0116
			$\eta$	0.0107	0.0380	0.0078	0.0093	0.0165	0.0096	−0.0009	0.0092
			$\alpha$	0.7128	2.1329	0.5623	0.9518	0.8316	1.7033	0.3253	0.4998
	0.85	0.75	${\delta }_0$	0.0906	0.0487	0.0519	0.0171	0.0608	0.0185	0.0430	0.0159
			$\eta$	0.0136	0.0425	0.0100	0.0108	0.0204	0.0113	−0.0004	0.0106
			$\alpha$	0.8068	2.1237	0.7125	1.3786	1.0452	2.4759	0.4248	0.7149
		0.9	${\delta }_0$	0.0568	0.0452	0.0400	0.0128	0.0474	0.0137	0.0325	0.0119
			$\eta$	−0.0091	0.0349	0.0017	0.0089	0.0106	0.0090	−0.0003	0.0088
			$\alpha$	0.4997	1.6666	0.4504	0.7996	0.6956	1.3985	0.2354	0.4424
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.0808	0.0350	0.0357	0.0097	0.0413	0.0103	0.0301	0.0092
			$\eta$	0.0206	0.0286	0.0139	0.0074	0.0203	0.0076	0.0075	0.0072
			$\alpha$	0.7674	1.5037	0.5743	0.7761	0.7752	1.2280	0.3950	0.4696
		0.9	${\delta }_0$	0.0463	0.0253	0.0301	0.0067	0.0349	0.0071	0.0254	0.0063
			$\eta$	−0.0057	0.0249	0.0021	0.0062	0.0077	0.0062	−0.0034	0.0061
			$\alpha$	0.3495	0.7793	0.3103	0.3492	0.4601	0.5727	0.1775	0.2140
	0.85	0.75	${\delta }_0$	0.0726	0.0346	0.0418	0.0097	0.0475	0.0104	0.0361	0.0091
			$\eta$	0.0102	0.0282	0.0076	0.0070	0.0142	0.0072	0.0010	0.0069
			$\alpha$	0.6001	1.2522	0.5000	0.6573	0.6963	1.0600	0.3251	0.3960
		0.9	${\delta }_0$	0.0330	0.0252	0.0239	0.0066	0.0286	0.0070	0.0192	0.0063
			$\eta$	0.0087	0.0231	0.0059	0.0058	0.0115	0.0059	0.0003	0.0057
			$\alpha$	0.2799	0.6892	0.2752	0.2907	0.4190	0.4712	0.1488	0.1852

,

Table 9. Point estimation via bias and MSE for parameters: Case 5.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=1.3,\mathit{\alpha }=1.2$				MLEs		SELE		LLF 1		LLF 2
Sample Sizes	p	r		Bias	MSE	Bias	MSE	Bias	MSE	Bias	MSE
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	0.1285	0.1063	0.0482	0.0270	0.0618	0.0295	0.0346	0.0250
			$\eta$	0.0205	0.0618	0.0135	0.0161	0.0265	0.0169	0.0005	0.0156
			$\alpha$	0.6828	1.2601	0.4685	0.5202	0.5678	0.7150	0.3762	0.3669
		0.9	${\delta }_0$	0.0815	0.0831	0.0452	0.0197	0.0568	0.0216	0.0335	0.0182
			$\eta$	0.0125	0.0506	0.0119	0.0134	0.0237	0.0141	0.0004	0.0130
			$\alpha$	0.3256	0.5574	0.2451	0.2077	0.3122	0.2847	0.1831	0.1506
	0.85	0.75	${\delta }_0$	0.1462	0.1135	0.0739	0.0296	0.0879	0.0328	0.0599	0.0267
			$\eta$	0.0019	0.0607	0.0073	0.0155	0.0207	0.0162	−0.0060	0.0152
			$\alpha$	0.5660	1.1381	0.3859	0.3910	0.4767	0.5507	0.3019	0.2720
		0.9	${\delta }_0$	0.0988	0.0798	0.0544	0.0208	0.0659	0.0228	0.0429	0.0191
			$\eta$	−0.0015	0.0525	0.0014	0.0135	0.0131	0.0139	−0.0051	0.0134
			$\alpha$	0.3401	0.5777	0.2438	0.2054	0.3080	0.2783	0.1841	0.1495
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	−0.0192	0.3193	−0.0683	0.0738	−0.0411	0.0750	−0.0952	0.0744
			$\eta$	0.0816	0.2016	0.0610	0.0660	0.0968	0.0753	0.0264	0.0596
			$\alpha$	0.8089	3.6924	0.4858	1.0226	0.6610	1.9272	0.3243	0.4953
		0.9	${\delta }_0$	0.0134	0.2758	0.0346	0.0555	0.0406	0.0618	0.0065	0.0510
			$\eta$	0.0297	0.1640	0.0268	0.0431	0.0581	0.0481	−0.0036	0.0404
			$\alpha$	0.7829	3.1387	0.4199	0.6323	0.5703	0.9511	0.2811	0.3989
	0.85	0.75	${\delta }_0$	0.3481	0.3889	0.0953	0.0741	0.1283	0.0853	0.0624	0.0653
			$\eta$	0.0242	0.1768	0.0329	0.0428	0.0656	0.0484	0.0013	0.0396
			$\alpha$	1.5839	6.5944	0.6398	1.1451	0.8330	1.7357	0.4580	0.7070
		0.9	${\delta }_0$	0.2655	0.3492	0.0910	0.0725	0.1136	0.0816	0.0607	0.0658
			$\eta$	0.0079	0.1555	0.0190	0.0404	0.0498	0.0448	−0.0011	0.0382
			$\alpha$	0.9861	3.7553	0.4579	0.7341	0.6210	1.1422	0.3088	0.4514
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.1022	0.0684	0.0416	0.0155	0.0504	0.0167	0.0328	0.0144
			$\eta$	0.0224	0.0420	0.0154	0.0107	0.0240	0.0111	0.0069	0.0103
			$\alpha$	0.4712	0.6788	0.3166	0.2321	0.3738	0.2987	0.2633	0.1790
		0.9	${\delta }_0$	0.0609	0.0514	0.0380	0.0112	0.0454	0.0122	0.0306	0.0105
			$\eta$	0.0028	0.0355	0.0048	0.0090	0.0123	0.0093	−0.0027	0.0089
			$\alpha$	0.2042	0.3121	0.1569	0.0918	0.1957	0.1175	0.1207	0.0721
	0.85	0.75	${\delta }_0$	0.0909	0.0610	0.0430	0.0138	0.0520	0.0150	0.0341	0.0127
			$\eta$	0.0133	0.0405	0.0122	0.0105	0.0210	0.0110	0.0035	0.0103
			$\alpha$	0.3209	0.4343	0.2151	0.1291	0.2654	0.1707	0.1684	0.0970
		0.9	${\delta }_0$	0.0651	0.0484	0.0420	0.0110	0.0495	0.0120	0.0346	0.0101
			$\eta$	0.0066	0.0322	0.0063	0.0082	0.0139	0.0084	−0.0012	0.0080
			$\alpha$	0.2021	0.2967	0.1583	0.0995	0.1984	0.1290	0.1209	0.0768

,

Table 10. Interval estimation via LCI and CP for parameters with different methods: Case 1.

