Bayesian Methods for Step-Stress Accelerated Test under Gamma Distribution with a Useful Reparametrization and an Industrial Data Application

Abstract

This paper presents a multiple step-stress accelerated life test using type II censoring. Assuming that the lifetimes of the test item follow the gamma distribution, the maximum likelihood estimation and Bayesian approaches are used to estimate the distribution parameters. In the Bayesian approach, new parametrizations can lead to new prior distributions and can be a useful technique to improve the efficiency and effectiveness of Bayesian modeling, particularly when dealing with complex or high-dimensional models. Therefore, in this paper, we present two sets of prior distributions for the parameters of the accelerated test where one of them is based on the reparametrization of the other. The performance of the proposed prior distributions and maximum likelihood approach are investigated and compared by examining the summaries and frequentist coverage probabilities of intervals. We introduce the Markov Chain Monte Carlo (MCMC) algorithms to generate samples from the posterior distributions in order to evaluate the estimators and intervals. Numerical simulations are conducted to examine the approach’s performance and one-sample lifetime data are presented to illustrate the proposed methodology.

1. Introduction

Step-stress accelerated lifetime testing (SSALT) is an important research area in reliability engineering and product development.

The SSALT allows for the faster evaluation of product reliability by accelerating the aging process. This helps to reduce the time and cost associated with traditional long-term testing methods, which may require months or even years to obtain reliable results. The SSALT approach aims to accelerate the degradation or failure of the specimen to obtain information about its performance under normal operating conditions more quickly.

In SSALT, the test units experience more than one level of stress progressively; that is, once they have sustained a prespecified duration under one stress level, they will be tested under an escalated stress level and so on.

Because the units will be tested under more than one stress level in SSALT, a model is needed to describe the effect of changing stress. A commonly used model in SSALT is the cumulative exposure model (CEM) introduced by [1], which assumes that the remaining life of units depends only on the current cumulative fraction failed and current stress regardless of how the fraction accumulated.

Censored data in which the specific failure timings of all units assigned to test are not known or all units assigned to test have not failed may arise in SSALT for a variety of reasons, including operational failure, device malfunction, expense and time restrictions.

Many works of various models, optimum designs and censored data in SSALT have been studied over the last few decades. Ref. [2] discussed inferential methods for SSALT under Weibull distributed lifetimes with type II censoring. Ref. [3] considered Weibull distributions in the design of simple SSALTs with type I censoring. Ref. [4] analyzed the step-stress model when the lifetimes follow the gamma distribution by considering different censoring schemes. Ref. [5] considered the analysis of simple step-stress accelerated life test data from a Lindley distribution under type I censoring. Ref. [6] proposed a generalized linear mixed-effect model (GLMM) to consider the random group effect in SSALT. Ref. [7] conducted a study on step-stress accelerated life testing for the Burr XII distribution under progressive type II censoring. Ref. [8] provided a procedure for determining optimal designs of simple step-stress accelerated life tests for one-shot devices.

Statistical inference for SSALT models has been developed for various modeling setups and underlying distributional assumptions, mainly through maximum likelihood or Bayesian procedures.

There is a vast amount of literature available on the estimation of the SSALT parameters using the maximum likelihood approach. For the analysis of failure time data, the Bayesian estimating approach has been frequently applied; however, little research has been done on Bayesian inference of the SSALT parameters. We can mention some studies like [9] which proposed a general Bayesian inference approach to the step-stress accelerated life test with type II censoring by assuming that the failure times at each stress level are exponentially distributed. Ref. [10] provided a Bayesian analysis of a simple SSALT under a Weibull lifetime distribution based on different censoring schemes. Ref. [11] proposed a general Bayesian analysis for SSALT and discussed its use in SSALT planning. Ref. [12] investigated the order-restricted Bayesian estimation for simple step-stress accelerated life tests. Ref. [13] designed Bayesian sampling plans for a simple step-stress of accelerated life test on censored data. Ref. [14] provided a Bayesian analysis for step-stress accelerated life testing under progressive interval censoring. More recently, ref. [15] considered a simple step-stress accelerated test assuming a cumulative exposure model with uncensored lifetime data following a Weibull distribution using Bayesian inference under maximal data information prior (MDIP) and gamma priors for the unknown parameters.

There are many practical applications for the Bayesian approach in reliability analysis, particularly when dealing with small sample sizes and the presence of censored data. An important consideration to be highlighted is that the Bayesian approach requires less sample data than traditional inference methods such as MLE. Bayesian methods offer the advantage of incorporating prior information, which can help mitigate the effects of small sample sizes and lead to more reliable parameter estimates. Additionally, techniques like reparametrization can further improve the performance of inference methods by addressing specific challenges associated with the modeling process.

The choice of prior distribution in Bayesian reliability analysis mainly for censored data problems is vital and challenging. The use of objective priors as Jeffreys and reference priors (see [16,17,18]) with censored data practically does not exist due to the dependence on the expected Fisher matrix, which cannot be obtained analytically. In these problems, the prior independence of the parameters is assumed, whose distributions are chosen as weakly informative (vague prior), that is, probability distributions with small or large values for the hyperparameters depending on the range of the model parameters, in order to be nearly flat over a large range of the parameter space. Ref. [19] provides Bayesian methods to find prior distributions to fully specify the model for the practical realities of reliability data. Ref. [20] presents a multiple SSALT with a gamma distribution for lifetime using type II censored data, in which the Bayesian estimators for the parameters are obtained based on different loss functions and a comparison with the usual maximum likelihood approach is carried out. We also refer to [21,22,23,24,25,26,27].

In this paper, we analyze the SSALT with samples from the gamma distribution under the CEM and type II censored data.

The gamma distribution plays a significant role in reliability analysis due to its flexibility and ability to model various types of failure rates. It is often used to analyze lifetime data where the time to failure is positively skewed, providing a better fit for many real-world data sets than other distributions. When modeling data that shows heavy tails (i.e., a higher probability of extreme values), the gamma distribution can provide a better fit.

The maximum likelihood estimation and Bayesian approaches are used to obtain the estimators for the SSALT model. We first consider the MLE and derive asymptotic confidence intervals of the model parameters. Applying Bayesian inference in statistical analysis requires the specification of a prior distribution for the model parameters. The specifications of the proposed prior distributions are intentionally chosen as vague so that the choice of prior will not heavily influence the posterior distribution. Furthermore, in the Bayesian context, it is important to check whether the use of different parametrizations improves the estimation of the parameters of interest or leads to some computational advantage, or both.

It is well known that the suitability and effectiveness of numerical and analytical methods for applying parametric Bayesian techniques largely depend on the chosen parametrization when defining the likelihood and prior distributions. There are numerous works in the literature dealing with reparametrization in Bayesian estimation. Ref. [28] discusses examples in censored and truncated data, mixture modeling, multivariate imputation, stochastic processes and multilevel models with Bayesian inference under reparametrizations. Ref. [29] proposes a new unconstrained parametrization in order to provide a natural prior specification as well as a simple implementation of a reversible-jump MCMC in a Bayesian Dirichlet mixture model for multivariate extremes. Ref. [30] studies the consequences of parametrization dependence of the noninformative Bayesian analysis and demonstrates how it can significantly affect the analysis, in particular on prediction, and can lead to strikingly different managerial decisions. Ref. [31] studies the benefits of reparametrization for the Poisson process characterization of extremes in a Bayesian context. They show that the orthogonal parametrization improves the performance of MCMC algorithms in terms of convergence and it also facilitates the derivation of priors, such as Jeffreys and an informative variant on the shape parameter using penalized complexity (PC) priors. Ref. [19] provides an overview of methods for selecting noninformative prior distributions for parameters of basic lifetime distributions in reliability theory. In their paper, the importance of reparametrization is discussed in the case of a small number of failures considering the lognormal and Weibull distributions.

In this paper, we propose a Bayesian approach to estimate the unknown parameters and we also identify an alternative and more accurate parametrization by comparing with the traditional parameters presented in [20]. We have shown that, with SSALT under heavy censoring (i.e., only a small fraction failing), the results with the proposed reparametrization are more accurate than the current methods presented in the literature.

