A Bayesian Approach to Step-Stress Partially Accelerated Life Testing for a Novel Lifetime Distribution

Abstract

In lifetime testing, the failure times of highly reliable products under normal usage conditions are often impractically long, making direct reliability assessment impractical. To overcome this, step-stress partially accelerated life testing is employed to reduce testing time while preserving data quality. This paper develops a Bayesian model based on Type II censored data, assuming that item lifetimes follow the Topp–Leone inverted Kumaraswamy distribution, a flexible alternative to classical lifetime models due to its ability to capture various hazard rate shapes and to model bounded and skewed lifetime data more effectively than traditional models observed in real-world reliability data. Bayes estimators of the model parameters and acceleration factor are derived under both symmetric (balanced squared error) and asymmetric (balanced linear exponential) loss functions using informative priors. The novelty of this work lies in the integration of the Topp–Leone inverted Kumaraswamy distribution within the Bayesian step-stress partially accelerated life testing framework, which has not been explored previously, offering improved modeling capability for complex lifetime data. The proposed method is validated through comprehensive simulation studies under various censoring schemes, demonstrating robustness and superior estimation performance compared to traditional models. A real-data application involving COVID-19 mortality data further illustrates the practical relevance and improved fit of the model. Overall, the results highlight the flexibility, efficiency, and applicability of the proposed Bayesian approach in reliability analysis.

1. Introduction

Increasing market competition, along with the growing expectations of customers, makes the production of highly reliable goods a necessity for manufacturers. However, with ever-decreasing time-to-market and ongoing improvements in manufacturing designs due to technological progress, acquiring timely lifetime information for highly reliable products or materials through standard testing has become increasingly difficult. In response to these challenges, accelerated life testing (ALT) or partially accelerated life testing (PALT) are frequently adopted in manufacturing to generate adequate failure data quickly and to understand its relationship with external stress. Utilizing such testing methods offers substantial savings in terms of time, labor, materials usage, and financial resources.

ALT is a broad class of experimental techniques designed to obtain failure data more rapidly by subjecting test items to elevated stress levels such as higher temperature, pressure, or load. These methods aim to model and predict reliability under normal operating conditions based on accelerated failure data. On the other hand, the accelerated failure time (AFT) model is a parametric approach commonly used in survival analysis, where the logarithm of the failure time is modeled as a linear function of covariates. While AFT models are often employed within ALT frameworks to establish a statistical relationship between stress levels and failure times, the two concepts are not interchangeable. In this work, we adopt a Bayesian ALT framework where the stress–lifetime relationship is captured through an acceleration factor, without directly specifying an AFT form. Nonetheless, the principles underlying AFT and its relevance to ALT are acknowledged, as both aim to explain time-to-failure behavior under varying conditions.

When life-stress models are unknown or unreliable, ALT data cannot accurately predict performance under normal use. Therefore, PALT, which exposes units to both standard and accelerated conditions, is a more fitting method in these situations. The foundational concepts of ALT were introduced and explored by refs. [1,2]. Furthermore, stress application in these tests can take various forms, such as constant, step, and progressive patterns.

In Step-Stress ALT (SS-ALT), test samples are exposed to increasing levels of stress applied sequentially. A sample begins under a specific constant stress for a predetermined time. If it survives and no failure occurs, the stress level is increased to a higher level for another specified period. This process of incrementally raising stress continues until the sample fails. Generally, all samples undergo the same standardized pattern of stress levels and testing durations. Detailed information regarding Step-Stress ALT can be found in the publications by refs. [3,4,5,6,7,8,9], among others.

Step stress-PALT (SS-PALT) involves initially testing a unit under normal operating conditions. If it survives a predetermined period or a certain number of failures, the testing is then run using accelerated conditions until the unit fails or the observation is stopped. This experimental approach aims to gather more failure information within a shorter timeframe without subjecting all units to high stress levels from the outset. Research has addressed Bayesian and non-Bayesian statistical methods for analyzing SS-PALT data under various censoring schemes, including Type I, Type II, and Type II progressive censoring (see refs. [10,11,12,13,14,15,16,17,18]).

Understanding product reliability and efficiently predicting lifespan are essential in modern engineering. Although ALT is widely used to accelerate failure data acquisition, its extrapolation to normal use conditions can be unreliable. In contrast, PALT, and specifically SS-PALT, offers a more realistic framework in many practical scenarios. This paper contributes to the reliability literature by investigating SS-PALT under Type II censoring within a Bayesian model. Recent advancements in reliability analysis have explored various data-driven and probabilistic modeling techniques to improve the prediction of failure behavior and remaining useful life. For instance, ref. [19] proposed a federated learning framework integrating adaptive sampling and ensemble methods to enhance the RUL prediction of aircraft engines, highlighting the importance of distributed data environments and model generalizability. Similarly, ref. [20] developed a multivariate Student-t process model tailored for tail-weighted and dependent degradation data, addressing challenges in modeling complex dependence structures and heavy-tailed behaviors.

A significant gap in existing research lies in the limited application of flexible lifetime distributions within Bayesian frameworks for PALT. To address this, a Bayesian SS-PALT model is introduced assuming that the lifetimes under use conditions follow the highly adaptable Topp–Leone Inverted Kumaraswamy (TL-IK) distribution. This distribution was selected due to its capacity to model a wide range of hazard shapes, including skewed and bathtub-like behaviors, which are often observed in real-world reliability data compared to traditional distributions.

Bayesian estimation enables the integration of prior information and offers advantages in handling complex censoring and uncertainty. In this study, Bayes estimators for model parameters and the acceleration factor are derived under both the symmetric balanced squared error (BSEL) and the asymmetric balanced linear exponential (BLL) functions, resulting in a more robust and informative estimation process.