${\mathit{\delta }}_0=0.6,\mathit{\eta }=0.5,\mathit{\alpha }=0.5$				MLEs				SELE				LLF 1				LLF 2
Sample Sizes	p	r		LACI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	1.3458	98.3%	0.0437	0.0431	0.5367	97.4%	0.0181	0.0181	0.5720	95.0%	0.0181	0.0182	0.5530	94.7%	0.0176	0.0172
			$\eta$	1.8242	97.0%	0.0590	0.0589	0.9452	97.8%	0.0317	0.0317	1.0725	94.2%	0.0354	0.0354	0.9755	94.1%	0.0318	0.0313
			$\alpha$	3.5243	98.3%	0.1165	0.1173	1.4008	96.8%	0.0484	0.0492	1.6844	95.0%	0.0556	0.0558	1.4077	94.3%	0.0454	0.0454
		0.9	${\delta }_0$	1.3630	98.1%	0.0439	0.0447	0.5249	98.3%	0.0179	0.0181	0.5808	94.9%	0.0182	0.0181	0.5608	95.0%	0.0173	0.0174
			$\eta$	1.7748	98.0%	0.0565	0.0570	0.8560	96.7%	0.0314	0.0311	1.0053	95.4%	0.0326	0.0330	0.9167	95.1%	0.0281	0.0283
			$\alpha$	3.1352	98.5%	0.0995	0.0956	1.2739	98.6%	0.0461	0.0463	1.5964	95.2%	0.0526	0.0512	1.3608	94.9%	0.0441	0.0434
	0.85	0.75	${\delta }_0$	1.5155	98.5%	0.0490	0.0491	0.5373	98.7%	0.0186	0.0183	0.5808	95.4%	0.0187	0.0185	0.5592	94.9%	0.0173	0.0175
			$\eta$	1.8899	96.9%	0.0586	0.0588	0.8392	97.0%	0.0305	0.0300	1.0007	95.4%	0.0321	0.0316	0.8972	95.1%	0.0288	0.0287
			$\alpha$	3.7194	98.5%	0.1184	0.1185	1.4148	98.6%	0.0488	0.0487	1.7281	93.7%	0.0526	0.0524	1.4199	94.0%	0.0471	0.0467
		0.9	${\delta }_0$	1.3279	97.8%	0.0428	0.0428	0.5356	98.6%	0.0187	0.0186	0.5829	95.3%	0.0186	0.0186	0.5588	95.3%	0.0190	0.0190
			$\eta$	1.7445	97.5%	0.0554	0.0552	0.8279	96.9%	0.0294	0.0293	0.9874	95.3%	0.0309	0.0309	0.8950	95.1%	0.0282	0.0282
			$\alpha$	3.2421	98.5%	0.1015	0.1016	1.2457	96.5%	0.0411	0.0412	1.5161	94.6%	0.0495	0.0486	1.2770	94.1%	0.0401	0.0404
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	0.7823	98.7%	0.0249	0.0249	0.3622	98.2%	0.0118	0.0118	0.3762	94.9%	0.0121	0.0122	0.3674	94.7%	0.0112	0.0112
			$\eta$	1.1938	97.5%	0.0363	0.0358	0.5927	96.5%	0.0205	0.0205	0.6518	95.4%	0.0203	0.0205	0.6164	95.7%	0.0198	0.0200
			$\alpha$	1.9973	98.5%	0.0637	0.0637	0.8888	98.9%	0.0310	0.0309	1.0871	94.5%	0.0333	0.0333	0.9524	94.7%	0.0314	0.0308
		0.9	${\delta }_0$	0.6579	98.9%	0.0211	0.0213	0.2883	98.7%	0.0097	0.0097	0.3136	95.7%	0.0099	0.0098	0.3049	95.5%	0.0096	0.0097
			$\eta$	1.0348	97.9%	0.0331	0.0329	0.5284	96.8%	0.0183	0.0184	0.5887	95.5%	0.0197	0.0199	0.5553	95.7%	0.0177	0.0177
			$\alpha$	1.4090	98.5%	0.0436	0.0437	0.6990	96.7%	0.0299	0.0284	1.1572	98.0%	0.0387	0.0359	0.7993	95.5%	0.0254	0.0256
	0.85	0.75	${\delta }_0$	0.7603	98.6%	0.0243	0.0246	0.3498	96.3%	0.0125	0.0122	0.3978	95.9%	0.0136	0.0133	0.3818	96.0%	0.0126	0.0125
			$\eta$	1.1237	97.6%	0.0359	0.0356	0.5829	98.8%	0.0201	0.0199	0.6377	95.0%	0.0199	0.0201	0.5987	94.9%	0.0197	0.0199
			$\alpha$	1.7910	98.5%	0.0568	0.0567	0.8572	97.0%	0.0500	0.0483	1.9733	96.6%	0.0866	0.0602	1.0285	95.5%	0.0339	0.0326
		0.9	${\delta }_0$	0.6809	96.5%	0.0213	0.0214	0.2873	96.3%	0.0106	0.0106	0.3243	96.1%	0.0101	0.0099	0.3151	96.0%	0.0100	0.0099
			$\eta$	1.0385	97.3%	0.0327	0.0335	0.5407	98.8%	0.0178	0.0177	0.5753	95.4%	0.0177	0.0185	0.5474	95.8%	0.0171	0.0172
			$\alpha$	1.4880	98.5%	0.0464	0.0451	0.6434	97.2%	0.0242	0.0245	0.8450	95.6%	0.0263	0.0267	0.7392	95.4%	0.0233	0.0232
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.5413	98.7%	0.0171	0.0171	0.2470	98.8%	0.0080	0.0079	0.2618	95.2%	0.0084	0.0084	0.2561	95.1%	0.0084	0.0085
			$\eta$	0.8762	97.5%	0.0286	0.0284	0.4462	96.5%	0.0155	0.0154	0.4805	95.4%	0.0162	0.0165	0.4612	95.6%	0.0145	0.0143
			$\alpha$	1.1824	98.5%	0.0364	0.0362	0.5690	96.9%	0.0225	0.0226	0.7131	95.9%	0.0221	0.0220	0.6335	95.6%	0.0204	0.0203
		0.9	${\delta }_0$	0.4968	98.3%	0.0156	0.0155	0.2315	98.2%	0.0079	0.0078	0.2475	95.3%	0.0073	0.0074	0.2426	95.3%	0.0076	0.0077
			$\eta$	0.8355	97.3%	0.0271	0.0269	0.4146	98.6%	0.0132	0.0133	0.4370	94.8%	0.0141	0.0139	0.4210	95.1%	0.0136	0.0137
			$\alpha$	1.0197	98.1%	0.0328	0.0329	0.4978	97.0%	0.0178	0.0181	0.6217	95.6%	0.0188	0.0194	0.5663	95.7%	0.0175	0.0175
	0.85	0.75	${\delta }_0$	0.5424	96.7%	0.0163	0.0163	0.2486	98.8%	0.0083	0.0085	0.2650	95.4%	0.0083	0.0083	0.2595	95.4%	0.0081	0.0081
			$\eta$	0.9333	97.5%	0.0300	0.0299	0.4659	96.3%	0.0157	0.0157	0.4935	95.5%	0.0159	0.0159	0.4748	95.3%	0.0153	0.0154
			$\alpha$	1.1709	98.6%	0.0362	0.0359	0.5656	96.9%	0.0198	0.0195	0.6949	95.1%	0.0218	0.0218	0.6300	94.9%	0.0199	0.0206
		0.9	${\delta }_0$	0.4837	98.4%	0.0151	0.0151	0.2150	96.3%	0.0074	0.0075	0.2269	95.5%	0.0073	0.0073	0.2223	95.4%	0.0072	0.0069
			$\eta$	0.8096	97.7%	0.0257	0.0259	0.4199	98.9%	0.0137	0.0136	0.4405	95.3%	0.0143	0.0144	0.4250	95.3%	0.0138	0.0138
			$\alpha$	0.9611	97.8%	0.0315	0.0312	0.4469	97.2%	0.0160	0.0158	0.5511	95.4%	0.0180	0.0178	0.5012	95.4%	0.0163	0.0163