The MCMC (Markov Chain Monte Carlo) techniques are also proposed to perform Bayesian inference to evaluate the marginal posterior of the main parameters and other unobserved quantities of interest derived from them.

The rest of this paper is organized as follows. Section 2 describes the SSALT under the cumulative exposure model and gamma lifetime model. In Section 3, we review the SSALT model under type II censored sample. Section 4 derives the maximum likelihood estimators under type II censored data and the confidence intervals of the unknown parameters. Bayesian analysis of the unknown parameters is provided in Section 5. An implementation of the MCMC Algorithm is considered in Section 6. We carry out simulations in Section 7 to investigate the performance of the proposed estimation approaches. In Section 8, a real-life data example is provided for illustrative purposes. Finally, concluding remarks are provided in Section 9. The Fisher information matrix of the model and its expectation are given in Appendix A and Appendix B, respectively.

2. Description of the Model

We assume that the failure time data come from the m-step-stress model with ordered stress levels $x_1<x_2<\dots <x_m$ ($m\ge 2$); let ${\tau }_1<{\tau }_2<\dots <{\tau }_{m-1}$be the pre-fixed times to change stress used in the SSALT.

According to the cumulative exposure model defined by ref. [1], the cumulative distribution function for the test is given by the following:

$$G\left(t\right)=\left\{\begin{array}{cc}{F}_1\left(t\right)\hfill & 0<t\le {\tau }_1\hfill \\ ⋮\hfill & ⋮\hfill \\ F_i(t-{\tau }_{i-1}+{\epsilon }_{i-1})\hfill & {\tau }_{i-1}<t\le {\tau }_i\hfill \\ ⋮\hfill & ⋮\hfill \\ F_m(t-{\tau }_{m-1}+{\epsilon }_{m-1})\hfill & {\tau }_{m-1}<t,\hfill \end{array}\right.$$

(1)

where $F_i$(t) is the cumulative distribution function (cdf) of the failure time at stress $x_i$, ${\tau }_i$is the time to change stress, and the equivalent starting time, ${\epsilon }_{i-1}$, is the solution of the equation

$$F_i\left({\epsilon }_{i-1}\right)=F_{i-1}({\tau }_{i-1}-{\tau }_{i-2}+{\epsilon }_{i-2}) , i=2, \dots , m.$$

(2)

The equivalent starting time, ${\epsilon }_{i-1}$, is the solution of Equation (2) given by

$${\epsilon }_{i-1}=\left({\tau }_{i-1}-{\tau }_{i-2}+{\epsilon }_{i-2}\right)\frac{{\theta }_i}{{\theta }_{i-1}} , i=2, \dots , m,$$

(3)

with ${\tau }_0={\epsilon }_0=0$.

Initially, n units are tested at a lower stress level $x_1$. The test is run until time${\tau }_1$, when the stress level is increased to $x_2$ and the life test continues until a pre-specified $r_2$ (≤n) number of failures are observed.

Suppose that the failure time $T$at each stress level $x_i$has a gamma distribution with common shape parameter $\alpha$ and scale parameter ${\theta }_i$, with probability density function (pdf) and cdf given by

$$f_i\left(t\right)=\frac{1}{\Gamma \left(\alpha \right){\theta }_i^{\alpha }}{t}^{\alpha -1}{e}^{-\frac{t}{{\theta }_i}} and F_i\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t\frac{1}{{\theta }_i^{\alpha }}{u}^{\alpha -1}{e}^{-\frac{\mathrm{u}}{{\theta }_i}}du,i=1,\dots ,m,$$

(4)

respectively, and $i=0$ denotes the normal operating condition. Thus, the cumulative distribution function (cdf) of the lifetime T for the SSALT is given by

$$G\left(t\right)=IG\left(t_i^{*}\right) , {\tau }_{i-1}<t\le {\tau }_i$$

(5)

and the corresponding density function of $T_j$ is given by

$$g\left(t\right)=\frac{1}{\Gamma \left(\alpha \right){\theta }_i}{t_i^{*}}^{\alpha -1}{e}^{-t_i^{*}} , {\tau }_{i-1}<t\le {\tau }_i,$$

(6)

where $t_i^{*}=\frac{t-{\tau }_{i-1}}{{\theta }_i}+ {\sum }_{j=1}^{i-1}\frac{{\Delta }_j}{{\theta }_j}$with ${\Delta }_l={\tau }_l-{\tau }_{l-1}$, for $i=1, \dots , m$ and ${\tau }_0=0$. Also, $IG\left(x\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^x{u}^{\alpha -1}{e}^{-u}du$ is the incomplete gamma ratio.

We consider a reparametrization of the m-step-stress model, in which ${\theta }_i$ is assumed to satisfy a log-linear link function of the form

$$log {\theta }_i={\beta }_0+{\beta }_1{x}_i, i=1,\dots , m,$$

(7)

where ${\beta }_0$and ${\beta }_1$ are unknown parameters and we need to develop inference only for these two parameters instead of for the original m parameters ${\theta }_i$, $i=1$, 2, …, m. Therefore, the failure time of this SSALT has the cdf $G\left(t\right)$given by

$$G\left(t\right)=IG\left({\zeta }_i\right) , {\tau }_{i-1}<t\le {\tau }_i,$$

(8)

$i=1, \dots , m$and ${\zeta }_i=(t-{\tau }_{i-1})e^{-({\beta }_0+{\beta }_1{x}_i)}+$${\sum }_{j=1}^{i-1}{\Delta }_j e^{-({\beta }_0+{\beta }_1{x}_i)}$.

The average lifetime that the unit is expected to operate $E\left(T\right)$at the SSALT under the gamma distribution is given in Appendix B.

Once the model has been formulated, then the parameters $\alpha$, ${\beta }_0$, and ${\beta }_1$ can be estimated using the maximum likelihood and Bayesian approaches.

3. SSALT Model under Type II Censored Sample

The type II censored sample is formed by terminating the life-testing experiment when a specified number of failures “r” are observed and the remaining “$n-r$” units are censored. Fixing the number of failures would make a test unit’s failure time random. So, the termination time would also be unknown prior to the experiment, which is a disadvantage of type II censoring. However, it has the advantage of yielding the required number of failures from the life test.

In a type II censoring scheme, only the “r” smallest ordered lifetimes in a random sample of size “n” are observed (fixed r, $1\le r\le n$).

The likelihood function based on “r” failure times, $t_{\left(1\right)}, \dots , t_{\left(r\right)}$, is given by

$$L=\frac{n!}{(n-r)!}f\left(t_{\left(1\right)}\right)f\left(t_{\left(2\right)}\right)\dots f\left(t_{\left(r\right)}\right){\left[S\left(t_{\left(r\right)}\right)\right]}^{n-r},$$

(9)

where $f\left(t\right)$ is the pdf of the random variable T, $S\left(t\right)=P\{T>t\}$is the reliability function and $t_{\left(r\right)}$ is the lifetime corresponding to the occurrence of the r-th failure.

For example, imagine a company manufacturing electronic capacitors that wants to estimate the life span and reliability of their capacitors under normal operating conditions. Testing under normal conditions would take too long, so they opt for SSAT combined with type II censoring. The objective is to estimate the mean time to failure (MTTF) and the reliability function of the capacitors. The company selects a sample size of 30 capacitors and decides to stop the test after 10 failures have occurred. The test begins by identifying the stress factors, such as temperature and voltage, and defining the stress levels. The capacitors are initially tested at 85 °C. Every 200 h, the temperature is increased by 10 °C. As the test progresses, the failure times for each capacitor are recorded. When 200 h have passed, the temperature is increased to 95 °C for the remaining capacitors that have not yet failed. This process continues, with the temperature being incrementally increased (e.g., to 105 °C, 115 °C, etc.), until 10 capacitors have failed, at which point the test is stopped according to type II censoring. Throughout the test, failure times at each stress level are collected, and the times at which the remaining capacitors were tested when the test was stopped are recorded as censored times. The failure times are then analyzed using a statistical model, such as the gamma model. An acceleration model adjusts for the different stress levels, allowing the data to be extrapolated back to normal operating conditions. This analysis provides an estimate of the MTTF and constructs the reliability function in normal operating conditions.