While most SS-PALT studies focus on classical lifetime distributions such as Weibull, exponential, or log-normal, the integration of the TL-IK distribution into a Bayesian SS-PALT framework is, to the best of our knowledge, novel. This research addresses that methodological gap by providing a more flexible and interpretable reliability model. The proposed approach is validated through extensive simulation studies and a real-data application, demonstrating improved performance and practical relevance compared to conventional methods.

The problem statement, methodology, and objectives of this paper are summarized as follows: in reliability testing, evaluating the lifetime of highly reliable items under normal use conditions is often impractical due to the excessive time required for failure observation. While full ALTs aim to mitigate this by exposing all items to high stress levels, the need to extrapolate results to normal conditions can lead to biased or unreliable estimates, especially when model assumptions are uncertain. PALT, particularly under step-stress conditions, provides a more robust alternative by combining data from both stress levels. However, there remains a need for efficient statistical methods that can handle censored data and model uncertainty within this framework, especially when using more flexible lifetime distributions.

A novel Bayesian estimation framework is introduced for SS-PALT under Type II censoring, assuming that lifetimes follow the TL-IK distribution, a flexible and recently studied model not previously applied in this testing context ref. [21]. Unlike existing approaches that primarily rely on traditional lifetime distributions or fully accelerated tests, the proposed method incorporates both BSEL and the asymmetric BLL functions within a Bayesian framework, providing more robust and adaptable estimators. A Bayesian approach is used to derive Bayes estimators for model parameters and the acceleration factor under informative prior distributions. The methodology is validated through extensive simulation studies under various censoring and stress scenarios, and its practical applicability is demonstrated using real-world COVID-19 data, highlighting both theoretical contribution and real-world relevance.

The objectives of this paper are to develop a Bayesian estimation framework for SS-PALT under Type II censoring and to derive Bayes estimators for the model parameters and the acceleration factor using informative priors under both the symmetric BSEL and the asymmetric BLL functions. Additionally, the paper aims to evaluate the performance of the proposed estimators through comprehensive simulation studies under various censoring schemes and stress conditions. Furthermore, it seeks to demonstrate the practical relevance and flexibility of the methodology through the analysis of a real-world COVID-19 dataset.

The structure of this paper is as follows: Section 2 details the model and its fundamental assumptions. Section 3 focuses on deriving Bayesian point estimates and credible intervals (CIs) for the unknown parameters and the acceleration factor in SS-PALT with Type II censored data. These derivations are based on both the BSEL function and BLL function, which serves as an asymmetric loss function. Section 4 provides a numerical example to support and validate the proposed Bayesian estimation methodology. Finally, Section 5 offers some general conclusions.

2. Description of the Model and Assumptions

The model details are presented in Section 2.1, where the basic assumptions are outlined, and the test procedure is described in Section 2.2.

2.1. Model Description

The reliability of parametric statistical inference depends on selecting a probability distribution that accurately represents the data and satisfies its underlying assumptions. Researchers have extensively studied statistical distributions, seeking and aiming to create models with more adaptable and desirable characteristics capable of capturing the diverse shapes observed in real-world data, including their density and failure rate patterns. Recently, research has mainly focused on developing new, wider families of distributions by generalizing existing ones to achieve a superior fit when modeling data. These extended distribution families are typically formed either by combining two or more existing distributions or by introducing additional parameters to a base distribution. The TL-IK distribution is a composite distribution of the TL $\left(\theta \right)$ and IK $(a,b)$ distributions. Its cumulative distribution function (cdf) and probability density function (pdf) are provided, respectively, by:

$$F\left(x|\underset{\_}{\vartheta }\right)={\left[\phi \left(a\right)\right]}^{\theta b}{\left\{2-{\left[\phi \left(a\right)\right]}^b\right\}}^{\theta }, 0<x<\infty , \left(\underset{\_}{\vartheta }>0\right),$$

(1)

where

$$\phi \left(a\right)=1-{\left(1+x\right)}^{-a}, \underset{\_}{\vartheta }={\left(a,b,\theta \right)}^{\prime },$$

(2)

and

$$f\left(x|\underset{\_}{\vartheta }\right)=2ab\theta {\left(1+x\right)}^{-\left(a+1\right)}{\left[\phi \left(a\right)\right]}^{\theta b-1}\left\{1-{\left[\phi \left(a\right)\right]}^b\right\}{\left\{2-{\left[\phi \left(a\right)\right]}^b\right\}}^{\theta -1}, 0<x<\infty ;\left(\underset{\_}{\vartheta }>0\right),$$

(3)

where $a,b$, and $\theta$ are shape parameters.

The reliability function (rf) and hazard rate function (hrf) are given, respectively, by:

$$R\left(x;\underset{\_}{\vartheta }\right)=1-{\left[\phi \left(a\right)\right]}^{\theta b}{\left\{2-{\left[\phi \left(a\right)\right]}^b\right\}}^{\theta }, 0<x<\infty ; \left(\underset{\_}{\vartheta }>0\right),$$

(4)

and

$$h\left(x;\underset{\_}{\vartheta }\right)={\displaystyle \frac{f\left(x\right)}{1-F\left(x\right)}}={\displaystyle \frac{2ab\theta {\left(1+x\right)}^{-\left(a+1\right)}{\left[\phi \left(a\right)\right]}^{\theta b-1}\left\{1-{\left[\phi \left(a\right)\right]}^b\right\}{\left\{2-{\left[\phi \left(a\right)\right]}^b\right\}}^{\theta -1}}{1-{\left[\phi \left(a\right)\right]}^{\theta b}{\left\{2-{\left[\phi \left(a\right)\right]}^b\right\}}^{\theta }}}, 0<x<\infty ; \left(\underset{\_}{\vartheta }>0\right).$$

(5)

Ref. [21] analyzed various statistical properties of the TL-IK distribution, including explicit expressions for its moments, the density functions of order statistics, the Rényi entropy, and the moment-generating function. The TL-IK distribution generalizes several well-known lifetime distributions such as the Topp–Leone–Lomax, Lomax, and Topp–Leone–log–logistic distributions, offering a high degree of modeling flexibility. Further applications of the TL-IK distribution in the context of accelerated life testing have been explored in recent works. For instance, ref. [22] developed maximum likelihood estimation and prediction methods for constant stress PALT using Type II censored data from the TL-IK distribution. ref. [23] extended this by presenting a Bayesian estimation and prediction framework for the same setup. More recently, ref. [18] investigated maximum likelihood estimation under SS-PALT schemes, again utilizing TL-IK distributed lifetimes with Type II censoring. These studies collectively demonstrate both the theoretical richness and practical relevance of the TL-IK distribution in reliability analysis, thereby justifying its use in our proposed Bayesian SS-PALT model.