,

Table 11. Interval estimation via LCI and CP for parameters with different methods: Case 2.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=0.5,\mathit{\alpha }=0.5$				MLEs				SELE				LLF 1				LLF 2
Sample Sizes	p	r		LACI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	3.0894	98.6%	0.0969	0.0969	1.2830	97.5%	0.0435	0.0433	1.4839	95.3%	0.0473	0.0475	1.3547	94.9%	0.0417	0.0418
			$\eta$	1.9416	98.0%	0.0626	0.0627	0.9152	96.8%	0.0330	0.0325	1.0700	95.0%	0.0344	0.0345	0.9710	94.4%	0.0296	0.0295
			$\alpha$	4.2271	98.6%	0.1390	0.1393	1.3156	98.9%	0.0611	0.0573	2.2813	97.7%	0.0765	0.0726	1.4349	95.1%	0.0448	0.0453
		0.9	${\delta }_0$	2.9486	98.6%	0.0935	0.0934	1.1692	98.4%	0.0396	0.0395	1.3177	94.9%	0.0416	0.0402	1.1996	94.7%	0.0376	0.0376
			$\eta$	1.7669	98.0%	0.0573	0.0575	0.8702	98.9%	0.0301	0.0302	1.0165	94.6%	0.0319	0.0324	0.9276	95.2%	0.0281	0.0285
			$\alpha$	3.5048	98.6%	0.1121	0.1116	1.0983	96.3%	0.0363	0.0364	1.3323	94.7%	0.0434	0.0432	1.1170	94.0%	0.0356	0.0353
	0.85	0.75	${\delta }_0$	3.3213	97.8%	0.1073	0.1073	1.3726	97.8%	0.0477	0.0477	1.5130	95.1%	0.0497	0.0497	1.3677	95.4%	0.0431	0.0432
			$\eta$	1.8985	98.8%	0.0611	0.0612	0.8391	96.9%	0.0295	0.0295	1.0268	95.4%	0.0343	0.0341	0.9296	95.0%	0.0293	0.0293
			$\alpha$	5.0493	98.6%	0.1577	0.1561	1.3368	97.2%	0.0456	0.0458	1.5895	95.1%	0.0502	0.0501	1.3381	94.7%	0.0428	0.0428
		0.9	${\delta }_0$	3.1588	98.6%	0.0988	0.1015	1.2876	98.4%	0.0452	0.0455	1.4351	95.2%	0.0446	0.0446	1.3186	94.8%	0.0424	0.0423
			$\eta$	1.7059	97.9%	0.0538	0.0538	0.9113	96.9%	0.0303	0.0308	1.0224	93.9%	0.0326	0.0329	0.9207	93.9%	0.0312	0.0307
			$\alpha$	4.1949	98.6%	0.1309	0.1321	1.3134	97.2%	0.0470	0.0464	1.6084	93.8%	0.0550	0.0542	1.3520	94.1%	0.0425	0.0424
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	1.8752	98.7%	0.0595	0.0596	0.8728	98.5%	0.0262	0.0263	0.9155	94.8%	0.0302	0.0301	0.8596	94.9%	0.0267	0.0266
			$\eta$	1.1487	97.7%	0.0339	0.0338	0.5808	96.6%	0.0201	0.0201	0.6400	95.6%	0.0207	0.0207	0.6021	95.6%	0.0205	0.0195
			$\alpha$	2.0821	98.6%	0.0619	0.0616	0.8702	98.7%	0.0319	0.0319	1.0686	95.0%	0.0352	0.0345	0.9265	94.7%	0.0283	0.0282
		0.9	${\delta }_0$	1.5845	98.4%	0.0497	0.0504	0.7635	98.3%	0.0233	0.0244	0.8052	95.0%	0.0247	0.0246	0.7567	95.1%	0.0234	0.0242
			$\eta$	1.0470	97.5%	0.0336	0.0336	0.5247	98.7%	0.0174	0.0174	0.5738	95.7%	0.0182	0.0187	0.5439	95.8%	0.0176	0.0175
			$\alpha$	1.4293	98.6%	0.0457	0.0458	0.7153	97.0%	0.0277	0.0287	0.9290	95.7%	0.0281	0.0277	0.8217	95.5%	0.0257	0.0258
	0.85	0.75	${\delta }_0$	1.9607	96.9%	0.0608	0.0611	0.8927	98.9%	0.0297	0.0300	0.9607	94.4%	0.0308	0.0302	0.9033	94.2%	0.0280	0.0270
			$\eta$	1.1265	97.4%	0.0354	0.0356	0.5704	96.9%	0.0211	0.0212	0.6190	95.1%	0.0195	0.0196	0.5845	95.2%	0.0178	0.0180
			$\alpha$	1.9211	98.6%	0.0616	0.0607	0.8812	97.2%	0.0312	0.0316	1.0653	94.0%	0.0314	0.0314	0.9379	94.1%	0.0297	0.0298
		0.9	${\delta }_0$	1.6667	96.3%	0.0522	0.0520	0.6926	98.7%	0.0241	0.0244	0.7855	95.6%	0.0253	0.0252	0.7326	95.9%	0.0231	0.0237
			$\eta$	1.0535	97.4%	0.0339	0.0337	0.5151	96.7%	0.0176	0.0177	0.5704	95.5%	0.0177	0.0176	0.5432	95.6%	0.0176	0.0176
			$\alpha$	1.4029	98.6%	0.0438	0.0440	0.6016	97.8%	0.0238	0.0238	0.7740	95.5%	0.0251	0.0251	0.6823	95.3%	0.0224	0.0220
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	1.3900	98.9%	0.0443	0.0445	0.6515	96.5%	0.0223	0.0223	0.7038	95.2%	0.0231	0.0233	0.6688	94.9%	0.0225	0.0223
			$\eta$	0.8957	97.1%	0.0281	0.0283	0.4618	98.9%	0.0154	0.0152	0.5004	95.4%	0.0156	0.0158	0.4809	95.9%	0.0165	0.0165
			$\alpha$	1.2422	98.6%	0.0397	0.0399	0.6190	96.9%	0.0217	0.0217	0.7305	94.6%	0.0224	0.0224	0.6623	94.6%	0.0210	0.0210
		0.9	${\delta }_0$	1.2461	98.7%	0.0405	0.0405	0.5632	98.4%	0.0182	0.0183	0.5831	94.7%	0.0187	0.0189	0.5540	94.9%	0.0181	0.0179
			$\eta$	0.8093	98.0%	0.0249	0.0253	0.4271	96.6%	0.0128	0.0130	0.4343	94.8%	0.0138	0.0138	0.4190	94.8%	0.0133	0.0134
			$\alpha$	0.9571	98.1%	0.0291	0.0289	0.4500	96.5%	0.0168	0.0171	0.5583	95.6%	0.0182	0.0178	0.5089	96.1%	0.0164	0.0165
	0.85	0.75	${\delta }_0$	1.4388	98.9%	0.0468	0.0468	0.6554	96.5%	0.0220	0.0222	0.7063	95.1%	0.0233	0.0234	0.6654	95.3%	0.0206	0.0215
			$\eta$	0.9096	97.6%	0.0294	0.0298	0.4635	97.8%	0.0147	0.0149	0.4974	95.8%	0.0154	0.0154	0.4742	95.7%	0.0140	0.0141
			$\alpha$	1.2094	98.2%	0.0377	0.0371	0.5700	97.2%	0.0212	0.0213	0.7080	95.3%	0.0218	0.0221	0.6383	95.4%	0.0201	0.0198
		0.9	${\delta }_0$	1.2302	98.7%	0.0397	0.0397	0.5669	98.4%	0.0181	0.0180	0.6045	95.1%	0.0182	0.0183	0.5744	94.9%	0.0182	0.0186
			$\eta$	0.8491	97.1%	0.0270	0.0265	0.4126	97.2%	0.0144	0.0144	0.4661	96.8%	0.0155	0.0153	0.4486	96.7%	0.0139	0.0138
			$\alpha$	0.9641	98.2%	0.0303	0.0305	0.4486	96.9%	0.0167	0.0170	0.5768	95.9%	0.0185	0.0184	0.5202	95.9%	0.0164	0.0163