In the m-step-stress model with type II censoring, we start with n identical units placed simultaneously on the SSALT. Each unit will be subjected to an initial stress level $x_1$. After that, the experiment is run until a fixed time denoted by ${\tau }_1$, at which time the stress level is changed to $x_2$, and the successive failure times are recorded. Then, at the fixed time ${\tau }_2$, the stress is increased to $x_3$ and so on. Based on this, the stress level starts with $x_1$ and changes to $x_2$, $x_3$, …, $x_m$ at fixed times ${\tau }_1$, ${\tau }_2$, …, ${\tau }_m$, respectively. The experiment is terminated when a fixed number of failures r are observed. Let $n_k$ be the number of units that fail between ${\tau }_{k-1}$ and ${\tau }_k$ at stress level $x_k$, for $k=1$, 2, …, m.

The observed censored sample is given by the ordered failure times denoted by

$$t_1<\dots <t_{n_1}<{\tau }_1\le t_{n_1+1}<\dots <t_{n_1+n_2}<{\tau }_2\le \dots <{\tau }_m\le t_{n_1+\dots +n_{m-1}+1}<\dots <t_r.$$

(10)

4. Maximal Likelihood Estimation

Considering the observed type II censored data given in (10), we can obtain the likelihood function, and then the maximum likelihood estimates (MLEs) of the unknown parameters $\alpha$, ${\beta }_0$ and ${\beta }_1$ from it. The likelihood function based on the censored data in (10) is given by

$$L(\alpha ,{\beta }_0,{\beta }_1 | \mathit{t})=\frac{n!}{(n-r_m)!} \prod _{k=1}^m\left( \prod _{i_k=r_{k-1}+1}^{r_k}{g}_k\left({\zeta }_{i_k}^{*}\right)\right){\left(1-IG\left({\zeta }_{r_m}^{*}\right)\right)}^{n-r_m},$$

(11)

where $r_{\mathrm{o}}=0$, $r_k={\sum }_{i=1}^k{n}_i ,r_m=r$, $\mathit{t}$ is the vector of observed failure time data and ${\zeta }_{i_k}^{*}=(t_{i_k}-{\tau }_{k-1})e^{-({\beta }_0+{\beta }_1{x}_k)}+ {\sum }_{j=1}^{k-1}({\tau }_j-{\tau }_{j-1})e^{-({\beta }_0+{\beta }_1{x}_j)}$, for $k=1, 2,\dots ,m$.

Let us assume, without loss of generality, that ${\tau }_j-{\tau }_{j-1}=\tau$, that is, the ${\tau }_1=\tau , {\tau }_2=2\tau$, …, ${\tau }_{k-1}=(k-1)\tau$.

Since we are considering the cumulative exposure model, then the likelihood function of $\alpha$, ${\beta }_0$ and ${\beta }_1$, based on the observed type II censored data, is given by

$$\begin{array}{ccc}\hfill L(\alpha ,{\beta }_0,{\beta }_1|\mathit{t})& \propto & \frac{1}{{\left(\Gamma \left(\alpha \right)\right)}^{r_m}}exp\left\{-\sum _{k=1}^m{n}_k\left({\beta }_0-{\beta }_1{x}_k\right)\right\}\hfill \\ & \times & \left[\prod _{k=1}^m\left(\prod _{i_k=r_{k-1}+1}^{r_k}{\zeta }_{i_k}^{* \alpha -1}{e}^{- {\zeta }_{i_k}}\right)\right]{\left[1-IG\left({\zeta }_{r_m}^{*}\right)\right]}^{n-r},\hfill \end{array}$$

(12)

It is convenient to work with the log-likelihood function rather than the likelihood function in (12), which is given by

$$\begin{array}{ccc}\hfill l(\alpha ,{\beta }_0,{\beta }_1|\mathit{t})& \propto & -r_{m}ln(\Gamma \left(\alpha \right))-\sum _{k=1}^m{n}_k({\beta }_0-{\beta }_1{x}_k)+(\alpha -1)\hfill \\ & \times & \sum _{k=1}^m\sum _{i_k=r_{k-1}+1}^{r_k}ln\left({\zeta }_{i_k}^{*}\right)-\sum _{k=1}^m\sum _{i_k=r_{k-1}+1}^{r_k}{\zeta }_{i_k}^{*}+(n-r_m)\hfill \\ & \times & ln\left(1-IG\left({\zeta }_{r_m}^{*}\right)\right).\hfill \end{array}$$

(13)

Differentiating the log-likelihood function in (13) with respect to $\alpha$, ${\beta }_0$ and ${\beta }_1$, we obtain the following likelihood equations, which need to be solved for finding the MLE of $\alpha$, ${\beta }_0$ and ${\beta }_1$.

The first partial derivatives are given by the following equations:

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial \alpha }l(\alpha ,{\beta }_0,{\beta }_1|\mathit{t})& =& -r_m\psi \left(\alpha \right)+ \sum _{k=1}^m\sum _{i_k=r_{k-1}+1}^{r_k}ln\left({\zeta }_{i_k}^{*}\right)+\frac{(n-r_m)}{1-IG\left({\zeta }_{r_m}^{*}\right)}\hfill \\ & \times & \left(\psi \left(\alpha \right)IG\left({\zeta }_{r_m}^{*}\right)-B_1\left({\zeta }_{r_m}^{*}\right)\right),\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\beta }_0}l(\alpha ,{\beta }_0,{\beta }_1|\mathit{t})& =& -\sum _{k=1}^m{n}_k+(\alpha -1)\sum _{k=1}^m\sum _{i_k=r_{k-1}+1}^{r_k}(-1)+\sum _{k=1}^m\sum _{i_k=r_{k-1}+1}^{r_k}{\zeta }_{i_k}^{*}\hfill \\ & -\hfill & \frac{(n-r_m){\left({\zeta }_{r_m}^{*}\right)}^{\alpha -1}{e}^{-{\zeta }_{r_m}^{*}}}{\Gamma \left(\alpha \right)(1-IG({\zeta }_{r_m}^{*}\left)\right)}\hfill \end{array}$$

(15)

and

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial {\beta }_1}lnL(\alpha ,{\beta }_0,{\beta }_1|t)& =& \sum _{k=1}^m{n}_k{x}_k+(\alpha -1)\sum _{k=1}^m\sum _{i_k=r_{k-1}+1}^{r_k}(-\frac{A_1\left(t_{i_k}\right)}{{\zeta }_{i_k}^{*}})\hfill \\ & -& \sum _{k=1}^m\sum _{i_k=r_{k-1}+1}^{r_k}\left(-A_1\left(t_{i_k}\right)\right)\hfill \\ & -& \frac{(n-r_m){\left({\zeta }_{r_m}^{*}\right)}^{\alpha -1}{e}^{-{\zeta }_{r_m}^{*}}}{\Gamma \left(\alpha \right)(1-IG({\zeta }_{r_m}^{*}\left)\right)}{A}_1(t_{r_m}).\hfill \end{array}$$

(16)

where $\psi \left(\alpha \right)=\frac{{\Gamma }^{\prime }\left(\alpha \right)}{\Gamma \left(\alpha \right)}$, $B_1\left(x\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^{x}ln\left(u\right)u^{\alpha -1}{e}^{-u}du$ and $A_1\left(t_{i_k}\right)=(t_{i_k}-{\tau }_{k-1})x_k$$e^{-({\beta }_0+{\beta }_1{x}_k)}+\tau e^{-{\beta }_0} {\sum }_{j=1}^{k-1}{x}_j{e}^{-{\beta }_1{x}_j}.$

The maximum likelihood estimators must be derived numerically because there is no obvious simplification of the non-linear likelihood equations. Here, a numerical likelihood maximization was carried out on the log-likelihood using R software.