2.2. Fundamental Assumptions and Testing Process

The underlying assumptions can be summarized as follows:

• $X$ is the lifetime of an item in its usual condition and has the TL-IK distribution.
• The random variables representing the time to failure for each item; $Y_{ij}; i=1, 2;$$j=1, 2, \dots , n_i$, are independent and identically distributed (i.i.d) random variables.
• Two distinct stress levels are employed: $z_1$ as a normal level and $z_2$ as a high level.
• The overall lifespan of the tested items, represented by $Y$, involves two phases: normal use and accelerated conditions. Consequently, the item’s lifetime with SS-PALT is

  $$Y=\left\{\begin{array}{c} X \mathrm{if} X\le y_{{1n}_1}\\ y_{{1n}_1}+{\beta }^{-1}\left(X-y_{{1n}_1}\right) \mathrm{if} X>y_{1n_1} \end{array}\right.,$$

  (6)

  where $y_{{1n}_1}$ represents the time of the last failure observed $n_1^{th}$ under normal operating conditions, and $X$ indicates the lifetime when the item is with stress level $z_1$.
• The TL-IK distribution describes the lifetime of a test unit at every considered stress level.

Test procedure

• Consider an experiment starts with $n$ identical items subjected to a normal level of stress $z_1$. The test is continued until $n_1$ are observed that these items fail at a specific point in time $y_{{1n}_1}$, while they are under this defined stress level $z_1$. The test will end if the number of failed items reaches a predetermined $n_1$ items, where $n_1={\pi }_{1}n$ and ${\pi }_1$ is a specific proportion of the total items being tested under normal use conditions, $0<{\pi }_1<1$.
• To observe failures in items that do not fail during usual use conditions $(n-n_1)$, they are placed under accelerated use conditions with high stress levels $z_2$ and run until failures $n_2$ are noticed at time $y_{{2n}_2}$. When $n_2={\pi }_{2}n$ and a predetermined proportion ${\pi }_2$ of the total test items are selected and subjected to accelerated testing where $0<{\pi }_2<1$ and $0<{\pi }_1+{\pi }_2<1$. The remaining items $n_c=n-n_1-n_2$ are then censored.

If an item survives a certain number of failures in the test, the stress is increased. The test continues at this higher stress level until either more failures occur or the observation is censored. This stress change effectively multiplies the item’s remaining lifetime by the inverse of an acceleration factor $\beta$, which $Y={\beta }^{-1}X,$ $\beta$ is the acceleration factor, representing the ratio of mean life at usual conditions to mean life at the accelerated condition and $\beta >1$.

As a result, the total lifetime of a test item consists of two consecutive phases: the usual use phase and the accelerated use phase, respectively (see Refs. [11,14]).

In a simple SS-PALT, under Type II censoring, the pdf of the total lifetime Y of an item is given by:

$$Y=\left\{\begin{array}{c}{f}_1\left(y;\underset{\_}{\vartheta }\right) \mathrm{if} y\le y_{{1n}_1} \\ \\ { f}_2\left(y;\underset{\_}{\mathsf{\Phi }}\right) \mathrm{if} y>y_{{1n}_1} \end{array},\right.$$

(7)

where $f_1\left(y\right),$ follows a TL-IK distribution with pdf,

$$\begin{gathered} f_1\left(y\right)=2ab\theta {\left(1+y_{1j}\right)}^{-\left(a+1\right)}{\left[{\phi }_1\left(a\right)\right]}^{\theta b-1}\left\{1-{\left[{\phi }_1\left(a\right)\right]}^b\right\}{\left\{2-{\left[{\phi }_1\left(a\right)\right]}^b\right\}}^{\theta -1}, \\ y\le y_{1n_1}; \left(\underset{\_}{\vartheta }>0\right), \end{gathered}$$

(8)

$f_2\left(y\right)$ is found through the transformation-variable method, based on $f_1\left(y\right)$ and the model from (3), and has the pdf

$$\begin{gathered} f_2\left(y;\underset{\_}{\mathsf{\Phi }}\right)=2ab\theta \beta {\left(D\right)}^{-\left(a+1\right)}{\left[{\phi }_1\left(a\beta \right)\right]}^{\theta b-1}\left\{1-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}{\left\{2-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}}^{\theta -1}, \\ y>y_{{1n}_1};\left(\underset{\_}{\vartheta }>0\right); \beta >1, \end{gathered}$$

(9)

where $\underset{\_}{\mathsf{\Phi }}={\left(\underset{\_}{\vartheta },\beta \right)}^{\prime }$, ${\phi }_1\left(a\right)=1-{\left(1+y_{1j}\right)}^{-a},$ ${\phi }_1\left(a\beta \right)=1-{\left(D\right)}^{-a},$

$$D=1+\beta \left(y_{2j}-y_{1n1}\right)+y_{1n1}$$

(10)

where $\underset{\_}{\vartheta }$ is given by (2).