,

Table 12. Interval estimation via LCI and CP for parameters with different methods: Case 3.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=2,\mathit{\alpha }=0.5$				MLEs				SELE				LLF 1				LLF 2
Sample Sizes	p	r		LACI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	3.1271	98.2%	0.1049	0.1023	1.3168	98.4%	0.0432	0.0433	1.4215	94.8%	0.0463	0.0460	1.2946	94.9%	0.0404	0.0408
			$\eta$	1.9638	97.4%	0.0647	0.0636	1.0178	97.6%	0.0341	0.0342	1.0894	95.3%	0.0334	0.0330	1.0533	95.4%	0.0334	0.0334
			$\alpha$	4.7020	98.2%	0.1524	0.1537	1.3062	96.8%	0.0447	0.0467	1.5474	94.3%	0.0489	0.0487	1.3022	95.0%	0.0416	0.0418
		0.9	${\delta }_0$	2.8622	98.2%	0.0866	0.0872	1.1486	98.6%	0.0406	0.0407	1.2809	95.1%	0.0409	0.0409	1.1872	94.9%	0.0359	0.0359
			$\eta$	1.9147	96.9%	0.0604	0.0610	0.9519	98.7%	0.0301	0.0298	0.9986	95.3%	0.0309	0.0312	0.9602	95.3%	0.0305	0.0306
			$\alpha$	3.8339	98.2%	0.1231	0.1221	1.1008	98.9%	0.0405	0.0409	1.3917	95.0%	0.0443	0.0450	1.1905	94.8%	0.0377	0.0365
	0.85	0.75	${\delta }_0$	3.0881	97.0%	0.0966	0.0974	1.2883	97.9%	0.0416	0.0419	1.3758	95.3%	0.0427	0.0425	1.2780	94.8%	0.0368	0.0366
			$\eta$	1.8427	97.5%	0.0593	0.0595	0.8874	98.3%	0.0288	0.0283	0.9275	95.1%	0.0291	0.0284	0.8805	95.3%	0.0277	0.0273
			$\alpha$	5.2875	98.2%	0.1626	0.1634	1.2079	96.7%	0.0457	0.0461	1.5345	95.3%	0.0506	0.0489	1.3316	95.0%	0.0438	0.0424
		0.9	${\delta }_0$	2.9351	98.2%	0.0923	0.0929	1.2060	98.8%	0.0402	0.0406	1.3333	95.2%	0.0419	0.0419	1.2274	95.1%	0.0372	0.0379
			$\eta$	1.7785	97.4%	0.0554	0.0553	0.8561	97.5%	0.0272	0.0272	0.8969	95.0%	0.0297	0.0301	0.8607	95.5%	0.0267	0.0273
			$\alpha$	4.1571	98.2%	0.1296	0.1293	1.1759	96.5%	0.0429	0.0423	1.4898	94.8%	0.0471	0.0469	1.2570	95.3%	0.0408	0.0404
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	1.8049	98.8%	0.0556	0.0544	0.8235	96.3%	0.0279	0.0279	0.9164	95.1%	0.0282	0.0282	0.8697	95.0%	0.0280	0.0285
			$\eta$	1.1846	97.0%	0.0378	0.0385	0.5924	98.2%	0.0185	0.0185	0.5992	94.8%	0.0178	0.0175	0.5866	95.0%	0.0185	0.0185
			$\alpha$	2.1558	98.2%	0.0679	0.0665	0.8485	96.9%	0.0307	0.0295	1.0470	94.5%	0.0338	0.0339	0.9323	95.5%	0.0285	0.0284
		0.9	${\delta }_0$	1.4990	98.3%	0.0467	0.0466	0.6769	96.8%	0.0238	0.0239	0.7689	95.4%	0.0250	0.0250	0.7302	95.6%	0.0246	0.0248
			$\eta$	1.0293	97.4%	0.0328	0.0330	0.4955	98.3%	0.0169	0.0170	0.5197	95.8%	0.0164	0.0164	0.5120	96.1%	0.0162	0.0161
			$\alpha$	1.4397	98.2%	0.0464	0.0460	0.6749	96.8%	0.0246	0.0246	0.8154	94.8%	0.0257	0.0258	0.7284	94.5%	0.0242	0.0245
	0.85	0.75	${\delta }_0$	1.7573	98.5%	0.0550	0.0540	0.8466	98.9%	0.0292	0.0289	0.9036	95.3%	0.0282	0.0286	0.8524	94.9%	0.0264	0.0263
			$\eta$	1.1714	98.1%	0.0366	0.0363	0.5594	98.0%	0.0187	0.0187	0.5905	95.8%	0.0185	0.0187	0.5802	95.9%	0.0188	0.0188
			$\alpha$	1.8111	98.2%	0.0593	0.0591	0.8328	96.5%	0.0297	0.0295	0.9910	94.4%	0.0321	0.0319	0.8758	94.6%	0.0292	0.0289
		0.9	${\delta }_0$	1.4897	98.8%	0.0484	0.0490	0.6826	98.6%	0.0237	0.0227	0.7741	95.7%	0.0256	0.0256	0.7292	96.3%	0.0234	0.0235
			$\eta$	1.0565	98.1%	0.0339	0.0338	0.5194	97.2%	0.0167	0.0167	0.5290	95.1%	0.0176	0.0176	0.5205	95.2%	0.0169	0.0166
			$\alpha$	1.4137	98.2%	0.0470	0.0466	0.6487	96.9%	0.0235	0.0238	0.7828	94.5%	0.0245	0.0246	0.6925	94.7%	0.0225	0.0229
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	1.3040	97.8%	0.0407	0.0405	0.5675	98.9%	0.0197	0.0196	0.6444	96.4%	0.0202	0.0204	0.6184	96.2%	0.0201	0.0202
			$\eta$	0.9035	97.3%	0.0292	0.0289	0.4415	97.9%	0.0145	0.0144	0.4533	95.2%	0.0148	0.0147	0.4483	95.3%	0.0143	0.0144
			$\alpha$	1.2557	98.2%	0.0387	0.0387	0.5749	97.2%	0.0204	0.0197	0.6450	94.8%	0.0197	0.0197	0.5935	94.8%	0.0188	0.0188
		0.9	${\delta }_0$	1.1319	98.5%	0.0370	0.0367	0.5182	96.6%	0.0173	0.0176	0.5710	95.5%	0.0190	0.0190	0.5430	95.5%	0.0177	0.0177
			$\eta$	0.7945	97.8%	0.0243	0.0244	0.3986	97.9%	0.0133	0.0133	0.4046	94.9%	0.0121	0.0121	0.4005	94.9%	0.0126	0.0125
			$\alpha$	0.9915	98.2%	0.0304	0.0308	0.4481	97.8%	0.0188	0.0185	0.5729	95.6%	0.0182	0.0183	0.5191	95.6%	0.0158	0.0162
	0.85	0.75	${\delta }_0$	1.3142	98.4%	0.0411	0.0410	0.6008	96.9%	0.0202	0.0202	0.6692	95.6%	0.0214	0.0217	0.6361	95.8%	0.0187	0.0185
			$\eta$	0.9158	97.7%	0.0285	0.0285	0.4661	98.5%	0.0151	0.0151	0.4712	94.8%	0.0145	0.0145	0.4656	95.1%	0.0152	0.0152
			$\alpha$	1.1662	98.2%	0.0360	0.0361	0.5374	96.7%	0.0231	0.0221	0.7073	96.0%	0.0235	0.0229	0.6385	95.9%	0.0201	0.0196
		0.9	${\delta }_0$	1.1362	97.9%	0.0337	0.0342	0.5005	98.4%	0.0165	0.0165	0.5239	95.0%	0.0163	0.0163	0.5027	95.0%	0.0158	0.0154
			$\eta$	0.8017	97.1%	0.0253	0.0253	0.3903	97.6%	0.0132	0.0132	0.4091	95.9%	0.0122	0.0123	0.4045	96.1%	0.0131	0.0133
			$\alpha$	0.9471	98.1%	0.0289	0.0289	0.4306	97.8%	0.0160	0.0158	0.5201	95.5%	0.0162	0.0166	0.4787	95.6%	0.0155	0.0154