Estimating confidence intervals for $\alpha$, ${\beta }_0$ and ${\beta }_1$ can be obtained by asymptotic normal approximation of the MLE in large samples, that is,

$$(\widehat{\alpha },{\widehat{\beta }}_0,{\widehat{\beta }}_1)\sim N_3((\alpha ,{\beta }_0,{\beta }_1), I^{-1}(\widehat{\alpha },{\widehat{\beta }}_0,{\widehat{\beta }}_1)),n\to \infty ,$$

(17)

where$I$($\alpha$, ${\beta }_0$, ${\beta }_1$) is the observed Fisher’s information matrix given by $I_{ij}$($\alpha$, ${\beta }_0$, ${\beta }_1$) = $-({\nabla }^{2}l$($\alpha$, ${\beta }_0$, ${\beta }_1$)). Elements of Fisher’s information matrix are given in Appendix A.

Approximate confidence intervals for the individual parameters $\alpha$, ${\beta }_0$ and ${\beta }_1$ with confidence coefficient 100(1$-\gamma$)% are given by

$$\widehat{\alpha }\pm z_{\frac{\gamma }{2}}\sqrt{V_{11}},$$

$${\widehat{\beta }}_0\pm z_{\frac{\gamma }{2}}\sqrt{V_{22}},$$

and

$${\widehat{\beta }}_1\pm z_{\frac{\gamma }{2}}\sqrt{V_{33}},$$

where V_ii is the diagonal element of $I^{-1}$($\widehat{\alpha }$, ${\widehat{\beta }}_0$, ${\widehat{\beta }}_1$) and $z_{\frac{\gamma }{2}}$ indicates the q-th upper percentile of the standard normal distribution.

5. Bayesian Estimation

In a Bayesian framework, the inference is based on the information provided by the posterior distribution of the parameters, denoted as $p(\alpha$, ${\beta }_0$, ${\beta }_1 |$$\mathit{t}$), which is obtained as the product of the likelihood function (12) and the prior $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ resulting in the following:

$$\begin{array}{c}\hfill \begin{array}{ccc}p(\alpha ,{\beta }_0,{\beta }_1|\mathit{t})\hfill & \propto \hfill & \pi (\alpha ,{\beta }_0,{\beta }_1)\frac{1}{{\left(\Gamma \left(\alpha \right)\right)}^{r_m}}exp\left\{-{\displaystyle \sum _{k=1}^m}{n}_k({\beta }_0+{\beta }_1{x}_k)\right\}\hfill \\ \\ & \times \hfill & \left[{\displaystyle \prod _{k=1}^m}\left({\displaystyle \prod _{i_k=r_{k-1}+1}^{r_k}}{\left({\zeta }_{i_k}^{*}\right)}^{\alpha -1}{e}^{-{\zeta }_{i_k}^{*}}\right)\right]{\left[1-IG\left({\zeta }_{r_m}^{*}\right)\right]}^{n-r_m}.\hfill \end{array}\end{array}$$

(18)

In this way, we need to choose an appropriate prior distribution for the parameters $\alpha$, ${\beta }_0$ and ${\beta }_1$ of the model, especially in the situations where we do not have expert opinion to build the prior. The specifications of these prior distributions can be intentionally noninformative so that the choice of prior will not heavily influence the posterior distribution.

We can assume that the parameters are independent variables a priori; then, the joint prior is given by $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)=\pi \left(\alpha \right)\pi \left({\beta }_0\right)\pi \left({\beta }_1\right)$.

Firstly, it is convenient to use normal priors for the regression coefficients. So, it is assumed

$${\beta }_0\sim N({\mu }_0,{\sigma }_0^2) and {\beta }_1\sim N({\mu }_1, {\sigma }_1^2),$$

(19)

where the hyperparameters ${\mu }_0$, ${\sigma }_0^2$, ${\mu }_1$and ${\sigma }_1^2$ are known.

Using normal priors with zero mean and large variance for regression coefficients in Bayesian inference offers several advantages. Firstly, such priors are considered noninformative or weakly informative. This means that, before seeing the data, we do not have strong beliefs about the values of the coefficients, allowing the data to primarily drive the posterior estimates. This approach is particularly useful when we want our conclusions to be heavily data driven rather than influenced by prior assumptions. A prior centered at zero is also highly interpretable. It implies that, a priori, the regression coefficients are expected to be around zero. This is often a reasonable assumption in practice, suggesting that, without data, we assume no effect of the predictors. This baseline assumption helps in building more realistic models that do not overestimate the importance of predictors without substantial evidence from the data. Moreover, while a large variance makes the prior weakly informative, it still provides some level of regularization. This regularization prevents overfitting by slightly shrinking the coefficient estimates towards zero, especially when the sample size is small or the predictors are highly collinear.

It is typically hard to specify informative beliefs about a scale parameter $\alpha$. As the parameter $\alpha$ has a positive support, we can assign a weakly informative prior like a gamma prior with small values of the shape and rate parameters. We consider the prior distribution

$$\alpha \sim \Gamma (a, b),$$

(20)

with a = b = 0.01 or 0.001 suggesting little prior information about the parameter $\alpha$.

A gamma distribution with very small shape (a) and rate (b) parameters (e.g., both set to 0.01 or 0.001) is nearly flat over a large range of the parameter space, that is, the distribution is spread out over a wide range, making it a weakly informative prior. This allows the data to play a dominant role in shaping the posterior distribution rather than the prior assumptions. This is particularly useful when prior knowledge is vague or non-existent about the parameter of interest, as it prevents the prior from overwhelming the likelihood derived from the data.

Therefore, based on the prior distribution for each parameter in (19) and (20), the joint prior density of the parameters $(\alpha$, ${\beta }_0$, ${\beta }_1)$ is given by

$$\pi (\alpha ,{\beta }_0,{\beta }_1)\propto {\alpha }^{\mathrm{a}-1}exp\left(-\frac{1}{2}\left[{\left(\frac{{\beta }_0-{\mu }_1}{{\sigma }_1}\right)}^2+{\left(\frac{{\beta }_1-{\mu }_2}{{\sigma }_2}\right)}^2\right]-\frac{\alpha }{b}\right).$$

(21)

By multiplying the likelihood function (12) by the prior (21), we obtain an expression for the posterior density given by

$$\begin{array}{c}\hfill \begin{array}{ccc}p(\alpha ,{\beta }_0,{\beta }_1 |\mathit{t})\hfill & \propto \hfill & \frac{{\alpha }^{\mathrm{a}-1}}{{\left(\Gamma \left(\alpha \right)\right)}^{r_m}}exp(-\frac{1}{2}\left[{\left(\frac{{\beta }_0-{\mu }_1}{{\sigma }_1}\right)}^2+{\left(\frac{{\beta }_1-{\mu }_2}{{\sigma }_2}\right)}^2\right]-\frac{\alpha }{b}\hfill \\ \\ & -\hfill & \sum _{k=1}^m{n}_k({\beta }_0+{\beta }_1{x}_k))\left[\prod _{k=1}^m\left( \prod _{i_k=r_{k-1}+1}^{r_k}{\zeta }_{i_k}^{* \alpha -1}{e}^{-{\zeta }_{i_k}^{*}}\right)\right]\hfill \\ \\ & \times \hfill & {\left[1-IG\left({\zeta }_{i_k}^{*}\right)\right]}^{n-r_m}.\hfill \end{array}\end{array}$$

(22)

When the inferential results are not very accurate, different parametrizations can improve the estimation. One important problem for all statisticians is to find a one-to-one transformation from the parameter vector $\mathit{\theta }$ to $\mathit{\psi }$ such that the likelihood or posterior density of $\mathit{\psi }$ is better behaved than in the parametrization $\mathit{\theta }$. However, in general, we have great difficulties in finding a reparametrization in the multiparameter case. In practical work, we usually try different parametrizations and based on the simulation study decide by the best parametrization in the multiparameter model.

In order to obtain better estimators under the Bayesian approach of view, we also use the reparametrization

$${\psi }_0=e^{{\beta }_0} \mathrm{and}{\psi }_1=e^{{\beta }_1}.$$

(23)

We could choose a probability distribution with a positive real support that is known to have heavy tails compared to another distribution with the same location and the corresponding priors designed for ${\psi }_0$and ${\psi }_1$ given by

$${\psi }_0\sim \Gamma (c_1,d_1) \mathrm{and}{\psi }_1\sim \Gamma (c_2,d_2),$$

(24)

where the hyperparameters “$c_i$” and “$d_i$”, $i=1,2$ are known.