For an item tested under accelerated conditions and having the pdf $f_2\left(y;\underset{\_}{\mathsf{\Phi }}\right)$ mentioned, its cdf, rf, and hrf are provided, respectively, below:

$$F\left(y;\underset{\_}{\mathsf{\Phi }}\right)={\left[{\phi }_c\left(a\beta \right)\right]}^{\theta b} {\left\{2-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}}^{\theta }, y>y_{{1n}_1};\left(\underset{\_}{\vartheta }>0\right); \beta >1,$$

(11)

$$R\left(y;\underset{\_}{\mathsf{\Phi }}\right)=1-{\left[{\phi }_c\left(a\beta \right)\right]}^{\theta b} {\left\{2-{\left[{\phi }_c\left(a\beta \right)\right]}^b\right\}}^{\theta }, y>y_{1n_1};\left(\underset{\_}{\vartheta }>0\right); \beta >1,$$

(12)

and

$$h\left(y;\underset{\_}{\mathsf{\Phi }}\right)={\displaystyle \frac{2ab\theta \beta {\left(D\right)}^{-\left(a+1\right)}{\left[{\phi }_1\left(a\beta \right)\right]}^{\theta b-1}\left\{1-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}{\left\{2-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}}^{\theta -1}}{1-{\left[{\phi }_c\left(a\beta \right)\right]}^{\theta b} {\left\{2-{\left[{\phi }_c\left(a\beta \right)\right]}^b\right\}}^{\theta }}}, y>y_{{1n}_1};\left(\underset{\_}{\vartheta }>0\right); \beta >1,$$

(13)

$${\phi }_c\left(a\beta \right)=1-{\left(D_1\right)}^{-a},D_1=1+\beta \left(y_{2n2}-y_{1n1}\right)+y_{1n1}.$$

(14)

We have observed the following values for the total lifetime:

$$y_{\left(1\right)}\le y_{\left(2\right)}\le \dots \le y_{\left(n_1\right)}\le y_{\left(n_1+1\right)}\le \dots \le y_{(n_1+n_2-1)}\le y_r$$

where $y_r$ corresponds to the lifetime ${(n}_1+n_2)th$ unit of the final unit to experience failure within this experiment.

3. Bayesian Estimation

This section details the derivation of the Bayes point estimators and CIs for the parameters of the TL-IK distribution using SS-PALT with Type II censored data. Two types of loss functions are employed: BSEL and the BLL functions.

Bayesian statistics serves as a useful tool for addressing inference problems, especially when small sample sizes limit the applicability of complex statistical methods. By allowing the incorporation of prior information alongside sample data, the Bayesian approach can help in situations with limited data (see ref. [24]).

Bayesian analysis models unknown parameters as random variables, allowing for the incorporation of prior information. This prior knowledge is updated with data to yield posterior information, which is used to estimate product behavior with usual conditions. The complex integrations inherent in this process necessitate the use of numerical methods like Markov Chain Monte Carlo (MCMC).

Regarding SS-PALT with Type II censored data, the likelihood function (LF) for the obtained observed and $n_c$ censored data is presented below.

$$\begin{gathered} L\left(\underset{\_}{\mathsf{\Phi }}| \underset{\_}{y}\right)\propto {\prod }_{j=1}^{n_1}ab\theta {\left(1+y_{1j}\right)}^{-\left(a+1\right)}{\left[{\phi }_1\left(a\right)\right]}^{\theta b-1}\left\{1-{\left[{\phi }_1\left(a\right)\right]}^b\right\}{\left\{2-{\left[{\phi }_1\left(a\right)\right]}^b\right\}}^{\theta -1} \\ \times {\prod }_{j=1}^{n_2}ab\theta \beta {\left(D\right)}^{-\left(a+1\right)}{\left[{\phi }_1\left(a\beta \right)\right]}^{\theta b-1}\left\{1-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}{\left\{2-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}}^{\theta -1} \\ \times {\prod }_{j=1}^{n_c}\left\{1-{\left[{\phi }_c\left(a\beta \right)\right]}^{\theta b} {\left\{2-{\left[{\phi }_c\left(a\beta \right)\right]}^b\right\}}^{\theta }\right\}, \end{gathered}$$

(15)

where, ${{\phi }_1\left(a\right), \phi }_1\left(a\beta \right),{D,\phi }_c\left(a\beta \right)$ and $D_1$ are given by (10) and (14), respectively.

The prior distributions for the parameters are considered to be conjugate informative priors. Furthermore, the components of the parameter vector $\underset{\_}{\mathsf{\Phi }}=\left(\underset{\_}{\vartheta },\beta \right)$ are independent, with each component following the gamma distribution, ${\mathsf{\Phi }}_i ~$ gamma $\left({\alpha }_i,{\gamma }_i\right)$.

The gamma distribution is suggested as the informative prior for the model parameters due to its flexibility, mathematical tractability, and common use in reliability analysis and Bayesian estimation, particularly for modeling positive-valued parameters such as scale or rate parameters. Its conjugacy with certain likelihood functions also facilitates analytical and computational convenience. Additionally, the gamma distribution allows the incorporation of prior knowledge or expert opinion through its shape and rate parameters, which aligns well with the context of accelerated life testing.

Here, ${\alpha }_i,{\gamma }_i$ represent the hyperparameters for $i=1,\dots ,4,$ of the respective prior distributions.

As a result, the joint prior distribution for the entire set of unknown parameters is characterized by the following joint pdf:

$$\pi \left(\underset{\_}{\mathsf{\Phi }}\right)\propto {\prod }_{i=1}^4{{\mathsf{\Phi }}_i}^{{\alpha }_i-1}\exp \left(-{\gamma }_i{\mathsf{\Phi }}_i\right), \beta >1; \left(\underset{\_}{\vartheta }, {\alpha }_i, {\gamma }_i\right)>0,$$

(16)

where ${\mathsf{\Phi }}_1=a,{\mathsf{\Phi }}_2=b,{\mathsf{\Phi }}_3=\theta$, ${\mathsf{\Phi }}_4=\beta$, and $\underset{\_}{\vartheta }={\left(a,b,\theta \right)}^{\prime }$.