,

Table 13. Interval estimation via LCI and CP for parameters with different methods: Case 4.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=2,\mathit{\alpha }=2.5$				MLEs				SELE				LLF 1				LLF 2
Sample Sizes	p	r		LACI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	1.4266	97.0%	0.0451	0.0448	0.7167	97.3%	0.0254	0.0257	0.8019	96.1%	0.0250	0.0251	0.7732	96.1%	0.0251	0.0255
			$\eta$	1.4389	97.7%	0.0457	0.0459	0.7443	98.0%	0.0244	0.0243	0.7796	95.2%	0.0247	0.0250	0.7549	95.3%	0.0237	0.0237
			$\alpha$	7.6979	98.0%	0.2651	0.2614	4.9648	96.8%	0.1968	0.1926	8.6808	96.1%	0.2765	0.2760	4.1990	94.3%	0.1346	0.1338
		0.9	${\delta }_0$	1.3476	97.4%	0.0409	0.0409	0.6708	98.7%	0.0216	0.0214	0.7235	95.9%	0.0236	0.0236	0.6897	95.7%	0.0218	0.0215
			$\eta$	1.3735	97.2%	0.0438	0.0438	0.7116	97.4%	0.0232	0.0234	0.7391	95.1%	0.0236	0.0235	0.7204	95.1%	0.0232	0.0238
			$\alpha$	7.5106	98.0%	0.2391	0.2399	4.2361	97.2%	0.1611	0.1611	6.6470	95.3%	0.2012	0.2011	3.6434	94.7%	0.1183	0.1192
	0.85	0.75	${\delta }_0$	1.3269	97.6%	0.0448	0.0446	0.7253	98.4%	0.0250	0.0247	0.7871	95.6%	0.0255	0.0255	0.7511	95.7%	0.0235	0.0235
			$\eta$	1.3391	97.1%	0.0414	0.0414	0.6711	98.5%	0.0216	0.0216	0.6780	94.9%	0.0213	0.0214	0.6597	94.7%	0.0213	0.0212
			$\alpha$	8.2669	98.1%	0.2512	0.2527	5.2214	96.9%	0.2047	0.2003	8.3778	94.5%	0.2679	0.2658	4.3818	94.6%	0.1362	0.1352
		0.9	${\delta }_0$	1.2525	97.7%	0.0400	0.0400	0.6453	96.9%	0.0220	0.0218	0.7024	95.5%	0.0209	0.0213	0.6703	95.5%	0.0213	0.0216
			$\eta$	1.3279	97.9%	0.0444	0.0453	0.6559	98.2%	0.0209	0.0205	0.6676	95.1%	0.0207	0.0209	0.6511	94.3%	0.0215	0.0216
			$\alpha$	7.6924	98.3%	0.2469	0.2497	4.6518	97.8%	0.1748	0.1745	6.9099	94.4%	0.2289	0.2279	3.7714	94.1%	0.1207	0.1216
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	0.7996	96.5%	0.0254	0.0256	0.4245	96.7%	0.0135	0.0135	0.4552	96.1%	0.0151	0.0151	0.4443	96.2%	0.0139	0.0143
			$\eta$	0.8685	97.0%	0.0273	0.0273	0.4189	96.5%	0.0141	0.0140	0.4254	94.5%	0.0140	0.0138	0.4213	95.1%	0.0134	0.0134
			$\alpha$	5.5851	98.1%	0.1922	0.1885	3.2683	97.2%	0.1129	0.1117	4.6485	95.6%	0.1481	0.1468	2.9258	94.8%	0.0893	0.0887
		0.9	${\delta }_0$	0.8036	97.5%	0.0274	0.0266	0.3804	98.6%	0.0130	0.0127	0.4105	95.6%	0.0128	0.0127	0.3989	95.6%	0.0129	0.0126
			$\eta$	0.7630	97.1%	0.0243	0.0240	0.3766	97.1%	0.0120	0.0120	0.3789	94.5%	0.0115	0.0118	0.3764	94.5%	0.0117	0.0117
			$\alpha$	4.9993	98.3%	0.1611	0.1579	2.6309	98.1%	0.1024	0.0992	3.9450	95.2%	0.1158	0.1154	2.4618	95.3%	0.0769	0.0770
	0.85	0.75	${\delta }_0$	0.7891	96.9%	0.0250	0.0249	0.4442	96.8%	0.0148	0.0147	0.4764	95.4%	0.0155	0.0155	0.4643	95.4%	0.0148	0.0148
			$\eta$	0.8069	97.4%	0.0230	0.0230	0.3908	98.0%	0.0124	0.0125	0.4091	95.7%	0.0128	0.0128	0.4033	95.8%	0.0132	0.0129
			$\alpha$	4.7597	98.3%	0.1454	0.1449	3.2451	98.1%	0.1146	0.1143	4.6130	94.9%	0.1512	0.1513	2.8671	95.2%	0.0930	0.0874
		0.9	${\delta }_0$	0.8038	97.4%	0.0252	0.0248	0.3976	98.9%	0.0131	0.0132	0.4201	95.0%	0.0136	0.0136	0.4089	95.0%	0.0134	0.0134
			$\eta$	0.7331	97.3%	0.0242	0.0246	0.3649	97.3%	0.0118	0.0117	0.3706	95.4%	0.0112	0.0111	0.3675	95.1%	0.0117	0.0117
			$\alpha$	4.6685	98.3%	0.1583	0.1586	2.4494	97.0%	0.0982	0.0972	3.7507	94.9%	0.1239	0.1217	2.4397	95.8%	0.0786	0.0789
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.6619	97.7%	0.0214	0.0215	0.3439	98.6%	0.0117	0.0117	0.3642	95.6%	0.0114	0.0114	0.3568	95.6%	0.0111	0.0110
			$\eta$	0.6588	97.0%	0.0206	0.0204	0.3279	98.8%	0.0109	0.0110	0.3330	94.3%	0.0103	0.0103	0.3309	94.5%	0.0106	0.0106
			$\alpha$	3.7511	98.9%	0.1173	0.1157	2.2375	96.9%	0.0825	0.0805	3.1056	95.9%	0.0954	0.0973	2.1964	95.8%	0.0696	0.0687
		0.9	${\delta }_0$	0.5962	97.8%	0.0194	0.0197	0.2895	98.3%	0.0091	0.0091	0.3009	95.3%	0.0101	0.0101	0.2945	95.5%	0.0090	0.0091
			$\eta$	0.6192	97.5%	0.0192	0.0195	0.2999	97.7%	0.0098	0.0100	0.3084	95.4%	0.0099	0.0098	0.3072	95.3%	0.0093	0.0094
			$\alpha$	3.1793	96.6%	0.1049	0.1034	1.7232	98.1%	0.0612	0.0617	2.3564	95.8%	0.0720	0.0721	1.6757	95.7%	0.0560	0.0565
	0.85	0.75	${\delta }_0$	0.6719	97.1%	0.0211	0.0213	0.3470	97.8%	0.0113	0.0111	0.3539	94.9%	0.0107	0.0109	0.3462	95.1%	0.0107	0.0106
			$\eta$	0.6577	97.8%	0.0212	0.0212	0.3318	96.9%	0.0099	0.0101	0.3270	94.5%	0.0104	0.0104	0.3255	94.4%	0.0099	0.0099
			$\alpha$	3.7044	96.7%	0.1154	0.1135	2.1893	98.6%	0.0801	0.0800	2.9742	95.2%	0.0908	0.0922	2.1132	95.7%	0.0658	0.0663
		0.9	${\delta }_0$	0.6089	98.0%	0.0192	0.0193	0.2890	97.8%	0.0096	0.0096	0.3084	95.8%	0.0095	0.0095	0.3027	96.0%	0.0097	0.0095
			$\eta$	0.5950	97.5%	0.0191	0.0190	0.2967	97.4%	0.0092	0.0095	0.2975	94.7%	0.0095	0.0096	0.2959	94.9%	0.0093	0.0092
			$\alpha$	3.0653	97.2%	0.1009	0.1007	1.7029	96.3%	0.0565	0.0565	2.1324	94.9%	0.0668	0.0670	1.5838	94.9%	0.0483	0.0478

and

Table 14. Interval estimation via LCI and CP for parameters with different methods: Case 5.