With this transformation, the proposed joint prior for the parameter vector ($\alpha$,${\psi }_0$,${\psi }_1$) is given by

$$\pi (\alpha ,{\psi }_0,{\psi }_1)\propto {\alpha }^{\mathrm{a}-1}{\psi }_0^{c_1-1}{\psi }_1^{d_1-1}exp\left(-\left(\frac{\alpha }{b}+\frac{{\psi }_0}{{\mathrm{d}}_1}+\frac{{\psi }_1}{{\mathrm{d}}_2}\right)\right).$$

(25)

Again, we consider absence of prior information under the gamma distribution for hyperparameter values 0.01.

In this new parametrization, the resulting posterior density is given by

$$\begin{array}{ccc}p(\alpha ,{\psi }_0,{\psi }_1 |\mathit{t})\hfill & \propto \hfill & \frac{{\alpha }^{\mathrm{a}-1{\psi }_0^{c_1-1}{\psi }_1^{d_1-1}}}{{\left(\Gamma \left(\alpha \right)\right)}^{r_m}}exp\left(-{\sum }_{k=1}^m{n}_k(ln {\psi }_0+x_k ln {\psi }_1)\right)\hfill \\ & \times & \left[ {\prod }_{k=1}^m\left( {\prod }_{i_k=r_{k-1}+1}^{r_k}{\zeta }_{i_k}^{*} \alpha -1e^{-{\zeta }_{i_k}^{*}}\right)\right]{\zeta }^{\ast \alpha -1}\hfill \\ & \times \hfill & {\left[1-IG\left({\zeta }_{i_k}^{*}\right)\right]}^{n-r_m}.\hfill \end{array}$$

(26)

Since the posteriors (22) and (26) are not a familiar probability distribution, one needs to use an MCMC algorithm to obtain simulated draws from the posterior.

6. Implementation of the MCMC Algorithm

As we are not able to find an analytic expression for marginal posterior distributions and hence to extract the characteristics of the parameters, such as the Bayes estimator and credible intervals, we need to appeal to the MCMC algorithm; specifically, we use the Metropolis–Hastings method to obtain a sample of values of ${\theta }_1=(\alpha ,{\beta }_0,{\beta }_1)$ and ${\theta }_2=(\alpha ,{\psi }_0,{\psi }_1)$from the joint posteriors (22) and (26) under both sets of prior distributions (21) and (25), respectively.

Specifically, we run an algorithm for simulating a long chain of draws from the posterior distribution, and base inferences on posterior summaries of the parameters or functionals of the parameters calculated from the samples.

In implementing the MCMC algorithm, a chain is run for R iterations with a burn-in period of $R_o$ (values were discarded) to decrease the effect of initial conditions and the remaining samples were used to calculate the statistics of the marginal posterior distributions.

We now describe the implementation of both the MCMC algorithms used in this work as given below.

6.1. MCMC Algorithm for Parameter Vector ${\mathit{\theta }}_{\mathbf{1}}=(\mathit{\alpha },{\mathit{\beta }}_{\mathbf{0}},{\mathit{\beta }}_{\mathbf{1}})$

Step (1) Choose starting values ${\alpha }^0$, ${\beta }_0^0$and ${\beta }_1^0$;

Step (2) Set $i=1$;

Step (3) Generate a new value ${\alpha }^{i+1}$conditional on the current ${\alpha }^i$from the gamma distribution $\Gamma$(${\alpha }^i$/$v_1$, $v_1$);

Step (4) The candidate ${\alpha }^{i+1}$ will be accepted with a probability given by the Metropolis ratio

$$prob1({\alpha }^i , {\alpha }^{i+1})=min\left\{1,\frac{\Gamma ({\alpha }^i/v_1,v_1) p\left({\alpha }^{i+1},{\beta }_0^i,{\beta }_1^i | \mathit{t} \right)}{\Gamma ({\alpha }^{i+1}/v_1,v_1) p\left({\alpha }^i,{\beta }_0^i,{\beta }_1^i | \mathit{t} \right)}\right\};$$

(27)

Step (5) Generate a uniform value $u_1$ on range 0 to 1, i.e., $u_1\sim U$[$0,$ 1];

Step (6) If $u_1<prob1({\alpha }^i$,${\alpha }^{i+1})$, accept the candidate value and set${\alpha }^{i+1}= {\alpha }^{i+1}$; otherwise, reject and set ${\alpha }^{i+1}= {\alpha }^i$;

Step (7) Generate the new value ${\beta }_0^{i+1}$from the normal distribution N(${\beta }_0^i$, $v_2$);

Step (8) The candidate ${\beta }_0^{i+1}$will be accepted with a probability given by the Metropolis ratio

$$prob2({\beta }_0^i, {\beta }_0^{i+1})=min\left\{1,\frac{p\left({\alpha }^{i+1},{\beta }_0^{i+1},{\beta }_1^i | \mathit{t} \right)}{p\left({\alpha }^{i+1},{\beta }_0^i,{\beta }_1^i | \mathit{t} \right)}\right\};$$

(28)

Step (9) Generate a uniform value $u_2$ on range 0 to 1, i.e., $u_2\sim U$[$0,$ 1];

Step (10) If $u_2<prob2({\beta }_0^i$,${\beta }_0^{i+1})$, accept the candidate value and set${\beta }_0^{i+1}= {\beta }_0^{i+1}$; otherwise, reject and set ${\beta }_0^{i+1}={\beta }_0^i$;

Step (11) Generate the new value ${\beta }_1^{i+1}$ from the normal distribution N(${\beta }_1^i$, $v_3$);

Step (12) The candidate normal distribution ${\beta }_1^{i+1}$will be accepted with a probability given by the Metropolis ratio

$$prob3({\beta }_1^i, {\beta }_1^{i+1})=min\left\{1,\frac{p\left({\alpha }^{i+1},{\beta }_0^{i+1},{\beta }_1^{i+1} | \mathit{t} \right)}{p\left({\alpha }^{i+1},{\beta }_0^{i+1},{\beta }_1^i | \mathit{t} \right)}\right\};$$

(29)

Step (13) Generate a uniform value $u_3$ on range 0 to 1, i.e., $u_3\sim U$[$0,$ 1];

Step (14) If $u_3<prob3({\beta }_1^i$,${\beta }_1^{i+1})$, accept the candidate value and set${\beta }_1^{i+1}= {\beta }_1^{i+1}$; otherwise, reject and set ${\beta }_1^{i+1}={\beta }_1^i$;

Step (15) Repeat Steps 3–14 R times;

Step (16) Obtain the Bayesian estimator of ${\mathit{\theta }}_{\mathbf{1}}=(\alpha ,{\beta }_0,{\beta }_1)$as

$$\widehat{\alpha }=\frac{1}{R-R_0}\sum _{i=R_0+1}^R{\alpha }^i , {\widehat{\beta }}_0=\frac{1}{R-R_0}\sum _{i=R_0+1}^R{\beta }_0^i , {\widehat{\beta }}_1=\frac{1}{R-R_0}\sum _{i=R_0+1}^R{\beta }_1^i,$$

(30)

where $R_0$ is the burn-in period of the Markov Chain;

Step (17) To compute the credible intervals of $\alpha ,$ ${\beta }_0$ and ${\beta }_1$, order the generated MCMC values for each parameter as ${\alpha }^{(R_o+1)}<\dots <{\alpha }^{\left(R\right)}, {\beta }_0^{(R_o+1)}<\dots <{\beta }_0^{\left(R\right)}$and${\beta }_1^{(R_o+1)}<\dots <{\beta }_1^{\left(R\right)}$; thus, the 100($1-\gamma )$% symmetric credible intervals of $\alpha ,$ ${\beta }_0$ and ${\beta }_1$ are given by

$$\left({\alpha }^{(R-R_o)\gamma /2},{\alpha }^{(R-R_o)(1-\gamma /2)}\right) , \left({\beta }_0^{(R-R_o)\gamma /2},{\beta }_0^{(R-R_o)(1-\gamma /2)}\right) , \left({\beta }_1^{(R-R_o)\gamma /2},{\beta }_1^{(R-R_o)(1-\gamma /2)}\right).$$

(31)

The proposed gamma distribution with parameters $\alpha /v_1$ was chosen to obtain a good mixing of the chains.