The joint posterior distribution for the parameter vector $\underset{\_}{\mathsf{\Phi }}$ is obtained by combining the likelihood function presented in (15) and the joint prior distribution given in (16), as shown below.

$$\begin{array}{c}\pi \left(\underset{\_}{\mathsf{\Phi }}| \underset{\_}{y}\right)=A{{{\mathsf{\Phi }}_i}^{{\alpha }_i-1}\exp \left(-{\gamma }_i{\mathsf{\Phi }}_i\right)\left(ab\theta \right)}^{n{(\pi }_1+{\pi }_2)}{\beta }^{n{\pi }_2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\{{\displaystyle \prod _{j=1}^{n_1}}{\left(1+y_{1j}\right)}^{-\left(a+1\right)}\left[{\phi }_1\left(a\right)\right]}^{\theta b-1}\left\{1-{\left[{\phi }_1\left(a\right)\right]}^b\right\}{\left\{2-{\left[{\phi }_1\left(a\right)\right]}^b\right\}}^{\theta -1}\}\\ \times \left\{\prod _{j=1}^{n_2}ab\theta \beta {\left(D\right)}^{-\left(a+1\right)}{\left[{\phi }_1\left(a\beta \right)\right]}^{\theta b-1}\left\{1-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}{\left\{2-{\left[{\phi }_1\left(a\beta \right)\right]}^b\right\}}^{\theta -1}\right\}\\ \times n_c\left\{1-{\left[{\phi }_c\left(a\beta \right)\right]}^{\theta b} {\left\{2-{\left[{\phi }_c\left(a\beta \right)\right]}^b\right\}}^{\theta }\right\},\end{array}$$

(17)

where ${{\phi }_1\left(a\right), \phi }_1\left(a\beta \right),D$ and ${{\phi }_c\left(a\beta \right),D}_1$ are defined by (10) and (14), respectively, and ${\alpha }_i,{\gamma }_i$ represent the hyperparameters of the prior distribution for $i=1,\dots ,4$. The normalizing constant, A, can be found using the subsequent equation.

$${\int }_{\underset{\_}{\mathsf{\Phi }}}\pi \left(\underset{\_}{\mathsf{\Phi }}| \underset{\_}{y}\right) d\underset{\_}{\mathsf{\Phi }}=1,$$

(18)

where

$${\int }_{\underset{\_}{\mathsf{\Phi }}}={\int }_a{\int }_b{\int }_{\theta }{\int }_{\beta } \mathrm{and} \underset{\_}{\mathrm{d}\mathsf{\Phi }}=da db d\theta d\beta$$

(19)

3.1. Bayesian Estimation with Balanced Loss Functions

The concept of the balanced loss function (BLF) originates with refs. [25,26] further developed this idea by introducing a more general class of BLF, where the extended form is given by:

$${\mathrm{L}}^{\mathrm{*}}\left({\theta ,\theta }^{\mathrm{*}}\right)=\omega l\left({\widehat{\theta }}_0, \theta \right)+\left(1-\omega \right)l\left({\theta ,\theta }^{\mathrm{*}}\right),$$

(20)

where $l\left({\theta ,\theta }^{\mathrm{*}}\right)$ denotes a general loss function, with ${\widehat{\theta }}_0$ being an estimator for $\theta$ and $\omega ϵ\left[0, 1\right]$ representing a weight. This BLF can be adapted to various loss functions such as SEL, absolute error, entropy, and linear exponential (LINEX), and it generalizes the SEL function. Ref. [27] proposed using the BSEL. Substituting $l\left({\theta ,\theta }^{\mathrm{*}}\right)={\left({\theta }^{\mathrm{*}}-\theta \right)}^2$ into (20) yields the BSEL function, which has the following form:

$$L^{*}\left({\theta ,\theta }^{*}\right)=\omega {\left({\theta }^{*}-{\widehat{\theta }}_0\right)}^2+\left(1-\omega \right){\left({\theta }^{*}-\theta \right)}^2,$$

when BSEL is employed, the resulting Bayes estimator ${\tilde{u}}_{\mathrm{B}\mathrm{S}\mathrm{E}\mathrm{L}}$ for a function $u \left(\theta \right)$ is expressed as:

$${\tilde{u}}_{\mathrm{B}\mathrm{S}\mathrm{E}\mathrm{L}}=\omega {\tilde{u}}_{\mathrm{M}\mathrm{L}}+\left(1-\omega \right){\tilde{u}}_{\mathrm{S}\mathrm{E}\mathrm{L}},$$

(21)

in which ${\tilde{u}}_{\mathrm{M}\mathrm{L}}$ denotes the ML estimator of $u \left(\theta \right)$, while ${\tilde{u}}_{\mathrm{S}\mathrm{E}\mathrm{L}}$ represents the corresponding Bayes estimate derived using the SEL function.

Moreover, the substitution of $l\left({\theta ,\theta }^{\mathrm{*}}\right)=e^{v\left({\theta }^{\mathrm{*}}-\theta \right)}-v\left({\theta }^{\mathrm{*}}-\theta \right)-1$ into (20) results in the determination of the BLL function as

$$L^{*}\left({\theta }^{*}, \theta \right)=\omega [e^{v\left({\theta }^{*}-{\widehat{\theta }}_0\right)}-v\left({\theta }^{*}-{\widehat{\theta }}_0\right)-1]+\left(1-\omega \right){[e}^{v\left({\theta }^{*}-\theta \right)}-v\left({\theta }^{*}-\theta \right)-1].$$

Here, the Bayes estimator utilizing the BLL function of $\theta$ is expressed as:

$$\widehat{\tilde{\theta }}\left(\mathrm{x}\right)=-{\displaystyle \frac{1}{v}}\ln \left[\omega e^{-v{\widehat{\theta }}_0}+\left(1-\omega \right)E\left(e^{-v\theta }|x\right)\right],$$

(22)

where the BLL function has a shape parameter denoted by $v\ne 0$.