${\mathit{\delta }}_0=1.5,\mathit{\eta }=1.3,\mathit{\alpha }=1.2$				MLEs				SELE				LLF 1				LLF 2
Sample Sizes	p	r		LACI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT	LCCI	CP	LBP	LBT
$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.75	${\delta }_0$	1.1754	97.9%	0.0368	0.0368	0.6338	96.6%	0.0198	0.0198	−0.6286	94.4%	0.0198	0.0199	−0.6049	94.3%	0.0185	0.0187
			$\eta$	0.9718	98.1%	0.0306	0.0307	0.4809	97.9%	0.0160	0.0161	−0.4994	95.0%	0.0161	0.0161	−0.4892	95.2%	0.0150	0.0152
			$\alpha$	3.4945	98.3%	0.1018	0.1019	1.8181	98.1%	0.0676	0.0675	−2.4577	94.9%	0.0763	0.0762	−1.8621	94.3%	0.0593	0.0595
		0.9	${\delta }_0$	1.0841	97.9%	0.0324	0.0323	0.5036	98.4%	0.0165	0.0166	−0.5312	95.1%	0.0172	0.0171	−0.5120	95.3%	0.0155	0.0154
			$\eta$	0.8808	97.4%	0.0288	0.0285	0.4400	98.1%	0.0139	0.0138	−0.4557	94.7%	0.0145	0.0145	−0.4468	95.3%	0.0143	0.0143
			$\alpha$	2.6352	98.3%	0.0848	0.0852	1.2833	98.9%	0.0504	0.0493	−1.6970	94.9%	0.0521	0.0534	−1.3422	95.3%	0.0434	0.0434
	0.85	0.75	${\delta }_0$	1.1905	98.4%	0.0371	0.0370	0.5699	98.7%	0.0195	0.0192	0.6216	95.3%	0.0197	0.0200	0.5963	95.6%	0.0189	0.0193
			$\eta$	0.9663	97.7%	0.0308	0.0307	0.4611	97.8%	0.0150	0.0150	0.4925	95.8%	0.0162	0.0163	0.4825	95.6%	0.0151	0.0151
			$\alpha$	3.5466	98.3%	0.1106	0.1111	1.6578	97.0%	0.0615	0.0612	2.2308	95.5%	0.0726	0.0730	1.6679	94.6%	0.0549	0.0558
		0.9	${\delta }_0$	1.0380	98.9%	0.0338	0.0342	0.5067	98.9%	0.0165	0.0166	0.5328	95.1%	0.0173	0.0173	0.5147	95.1%	0.0163	0.0162
			$\eta$	0.8981	97.6%	0.0272	0.0270	0.4529	97.8%	0.0143	0.0140	0.4604	94.9%	0.0149	0.0150	0.4518	95.0%	0.0134	0.0132
			$\alpha$	2.6659	98.3%	0.0886	0.0893	1.3037	97.0%	0.0461	0.0457	1.6799	95.4%	0.0530	0.0509	1.3337	95.3%	0.0432	0.0432
$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.75	${\delta }_0$	2.2151	98.0%	0.0706	0.0706	0.9596	98.4%	0.0327	0.0327	1.0620	95.1%	0.0333	0.0332	1.0022	95.0%	0.0330	0.0328
			$\eta$	1.7317	96.9%	0.0561	0.0562	0.9394	98.6%	0.0298	0.0297	1.0070	95.6%	0.0323	0.0320	0.9522	95.3%	0.0296	0.0296
			$\alpha$	6.8361	98.3%	0.2197	0.2169	2.6484	96.3%	0.1171	0.1176	4.7879	97.8%	0.1617	0.1600	2.4497	94.2%	0.0778	0.0771
		0.9	${\delta }_0$	1.9912	98.2%	0.0624	0.0637	0.8345	98.5%	0.0287	0.0288	0.9435	96.2%	0.0319	0.0318	0.8850	95.6%	0.0282	0.0283
			$\eta$	1.5842	97.9%	0.0498	0.0499	0.8141	98.1%	0.0257	0.0252	0.8291	94.3%	0.0252	0.0256	0.7885	93.9%	0.0252	0.0253