6.2. MCMC Algorithm for Parameter Vector ${\mathit{\theta }}_{\mathbf{2}}=(\mathit{\alpha },{\psi }_{\mathbf{0}},{\psi }_{\mathbf{1}})$

Step (1) Choose starting values ${\alpha }^0$, ${\psi }_0^0$and ${\psi }_1^0$;

Step (2) Set $i=1$;

Step (3) Generate a new value ${\alpha }^{i+1}$conditional on the current ${\alpha }^i$from the gamma distribution $\Gamma$(${\alpha }^i$/$v_1$, $v_1$);

Step (4) The candidate ${\alpha }^{i+1}$ will be accepted with a probability given by the Metropolis ratio

$$prob1({\alpha }^i , {\alpha }^{i+1})=min\left\{1,\frac{\Gamma ({\alpha }^i/v_1,v_1) p\left({\alpha }^{i+1},{\psi }_0^i,{\psi }_1^i | \mathit{t} \right)}{\Gamma ({\alpha }^{i+1}/v_1,v_1) p\left({\alpha }^i,{\psi }_0^i,{\psi }_1^i | \mathit{t} \right)}\right\};$$

(32)

Step (5) Generate a uniform value $u_1$ on range 0 to 1, i.e., $u_1\sim U$[$0,$ 1];

Step (6) If $u_1<prob1({\alpha }^i$,${\alpha }^{i+1})$, accept the candidate value and set${\alpha }^{i+1}= {\alpha }^{i+1}$; otherwise, reject and set ${\alpha }^{i+1}= {\alpha }^i$;

Step (7) Generate the new value ${\psi }_0^{i+1}$ from the gamma distribution $\Gamma$(${\psi }_0^i$/$v_2$, $v_2$);

Step (8) The candidate ${\psi }_0^{i+1}$ will be accepted with a probability given by the Metropolis ratio

$$prob2({\psi }_0^i , {\psi }_0^{i+1})=min\left\{1,\frac{\Gamma ({\psi }_0^i/v_2,v_2) p\left({\alpha }^{i+1},{\psi }_0^{i+1},{\psi }_1^i | \mathit{t} \right)}{\Gamma ({\psi }_0^{i+1}/v_2,v_2) p\left({\alpha }^{i+1},{\psi }_0^i,{\psi }_1^i | \mathit{t} \right)}\right\};$$

(33)

Step (9) Generate a uniform value $u_2$ on range 0 to 1, i.e., $u_2\sim U$[$0,$ 1];

Step (10) If $u_2<prob2({\psi }_0^i$,${\psi }_0^{i+1})$, accept the candidate value and set${\psi }_0^{i+1}={\psi }_0^{i+1}$; otherwise, reject and set ${\psi }_0^{i+1}={\psi }_0^i$;

Step (11) Generate the new value ${\psi }_1^{i+1}$ from the gamma distribution $\Gamma$(${\psi }_1^i$/$v_3$, $v_3$);

Step (12) The candidate ${\psi }_1^{i+1}$ will be accepted with a probability given by the Metropolis ratio

$$prob3({\psi }_1^i , {\psi }_1^{i+1})=min\left\{1,\frac{\Gamma ({\psi }_1^i/v_3,v_3) p\left({\alpha }^{i+1},{\psi }_0^{i+1},{\psi }_1^{i+1} | \mathit{t} \right)}{\Gamma ({\psi }_1^{i+1}/v_3,v_3) p\left({\alpha }^{i+1},{\psi }_0^{i+1},{\psi }_1^i | \mathit{t} \right)}\right\};$$

(34)

Step (13) Generate a uniform value $u_3$ on range 0 to 1, i.e., $u_3\sim U$[$0,$ 1];

Step (14) If $u_3<prob3({\psi }_1^i$,${\psi }_1^{i+1})$, accept the candidate value and set${\psi }_1^{i+1}={\psi }_1^{i+1}$; otherwise, reject and set ${\psi }_1^{i+1}={\psi }_1^i$;

Step (15) Repeat Steps 3–14 R times;

Step (16) Obtain the Bayesian estimator of ${\mathit{\theta }}_{\mathbf{2}}=(\alpha ,{\psi }_0,{\psi }_1)$as

$$\widehat{\alpha }=\frac{1}{R-R_0}\sum _{i=R_0+1}^R{\alpha }^i , {\widehat{\psi }}_0=\frac{1}{R-R_0}\sum _{i=R_0+1}^R{\psi }_0^i , {\widehat{\psi }}_1=\frac{1}{R-R_0}\sum _{i=R_0+1}^R{\psi }_1^i,$$

(35)

where $R_0$ is the burn-in period of the Markov Chain;

Step (17) To compute the credible intervals of $\alpha ,$ ${\psi }_0$ and ${\psi }_1$, order the generated MCMC values for each parameter as ${\alpha }^{(R_o+1)}<\dots <{\alpha }^{\left(R\right)}, {\psi }_0^{(R_o+1)}<\dots <{\psi }_0^{\left(R\right)}$and${\psi }_1^{(R_o+1)}<\dots <{\psi }_1^{\left(R\right)}$; thus, the 100($1-\gamma )$% symmetric credible intervals of $\alpha ,$ ${\psi }_0$ and ${\psi }_1$ are given by

$$\left({\alpha }^{(R-R_o)\gamma /2},{\alpha }^{(R-R_o)(1-\gamma /2)}\right) , \left({\psi }_0^{(R-R_o)\gamma /2},{\psi }_0^{(R-R_o)(1-\gamma /2)}\right) , \left({\psi }_1^{(R-R_o)\gamma /2},{\psi }_1^{(R-R_o)(1-\gamma /2)}\right).$$

(36)

The proposed gamma distribution with parameters $\alpha /v_1$ and ${\psi }_i/v_i$ and $v_i$, $i=2,3,$ was chosen to obtain a good mixing of the chains.

For more details of MCMC in a variety of ways to construct these chains, see, for example, refs. [32,33].

7. Simulation Study

In this section, we assess the performance of the proposed priors and also evaluate the performance between the maximum likelihood and Bayesian approaches through Monte Carlo simulation.

We consider the noninformative priors for the parameters $\alpha$, ${\beta }_0$ and ${\beta }_1$ given in (21) and the prior under reparametrization $\alpha$, ${\psi }_0$ and ${\psi }_1$ given in (25).

The simulation study is based on $N=1000$ generated data sets from the step-stress model (8) for different sample sizes (n) and a highly censored setting. The chosen values of the considered sample sizes are $n=$(14, 20, 34, 50, 100) and the two percentages of the censored data equal 50% and 70%.

Table 1. Number of observed failures (r) for each percentage of censored data.

n	14	20	34	50	100
$r$for 50% of censored data	7	10	17	25	50
r for 70% of censored data	4	6	10	15	30

displays the various values of the sample size (n) and the number of observed failures (r) for each percentage of censored data.

Note from Table 1 that $(n, r)=(14,4)$ means a significant number of censored data in a small sample considered in the experiment under SSALT.

The values for the parameters are chosen to be $\alpha =2$, ${\beta }_0=4$ and ${\beta }_1=2$, and the levels of stress are $x_1=1$, $x_2=1.5$ and $x_3=2.5$. In this simulation, the pre-specified times ${\tau }_1$ and ${\tau }_2$ for the stress change are fixed according to the sample size and percentage of censored data.

The simulation presents some frequentist properties of the estimators of the model parameters $\alpha$, ${\beta }_0$ and ${\beta }_1$, such as the bias and mean square error (MSE) given by

$$bias=\frac{1}{N}\sum _{i=1}^N({\widehat{e}}_i-e_i) \mathrm{and}MSE=\frac{1}{N}\sum _{i=1}^N{({\widehat{e}}_i-e_i)}^2,$$

(37)

respectively, and the frequentist coverage probability (CP) of 95% confidence and credible intervals for different sample sizes.