As a result, the estimator of a function derived using BLF takes the form of a mixture between the function’s ML estimator and its Bayes estimators obtained under any chosen loss function. It is also possible to replace the ML estimator with alternative estimators, for instance, the least-squares estimator. Using both symmetric and asymmetric BLF, Bayes estimators have been obtained for a wide range of distributions in the literature (see refs. [28,29,30]).

The symmetric BSEL and asymmetric BLL functions were chosen to provide a comprehensive evaluation of the Bayes estimators under different decision-making scenarios. The BSEL is widely used due to its symmetry and mathematical convenience, reflecting equal penalties for overestimation and underestimation. However, in many real-world applications, estimation errors may not have equal consequences, such as maintenance scheduling or warranty cost estimation; the underestimation of failure times can be more critical than overestimation (or vice versa). To address this, we also considered that the BLL function provides a more realistic framework by allowing asymmetric treatment of estimation errors and is particularly suitable when overestimation and underestimation have different costs or risks. Including both loss functions enhances the robustness and practical relevance of the analysis.

The purpose of this paper is to demonstrate Bayesian estimation for the parameters and the acceleration factor $\beta$ of the TL-IK $(a,b,\theta ,\beta )$ distribution by considering the BSEL and BLL functions.

3.2. Bayes Estimators with Balanced Squared Error Loss Function

The Bayes estimators of the parameters with the BSEL function are derived from (17) and (21), respectively, as:

$${{\tilde{\mathsf{\Phi }}}_i}_{BSEL}=\omega {\widehat{\mathsf{\Phi }}}_{iML}+(1-\omega ){\int }_{\underset{\_}{\mathsf{\Phi }}}{\mathsf{\Phi }}_i\pi \left({\mathsf{\Phi }}_i|\underset{\_}{y}\right) d\underset{\_}{\mathsf{\Phi }}$$

(23)

where $\pi \left({\mathsf{\Phi }}_i| \underset{\_}{y}\right)$ is given by (17), and ${\int }_{\underset{\_}{\mathsf{\Phi }}}, d\underset{\_}{\mathsf{\Phi }},$ is represented by (19).

One can find the Bayes estimates for the parameters when using the BSEL function by performing specific replacements in ${\mathsf{\Phi }}_i$ by $a, b, \theta ,$ and $\beta$ (23).

3.3. Bayes Estimators with Balanced Linear Exponential Loss Function

The Bayes estimators for the parameters with the BLL function are determined from (17) and (22), respectively, as:

$${{\widehat{\tilde{\mathsf{\Phi }}}}_i}_{\mathrm{B}\mathrm{L}\mathrm{L}}={\displaystyle \frac{-1}{v}}\mathrm{l}\mathrm{n}\{\omega \exp \left(-v{\widehat{\mathsf{\Phi }}}_{iML}\right)+(1-\omega ){\int }_{\underset{\_}{\mathsf{\Phi }}}\exp \left(-{\mathsf{\Phi }}_{i}v\right)\pi \left({\mathsf{\Phi }}_i|\underset{\_}{y}\right) d\underset{\_}{\mathsf{\Phi }}.$$

(24)

One can find the Bayes estimates for the parameters when using the BLL function by performing specific replacements in ${\mathsf{\Phi }}_i$ by $a, b, \theta ,$ and $\beta$ (24).

3.4. Credible Intervals

In general, $\left[\mathrm{L}\left(\underset{\_}{y}\right)<{\mathsf{\Phi }}_i<\mathrm{U}\left(\underset{\_}{y}\right)\right]$ is a 100 (1−τ) % CIs for ${\mathsf{\Phi }}_{i },$where ${\mathsf{\Phi }}_i=$ $a or b or \theta$ or $\beta$ if

$$P\left(\left(L\right(\underset{\_}{y})<{\mathsf{\Phi }}_i<U(\underset{\_}{y})|\underset{\_}{y}\right)={\int }_{\mathrm{L}\left(\underset{\_}{y}\right)}^{\mathrm{U}\left(\underset{\_}{y}\right)}\pi \left({\mathsf{\Phi }}_i|\underset{\_}{y}\right) d{\mathsf{\Phi }}_i=1-\tau ,$$

(25)

The lower and upper bounds $\left[L\left(\underset{\_}{y}\right), U\left(\underset{\_}{y}\right)\right]$ can be obtained by solving (25) as follows:

$$P({\mathsf{\Phi }}_i>L(\underset{\_}{y})\left|\underset{\_}{y}\right)=1-{\displaystyle \frac{\tau }{2}},$$

(26)

and

$$P({\mathsf{\Phi }}_i>U(\underset{\_}{y})\left|\underset{\_}{x} ,\underset{\_}{y}\right)={\displaystyle \frac{\tau }{2}}.$$

(27)

Equations (26) and (27) can be numerically solved simultaneously to obtain the lower and upper bounds. The CIs were derived by approximating the posterior distributions using the MCMC method, implemented in the R programming language. MCMC sampling was employed to generate posterior samples of the parameters, from which CIs were constructed by taking appropriate quantiles of these samples.

4. Statistical Illustration

This section aims to investigate the precision of the numerical results of estimation and prediction on the basis of simulated and real data.

4.1. Modeling and Simulation Procedure

A simulation study is performed in this subsection to evaluate the effectiveness of the introduced Bayes estimates based on data sampled from the TL-IK $(a, b, \theta )$ distribution considering the SS-PALT. Employing Type II censoring, the estimated risk (ER) and parameter CIs were determined.

The Adaptive Metropolis (AM) algorithm, a specialized MCMC algorithm, is a variant of the Metropolis–Hastings algorithm, established by Ref. [31]. It improves the efficiency of standard Metropolis–Hastings by adapting the proposal distribution during the sampling process based on the past history of the chain. The R package version 4.5.0 was employed to conduct the simulation studies, which serve to visualize and clarify the findings.