			$\alpha$	6.1411	98.3%	0.1882	0.1900	2.3551	96.6%	0.0865	0.0877	3.1027	94.2%	0.0993	0.0968	2.2182	94.5%	0.0682	0.0683
	0.85	0.75	${\delta }_0$	2.0296	98.6%	0.0636	0.0642	0.9524	98.4%	0.0315	0.0309	1.0292	94.9%	0.0325	0.0323	0.9720	95.1%	0.0312	0.0304
			$\eta$	1.6464	96.7%	0.0520	0.0518	0.7986	96.5%	0.0254	0.0257	0.8234	94.9%	0.0264	0.0262	0.7804	94.9%	0.0246	0.0248
			$\alpha$	7.9275	98.3%	0.2437	0.2439	3.0210	96.7%	0.1032	0.1047	4.0032	94.4%	0.1312	0.1315	2.7654	94.5%	0.0895	0.0893
		0.9	${\delta }_0$	2.0704	98.6%	0.0669	0.0663	0.8974	98.6%	0.0304	0.0304	1.0196	95.6%	0.0318	0.0322	0.9630	95.5%	0.0299	0.0298
			$\eta$	1.5462	97.7%	0.0496	0.0508	0.7749	97.2%	0.0242	0.0244	0.8068	95.0%	0.0255	0.0250	0.7649	94.7%	0.0248	0.0247
			$\alpha$	6.5427	98.3%	0.1942	0.1940	2.5121	96.8%	0.0936	0.0936	3.4113	93.9%	0.1064	0.1052	2.3402	93.9%	0.0771	0.0755
$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.75	${\delta }_0$	0.9442	98.6%	0.0295	0.0300	0.4389	98.5%	0.0149	0.0149	0.4659	95.0%	0.0147	0.0149	0.4530	95.4%	0.0143	0.0145
			$\eta$	0.7987	97.3%	0.0256	0.0256	0.4081	97.0%	0.0124	0.0129	0.4031	94.0%	0.0130	0.0129	0.3975	94.4%	0.0130	0.0128
			$\alpha$	2.6505	98.3%	0.0845	0.0846	1.2933	96.3%	0.0468	0.0463	1.5639	94.4%	0.0482	0.0483	1.2991	94.9%	0.0431	0.0430
		0.9	${\delta }_0$	0.8564	97.9%	0.0277	0.0278	0.3807	98.1%	0.0121	0.0120	0.3940	94.9%	0.0124	0.0125	0.3828	95.0%	0.0122	0.0121
			$\eta$	0.7390	97.5%	0.0235	0.0233	0.3695	97.9%	0.0124	0.0124	0.3748	95.3%	0.0116	0.0119	0.3704	95.3%	0.0114	0.0116
			$\alpha$	2.0393	98.3%	0.0629	0.0629	0.8710	96.9%	0.0321	0.0329	1.1036	95.6%	0.0352	0.0352	0.9405	95.7%	0.0304	0.0306
	0.85	0.75	${\delta }_0$	0.9003	98.6%	0.0277	0.0283	0.4169	98.1%	0.0133	0.0133	0.4349	95.2%	0.0135	0.0136	0.4219	95.4%	0.0130	0.0130
			$\eta$	0.7878	97.7%	0.0252	0.0253	0.4013	98.1%	0.0129	0.0129	0.4024	94.9%	0.0118	0.0118	0.3974	94.9%	0.0130	0.0131
			$\alpha$	2.2577	98.1%	0.0730	0.0734	0.9993	96.9%	0.0362	0.0363	1.2420	95.4%	0.0394	0.0392	1.0278	95.7%	0.0325	0.0327
		0.9	${\delta }_0$	0.8246	98.0%	0.0251	0.0252	0.3657	96.9%	0.0119	0.0118	0.3827	95.2%	0.0119	0.0120	0.3705	95.3%	0.0121	0.0121
			$\eta$	0.7031	97.7%	0.0223	0.0224	0.3506	98.3%	0.0115	0.0115	0.3562	95.2%	0.0113	0.0114	0.3518	95.1%	0.0108	0.0106
			$\alpha$	1.9840	98.3%	0.0656	0.0647	0.9102	98.1%	0.0345	0.0343	1.1744	95.7%	0.0366	0.0366	0.9783	95.3%	0.0308	0.0311

highlight several key points:

• Generally, the error in both MLEs and Bayesian estimates decreases as the dataset size increases, (except for some variations). This is likely because using more data provides a clearer picture of the underlying relationships.
• In most cases, Bayesian estimates outperform MLEs in terms of accuracy (measured by MSE). This suggests that incorporating prior information about the parameters can be beneficial.
• Among the Bayesian approaches, estimates obtained using the LINEX loss function with c = 2 yield the most accurate results (lowest MSE) for the parameters.
• The width of both approximate and Bayesian confidence intervals generally shrinks as the data size increases (with some exceptions due to data variation). This indicates a more precise range for the true parameter values.
• Compared to approximate confidence intervals, Bayesian credible intervals tend to more accurately capture the true parameter values within the specified confidence level.
• Bootstrap techniques tend to be more accurate in terms of capturing the true parameter values within the specified confidence level.
• The Bayesian credible intervals have a higher probability of covering the true parameter values compared to approximate confidence intervals. This reinforces the potential advantages of the Bayesian approach for reliable parameter estimation.

Table 15. Optimal censoring plan optimization.

r			0.75			0.9
Case	Sample Sizes	p	O1	O2	O3	O1	O2	O3
5	$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	0.9884	0.0005614	116.1080	0.6037	0.0002511	140.6561
		0.85	0.9188	0.0005545	112.3828	0.6105	0.0002578	134.7097
	$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	4.1759	0.0152035	445.6076	3.4573	0.0112355	84.4429
		0.85	6.1962	0.0267276	92.2474	3.7519	0.0136861	75.9936
	$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.4910	0.0001091	182.1843	0.3223	0.0000515	219.5484
		0.85	0.4279	0.0000978	180.6832	0.3208	0.0000517	217.1690
4	$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	7.1771	0.0149412	88.0122	6.6356	0.0123196	67.2185
		0.85	9.8186	0.0236595	52.5360	7.3986	0.0145480	56.5925
	$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	2.6818	0.0008627	139.7062	1.9776	0.0004962	159.7187
		0.85	2.2287	0.0008058	132.7276	1.7438	0.0004534	158.4398
	$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	1.3082	0.0001820	216.9711	0.8967	0.0000929	253.6131
		0.85	1.2211	0.0001766	213.3788	0.8630	0.0000897	254.8714
3	$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	1.6751	0.0136076	426.6626	1.7406	0.0085678	226.1592
		0.85	2.8297	0.0216563	149.9145	1.9004	0.0109613	146.8757
	$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.4956	0.0003454	220.1755	0.3377	0.0001420	266.8394
		0.85	0.4546	0.0003131	213.0106	0.3294	0.0001384	252.9630
	$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.2485	0.0000545	305.6729	0.1872	0.0000252	382.3596
		0.85	0.2352	0.0000491	330.0816	0.1844	0.0000243	408.9906
2	$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	1.7552	0.0114175	469.1266	1.6740	0.0077327	304.0407
		0.85	2.8980	0.0209264	143.1245	1.9560	0.0106086	162.8618
	$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.4926	0.0003214	261.4514	0.3648	0.0001474	288.7144
		0.85	0.4943	0.0003412	227.5894	0.3548	0.0001392	292.3549
	$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.2590	0.0000539	319.1510	0.2013	0.0000267	393.0197
		0.85	0.2559	0.0000533	337.1645	0.1974	0.0000254	402.9686
1	$(n_1=10,n_2=10,n_3=8,n_4=8)$	0.25	1.2615	0.0015968	2428.3892	1.0770	0.0011363	1286.5635
		0.85	1.7256	0.0027836	439.3270	1.0898	0.0014481	463.0750
	$(n_1=30,n_2=30,n_3=25,n_4=20)$	0.25	0.3349	0.0000523	653.1210	0.2203	0.0000227	730.4343
		0.85	0.3117	0.0000514	608.6716	0.2178	0.0000231	661.5870
	$(n_1=50,n_2=50,n_3=40,n_4=30)$	0.25	0.1579	0.0000081	847.6985	0.1206	0.0000042	990.4194
		0.85	0.1522	0.0000079	835.8079	0.1187	0.0000041	990.4676

uses optimality measures to determine the best parameter of binomial removal for this scheme.

9. Summary and Conclusions

This study addressed point and interval estimations for item lifetimes under use conditions following the MOLBE distribution within a progressive-stress ALT framework, utilizing progressive type-II censoring. We employed MLEs and Bayesian methods (under LINEX and SE loss functions) for parameter estimation based on progressive type-II censoring with binomial removal. For the CIs estimated, asymptotic and credible intervals were both obtained. The MCMC technique was used to derive Bayesian estimates of the model parameters. A simulation study was conducted to evaluate the accuracy of the estimates and to compare the CI outputs. Two techniques of bootstrap CI were obtained. Additionally, we examined the three distinct optimum test methods for the suggested model using various optimal criteria. A real dataset was analyzed to test the efficiency of the proposed estimation methods. The results indicate that the MOLBE distribution effectively fits the data and that the estimation methods perform well under progressive-stress ALT with progressive type-II censoring. We recommend using the Bayes estimation technique under the LLF loss function with c values close to 0.5 for estimating the model parameters.

In many different sectors, accelerated life testing has been used to rapidly collect failure time data for test units in a significantly shorter period than testing under typical operating circumstances. A ramp stress ALT under type-II UPHC is taken into consideration in this article when the lifetime of test units follows a truncated Cauchy power exponential distribution. The scale parameter of the distribution follows the inverse power law, and the cumulative exposure model accounts for the effect of fluctuating stress. Using type-II UPHC, the maximum likelihood estimates are compared with the Bayesian estimates of the unknown parameters based on symmetric and asymmetric loss functions via MCMC. We also provide some interval estimators of the unknown parameters, including asymptotic intervals, bootstrap intervals, and highest posterior density intervals. Simulations are used to compare the accuracy of the maximum likelihood estimates with the Bayesian estimates, as well as to evaluate the effectiveness of the proposed confidence intervals for various parameter values and sample sizes. Finally, analysis of real data was examined.