As the marginal posterior distributions cannot be expressed in closed form, we apply the MCMC algorithm to obtain the point estimators and credible intervals for the parameters. The chain is run for 105,000 iterations with a burn-in period of 5000. For each simulated data set, we obtain the posterior summaries of the parameters

Figure 1 and Figure 2 show how the average biases and the MSE, respectively, vary with respect to n and the percentage of censored data for the parameters $\alpha$, ${\beta }_0$ and ${\beta }_1$ by considering MLE and Bayesian inference under the two proposed priors.

Figure 1 and Figure 2 show some differences between the Bayesian reparametrization approach and the other methods. In addition, it is important to observe that the point estimations for the three parameters are quite affected by the high proportion of censored data and the small sample size. On the other hand, if the sample size n is large, there is practically no difference, as expected. Furthermore, it is noted that both average bias and MSE decrease greatly when the percentage of censored data is 50% for the three parameters and any used approach. This result is actually expected when the amount of censored data tends to decrease, and mainly when increasing the sample size.

Another conclusion from Figure 1 and Figure 2 is that the MLE does not perform as well as the Bayesian method for estimating the three-parameter model especially when n is small. However, this difference disappears when n tends to a higher value.

In the comparison between the two proposed priors, that is, $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ and $\pi (\alpha$,${\psi }_0$, ${\psi }_1)$, we observed that the reparametrized prior provides better accuracy in the estimation of the parameters $(\alpha$, ${\beta }_0$, ${\beta }_1)$ in both metrics, bias and MSE. This difference is more relevant in estimating regression coefficients when n is small under a high percentage of censored data, where we can observe that it has the highest bias compared to the other methods.

Figure 3 shows the coverage probability (CP) of the 95% intervals for the parameters $\alpha$, ${\beta }_0$ and ${\beta }_1$. The confidence intervals for the parameters based on the MLE are obtained by the asymptotically normal approximation.

Based on the criterion of coverage probability CP, we can observe both the MLE and $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ prior performs poorly, that is, CP is too low the nominal value of the intervals, for the three parameters in comparison with the reparametrized prior $\pi (\alpha$, ${\psi }_0$, ${\psi }_1)$ when n and r assume small values. Moreover, the plots show that the CP is often below the nominal level for the three-parameter model but there is a convergence to 95% level when n increases. Thus, we can observe a remarkable influence of the large percentage of censored data on these estimation approaches for SSALT. However, they tend to be similar when the number of censored data becomes small and $n$large.

It is worth highlighting that the CP value under the reparametrized prior $\pi (\alpha$, ${\psi }_0$, ${\psi }_1)$ remains close to the nominal value of the interval for the three estimated parameters regardless of the values of n and r.

Therefore, we can conclude that the application of reparametrizations ${\psi }_0$ and ${\psi }_1$ improves upon the estimation compared with the others approaches for all pairs of values $(n,r)$.

8. Illustrative Simulated Example

In this section, we consider the data generated with $n=50$ observations tested with the test stopped after the 25th failure ($r=25$). We consider the parameters $\alpha =3$, ${\beta }_0=4$ and ${\beta }_1=0.3$. There are three different levels of stresses, $x_1=3$, $x_2=6$ and $x_3=10$. The pre-specified times for the stress change are ${\tau }_1=15$ and ${\tau }_2=25$. The simulated failure times in the first level of stress are [9.93820, 11.72592, 14.43394, 14.47979], in the second level of stress are [15.94424, 16.36506, 17.00182, 17.81110, 18.68743, 19.63954, 19.97476, 20.38834, 20.99103, 23.66255, 23.72308, 24.35782] and in the third level of stress are [25.46746, 25.50180, 25.67916, 25.98097, 26.02556, 26.08623, 26.27048, 26.27455, 26.60130].

We now compare the performance of the MLE and Bayesian procedures under the noninformative priors $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ and $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$).

The comparison is carried out by examining the point estimators, intervals and plots of the posterior densities and survival functions.

Table 2. Summaries and 95% credible intervals for $\alpha$.

Approach	$\widehat{\mathit{\alpha }}$	S.D.	95% C.I.
MLE	2.9332	1.5353	(−0.0759, 5.9423)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	3.2830	1.7353	(1.0797, 7.6721)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	3.2371	1.2206	(1.5515, 6.1635)

,

Table 3. Summaries and 95% credible intervals for ${\beta }_0$.

Approach	${\widehat{\mathit{\beta }}}_0$	S.D.	95% C.I.
MLE	3.7036	1.2432	(1.2669, 6.1402)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	3.7516	1.2489	(1.5060, 6.3415)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	3.5917	0.8231	(2.0252, 5.2386)

and

Table 4. Summaries and 95% credible intervals for ${\beta }_1$.

Approach	${\widehat{\mathit{\beta }}}_1$	S.D.	95% C.I.
MLE	0.2654	0.1182	(0.0337, 0.4972)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	0.2584	0.1186	(0.0362, 0.4965)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	0.2462	0.0843	(0.0804, 0.4034)

present the estimator, standard deviation and 95% credible interval.

The results of Table 2, Table 3 and Table 4 show that there is little difference between the performances of the MLE and Bayesian estimators, while the reparametrized prior gives the smallest standard deviation and a remarkably narrowest 95% credible interval for the parameters. Therefore, the prior distribution under reparametrization tends to have lower uncertainty compared to the other two approaches. Note that the lower endpoint of the 95% interval for $\alpha$ obtained by the MLE approach gives a negative value although this parameter assumes positive values. This is not surprising as, with small sample sizes, methods can exhibit significant bias. For instance, it is recognized that the MLE of the shape parameter for the Weibull distribution can be heavily biased with small sample sizes, a bias that may escalate with increased censoring. Such bias can lead to substantial discrepancies in analysis. Furthermore, there are instances where the MLE’s asymptotic properties do not hold, leading to pathological situations. Surely, these situations can explain the negative result of the lower limit of the confidence interval.

For comparison of the two priors $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ and $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$) proposed in this paper, the marginal posterior densities for $\alpha$, ${\beta }_0$ and ${\beta }_1$ are plotted in Figure 4.

From Figure 4, the marginal posterior densities by using $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ and $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$) are practical the same in their tails; however, for values close to their maximum point, they are quite different where the posterior under $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$) shows a sharp peak from under $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ due to a smaller standard deviation.

To visualize the performance of the estimation, the survival functions $S\left(t\right)$, given by $S\left(t\right)=1-G\left(t\right)$, are plotted in Figure 5 based on the MLE and Bayesian approaches.

From Figure 5, we observe that there is no difference among the plots of the estimated survival functions with the MLE and Bayesian approaches. However, the estimated survival functions do not provide a close fitting to the true function mainly between the change stress ${\tau }_1=15$ and ${\tau }_2=25$.

9. Real-Life Data Illustration

In this section, we analyzed a set of industrial data given in ref. [9]. The experiment was performed to hasten failures for a particular automotive part under four levels of stress. The combination of applied stresses is quantified in terms of a “percentage stress” as compared with the use condition; that is, if the use condition is defined as 100%, other stress levels are quantified as percentages over the use condition, such as 125%, 175%, 200% and 250%. The data consist of 10 failures out of 12 test units for four stress levels, and the numbers of failures at each stress level are 0, 2, 3 and 5, respectively. These units are tested at stress level 1 for 200 h, stress level 2 for 100 h, stress level 3 for 50 h and stress level 4 for 25 h; therefore, ${\tau }_1=200$, ${\tau }_2=300$, ${\tau }_3=350$ and ${\tau }_4=375$. The test is terminated after obtaining the 10th failure at 375 h.

Table 5. Failure data set for an automotive part.

Number	Failure Time (h)	Final Stress Level	Stress V (%)
1	252	2	175
2	280	2	175
3	320	3	200
4	328+	3	200
5	335	3	200
6	354	4	250
7	361	4	250
8	362	4	250
9	368	4	250
10	375	4	250
11	375+	4	250
12	375+	4	250

presents the data set for the fourth-level SSALT with type II censoring.