The steps of the AM algorithm are outlined below:

• Step 1. Choose an initial $\left(l\times 1\right)$ vector of values for ${\underset{\_}{\psi }}^{\left(0\right)}$.
• Step 2. At each iteration $\xi ,$ where $\xi =1, 2, \dots , h$, a proposed value ${\underset{\_}{\psi }}^{*}$ is sampled from the candidate distribution $j_{\xi }\left({\underset{\_}{\psi }}^{*}|{\underset{\_}{\psi }}^{\left(0\right)}, {\underset{\_}{\psi }}^{\left(1\right)}, \dots , {\underset{\_}{\psi }}^{\left(\xi -1\right)}\right)$. The algorithm utilizes a Gaussian candidate distribution with the current point as its mean ${\underset{\_}{\psi }}^{\left(\xi -1\right)}$, with a covariance matrix $C_{\xi }$ that depends on the sequence of previous points $C_{\xi }\left({\underset{\_}{\psi }}^{\left(0\right)}, {\underset{\_}{\psi }}^{\left(1\right)}, \dots , {\underset{\_}{\psi }}^{\left(\xi -1\right)}\right).$
• Step 3. Evaluate the acceptance rate:

  $$\mathrm{A}\mathrm{R}={\displaystyle \frac{\pi \left({\underset{\_}{\psi }}^{\mathrm{*}}|\underset{\_}{x}\right)}{\pi \left({\underset{\_}{\psi }}^{\left(\xi -1\right)}|\underset{\_}{x}\right)}},$$

  (28)

  the posterior distribution $\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)$ is considered without including the normalization constant.
• Step 4. Retain ${\underset{\_}{\psi }}^{\mathrm{*}}$ as ${\underset{\_}{\psi }}^{\left(\xi \right)}$ with probability $\min \left(AR, 1\right)$. If ${\underset{\_}{\psi }}^{\mathrm{*}}$ is rejected, then set ${\underset{\_}{\psi }}^{\left(\xi \right)}={\underset{\_}{\psi }}^{\left(\xi -1\right)}$. This acceptance decision can be implemented by simulating a random variable $U$ from a uniform distribution. If the value $u$ is less than or equal to $AR$, then ${\underset{\_}{\psi }}^{\left(\xi \right)}$ at iteration $\xi$ is updated to ${\underset{\_}{\psi }}^{*}$; otherwise, it retains the value from the previous iteration, ${\underset{\_}{\psi }}^{\left(\xi \right)}={\underset{\_}{\psi }}^{\left(\xi -1\right)}.$
• Step 5. Execute Steps 2–4 repeatedly for $h$ iterations, where $h$ should be substantial enough to ensure the stability of the results.
• Step 6. A warm-up phase is applied to mitigate the influence of initial values, during which the first $M$ simulated $\underset{\_}{\psi }$ values are rejected. Using the AM algorithm, the Bayes estimates of ${\psi }_j$, $j=\mathrm{1,2},\mathrm{3,4}$, can be derived within terms of both the BSEL and BLL functions as follows:
• where $\left({\psi }_{1l}, {\psi }_{2l}, {\psi }_{3l},{\psi }_{4l}\right)=\left({\alpha }_l,{\beta }_l,{\lambda }_l,{\theta }_l\right), l=1, 2, \dots , h$, are drawn from the posterior distribution, with $M$ denoting the warm-up phase

The simulation procedure for Type II censored data involves the following steps:

• Step 1: Random samples of size $n$ were generated from the TL-IK $\left(a, b, \theta \right)$ distribution using the population parameter values of $\underset{\_}{\vartheta }$.
  • According to ref. [32], one can obtain the quantile function of TL-IK to generate data from the TL-IK distribution using the uniform distribution, for which the transformation is:

    $$x=\left[{\left(1-{\left(u\right)}^{{\displaystyle \frac{1}{b}}}\right)}^{-{\displaystyle \frac{1}{a}}}-1\right]\left(1-\sqrt{1-{\left(u\right)}^{{\displaystyle \frac{1}{\theta }}}}\right), 0<u<1$$

    where $u_1,u_2,\dots ,u_n$ represents random samples drawn from a uniform distribution between 0 and 1.
• Step 2: The experiment was conducted with Type II censoring, a scheme where termination occurs after a pre-defined number of failures is observed. Initially, each of the n test items is tested subject to usual use conditions. The test is designed to end when the number of failures reaches $n_1={\pi }_{1}n$ a pre-specified value, calculated as a proportion ${\pi }_1=20\%$ of the total items originally placed employing usual use. Subsequently, the ($80\% n$) of the original $n$ items that have not failed are tested under accelerated use conditions. This phase of the test ends when the number of failures $n_2$ reaches a pre-specified proportion $n_2={\pi }_{2}n$ of the total items under accelerated stress. ${\pi }_2=50\%$ is the pre-determined failure proportion for this accelerated testing.
  • Simulation results of the Bayes estimates under the SEL and LINEX loss functions are available in

    Table 1. The averages, ERs, and 95% CIs for the parameters $a, b, \theta , \beta .$ using informative prior under SEL function based on Type II censoring (N = 10,000,${ \pi }_1=20\%, {\pi }_2=20\% \mathit{and} {\pi }_1=20\%, {\pi }_2=50\%$) (Case 1, $a=1.2, b=1.3, \theta =0.8, \beta =1.1)$.