We converted it to a log-linear life-stress function as

$$log {\theta }_i={\beta }_0+{\beta }_1{x}_i, i=1,\dots , 4,$$

(38)

where $x_0=log\left(100\right), x_1=log\left(125\right), x_2=log\left(175\right), x_3=log\left(200\right)$ and $x_4=log\left(250\right).$

To compute the proposed estimators in this paper, we consider the gamma prior $\Gamma (a, b)$for parameter $\alpha$ and normal priors $N(\mu , {\sigma }^2)$for the parameters ${\beta }_0$and ${\beta }_1$, and it is assumed to be gamma prior $\Gamma (a, b)$for the reparametrization parameters ${\psi }_0=e^{{\beta }_0}$and ${\psi }_1=e^{{\beta }_1}$. We assume that we do not have any prior information on $\alpha$, ${\beta }_0$and ${\beta }_1$; so, we use the noninformative priors, that is, $a=b=0.001, \mu =0$and ${\sigma }^2={100}^2$.

We generate 105,000 MCMC samples of the joint posterior and then compute the estimators and intervals with respect to each set of prior distribution. We also calculate the MLE for the parameters $\alpha$, ${\beta }_0$and ${\beta }_1$ under asymptotic properties of MLE. All the results are given in

Table 6. Summaries and 95% credible intervals for $\alpha$.

Approach	$\widehat{\mathit{\alpha }}$	S.D.	95% C.I.
MLE	4.6328	5.5520	(−6.2490, 15.5146)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	7.1179	3.7401	(2.6488, 16.8201)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	7.3297	3.5083	(2.8334, 15.6736)

,

Table 7. Summaries and 95% credible intervals for ${\beta }_0$.

Approach	${\widehat{\mathit{\beta }}}_0$	S.D.	95% C.I.
MLE	28.8693	16.6103	(−3.6868, 61.4255)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	14.1308	2.0441	(10.6664, 19.2661)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	13.5277	0.7985	(11.2770, 13.9638)

and

Table 8. Summaries and 95% credible intervals for ${\beta }_1$.

Approach	${\widehat{\mathit{\beta }}}_1$	S.D.	95% C.I.
MLE	4.7993	2.9557	(−0.9939, 10.5926)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	3.4067	0.3647	(2.7687, 4.2853)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	3.2291	0.1659	(2.8062, 3.4268)

.

From Table 6, Table 7 and Table 8, clearly, all the Bayesian estimators are not very close to each other but they are quite different from the MLE. Bayesian intervals are shorter than the confidence intervals based on the MLE approach. In addition, the confidence interval presents negative values for the lower limit of the intervals showing that for SSALT with a small sample size and many steps the MLE is not appropriate. Besides the small sample, note that for step 1 of the SSALT there was no one observation, which further compromises the results under the MLE approach.

An inspection of Table 6 reveals that both priors set present practically no difference in the results for estimates for the parameter $\alpha$, that is, the transformation applied to the regression parameters ${\beta }_0$and ${\beta }_1$ does not affect the estimation of $\alpha$. However, from Table 7 and Table 8, we can see that the prior $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$) performs slightly better than $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ on the estimation of parameter ${\beta }_1$ while $\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$ performs most poorly for the estimation of parameter ${\beta }_0$due to resulting a higher standard deviation in comparison with prior $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$).

The graphs in Figure 6 represent the posterior densities for each parameter of the SSALT model constructed from the sample selected for the parameters $\alpha$, ${\beta }_0$and ${\beta }_1$. As expected, the posterior of $\alpha$ does not change with reparametrization. On the other hand, this figure indicates that the differences between the priors for ${\beta }_0$and ${\beta }_1$ under the reparametrization are noticeable and suggest that the choice of prior distribution has a substantial impact on the results for these parameters.

As the stress levels fixed in the accelerated test becomes higher, the required test duration decreases, that is, failures will occur more quickly, but the uncertainty involved in extrapolation increases. Therefore, it is expected that the mean time (MTTF) $t_m$to failure of the components under test will be longer under normal conditions of use. The confidence intervals, for these conditions, provide measures of these uncertainties in the extrapolation. In this way, the posterior summaries of and credible intervals for $t_m$of the units in test are considered under the proposed approaches and given in

Table 9. Summaries and 95% credible intervals for MTTF ($t_m$).

Approach	$\mathit{M}\widehat{\mathit{T}}\mathit{TF}$	S.D.	95% C.I.
MLE	4027.3205	5222.7941	(−6209.3560, 14,264.0000)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	1858.6666	16.8183	(1130.3380, 3086.6700)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	1590.0950	6.6579	$(1186.0800, 1976.0960)$

.

In the case of a small data set, MLE may yield parameter estimates with large standard deviations due to the limited amount of information available, resulting in wider confidence intervals. By specifying an appropriate prior distribution, we can potentially reduce the uncertainty in parameter estimates, resulting in smaller standard deviations and narrower credible intervals. A comparison of the point estimates and credible intervals provided by the priors shows there is a difference between them. The prior under reparametrization presents a small standard deviation and shorter interval.

For comparison of both the priors proposed in this example, the posterior densities for the expected lifetime are plotted in Figure 7. Examining Figure 7, we observe that the two classes of priors provide quite different posterior densities.

Now we compute the Bayes estimators of the reliability functions $R\left(t\right)$for $t=2650$. Assuming there is no prior information, we consider the proposed priors studied in this paper.

The resulting Bayesian estimates for the $R\left(t\right)$ are given in

Table 10. Summaries and 95% credible intervals for reliability $R\left(2650\right)$.

Approach	$\widehat{\mathit{R}}\left(2650\right)$	S.D.	95% C.I.
MLE	0.7523	0.2516	(0.2591, 1.2454)
$\pi (\alpha$, ${\beta }_0$, ${\beta }_1)$	0.1867	0.1633	(0.0033, 0.5918)
$\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$)	0.2066	0.0887	(0.0452, 0.3733)

and the posterior densities under the two set of priors are displayed in Figure 8.

According to the results shown in Table 10, the probability of the lifetime of a particular automotive is greater than 2650, depending on the estimation approach applied. It seems coherent that the MLE approach, which provides a higher expected failure time than the Bayesian method, also provides a higher reliability for the time $t_0$, which is smaller than the expected failure time under MLE. It is also observed that the $\pi (\alpha$, ${\beta }_0$, ${\beta }_1$) and $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$) priors adopted for solving this problem have produced almost similar results, although the prior $\pi$($\alpha$, ${\psi }_0$, ${\psi }_1$) results in a much smaller standard deviation, consequently providing a shorter credibility interval than that provided by prior $\pi (\alpha$, ${\beta }_0$, ${\beta }_1$). This comparison between these priors is also confirmed by Figure 8 with a plot of posterior densities for the parameter $R\left(t_0\right)$.

10. Conclusions and Discussion

In this paper, we have presented a Bayesian analysis of the multiple SSALT model under a gamma distribution and type II censored data based on two sets of prior distributions for the parameters. We also present a study comparing the maximum likelihood (MLE) and Bayesian approaches for estimating the parameters of the SSALT model. An extensive simulation has been conducted for different sample sizes and censoring proportions to evaluate the performance of the Bayesian and MLE approaches. The simulation results demonstrate that the Bayesian estimators give better results than MLE with respect to MSE, bias and coverage probability.

One important problem for all statisticians is finding a one-to-one transformation of parameters such that the likelihood function or posterior density is better behaved than in the original parametrization. Reparametrization involves transforming the parameters of interest into new parameters that may lead to more stable or efficient estimation. This technique can generally improve the performance of both frequentist and Bayesian inference methods. In a Bayesian framework, this transformation of parameters typically suggests a new family of prior distributions and the effect of this change on the Bayesian estimation is considered.

In this paper, we have presented a useful reparametrization that significantly improves the accuracy of the estimation for small or moderate sample sizes even under censored data. We have shown that the proposed reparametrization under new priors has led to better estimation compared to the original parametrization. Therefore, the choice of an appropriate parametrization for a statistical model is an important contribution for Bayesian inference.

The real-world data example presented showed the convenience and efficiency of using the SSALT model under the proposed Bayesian analysis.