    n	$\mathit{\pi }$	Parameters	Averages	ERs	UL	LL	length
    30	${\pi }_1=20\%$ ${\pi }_2=20\%$	$a$	1.19756	6.80101 × 10−6	1.20242	1.19591	0.00356
    		$b$	1.29837	4.26106 × 10−6	1.30029	1.29615	0.00414
    		$\theta$	0.79713	9.10427 × 10−6	0.80372	0.79594	0.00361
    		$\beta$	1.10143	3.79705 × 10−6	1.10326	1.09895	0.00431
    	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19714	1.17365 × 10−5	1.20078	1.19426	0.00566
    		$b$	1.30292	1.27480 × 10−5	1.30579	1.29949	0.00629
    		$\theta$	0.79670	1.29232 × 10−5	0.79978	0.79486	0.00491
    		$\beta$	1.10379	1.67931 × 10−5	1.10618	1.09994	0.00623
    60	${\pi }_1=20\%$ ${\pi }_2=20\%$	$a$	1.20111	1.90768 × 10−6	1.20110	1.19945	0.00278
    		$b$	1.29839	3.14734 × 10−6	1.29952	1.29681	0.00271
    		$\theta$	0.79998	1.26245 × 10−6	0.80029	0.79878	0.00232
    		$\beta$	1.10119	2.29388 × 10−6	1.10265	1.09919	0.00346
    	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19757	7.19757 × 10−6	1.20088	1.19603	0.00342
    		$b$	1.30205	4.95889 × 10−6	1.30303	1.29999	0.00304
    		$\theta$	0.79799	4.68001 × 10−6	0.80093	0.79652	0.00302
    		$\beta$	1.10273	9.50241 × 10−6	1.10459	1.09998	0.00461
    100	${\pi }_1=20\%$ ${\pi }_2=20\%$	$a$	1.19915	9.45226 × 10−7	1.20159	1.19830	0.00152
    		$b$	1.29982	7.70518 × 10−7	1.30076	1.29909	0.00167
    		$\theta$	0.79968	6.93797 × 10−7	0.80009	0.79869	0.00162
    		$\beta$	1.09983	6.75605 × 10−7	1.10048	1.09865	0.00183
    	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19727	3.16548 × 10−6	1.20074	1.19593	0.00332
    		$b$	1.29917	1.04986 × 10−6	1.29994	1.29783	0.00211
    		$\theta$	0.79836	3.19707 × 10−6	0.80066	0.79717	0.00266
    		$\beta$	1.10209	5.36709 × 10−6	1.10349	1.10001	0.00348

    ,

    Table 2. The averages, ERs, and 95% CIs for the parameters $a, b, \theta , \beta$ using informative prior under the LINEX loss function based on Type II censoring (N = 10,000, $\pi =0.20, {\pi }_1=20\%,{\pi }_2=20\%$) (Case 1, $a=1.2, b=1.3, \theta =0.8, \beta =1.1, v=-2)$.

    n	$\mathit{\pi }$	Parameters	Averages	ERs	UL	LL	Length
    30	${\pi }_1=20\%$ ${\pi }_2=20\%$	$a$	1.20283	9.69319 × 10−6	1.20435	1.19973	0.00462
    		$b$	1.30347	1.38962 × 10−5	1.30491	1.29989	0.00502
    		$\theta$	1.10157	3.35660 × 10−6	1.10350	1.10005	0.00345
    		$\beta$	0.79836	3.18884 × 10−6	0.79979	0.79711	0.00269
    30	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19685	1.21794 × 10−5	1.19968	1.19442	0.00526
    		$b$	1.30381	1.70145 × 10−5	1.30573	1.29989	0.00584
    		$\theta$	0.80346	1.37878 × 10−5	0.80489	0.79987	0.0052
    		$\beta$	1.09747	7.87638 × 10−6	1.09961	1.09588	0.00372
    60	${\pi }_1=20\%$ ${\pi }_2=20\%$	$a$	1.20044	8.04963 × 10−7	1.20186	1.19897	0.00289
    		$b$	1.30055	8.50135 × 10−7	1.30152	1.29889	0.00263
    		$\theta$	0.80049	9.20227 × 10−7	0.80196	0.79886	0.00309
    		$\beta$	1.09943	8.84456 × 10−7	1.10044	1.09796	0.00248
    60	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19741	8.09336 × 10−6	1.19981	1.19564	0.00416
    		$b$	1.29814	4.47227 × 10−6	1.29988	1.29633	0.00355
    		$\theta$	0.79818	4.00739 × 10−6	0.79989	0.79642	0.00348
    		$\beta$	1.10188	6.38349 × 10−6	1.10263	1.09982	0.00281
    100	${\pi }_1=20\%$ ${\pi }_2=20\%$	$a$	1.19987	2.86827 × 10−7	1.20081	1.19886	0.00195
    		$b$	1.30108	3.20254 × 10−7	1.30183	1.29991	0.00193
    		$\theta$	0.80028	2.73722 × 10−7	0.80097	0.79929	0.00168
    		$\beta$	1.09984	1.81825 × 10−7	1.10045	1.09849	0.00196
    100	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19942	1.49934 × 10−6	1.20070	1.19698	0.00373
    		$b$	1.29877	2.59439 × 10−6	1.30036	1.29713	0.00323
    		$\theta$	0.80117	1.78183 × 10−6	0.80223	0.79999	0.00223
    		$\beta$	1.10103	1.53018 × 10−6	1.10209	1.09954	0.0025

    ,

    Table 3. The averages, ERs, and 95% CIs for the parameters $a, b, \theta , \beta$ using informative prior under the SEL and LINEX loss function based on Type II censoring (N = 10,000$, {\pi }_1=20\%,{\pi }_2=50\%$) (Case 2, $a=1.2, b=1.3, \theta =0.8, \beta =2.6)$.

    n	$\mathit{\pi }$	Parameters	Averages	ERs	UL	LL	Length
    30	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19741	7.52779 × 10−6	1.20081	1.19631	0.00349
    		$b$	1.29750	7.71457 × 10−6	1.29978	1.29518	0.00460
    		$\theta$	0.79670	1.29232 × 10−5	0.79978	0.79486	0.00491
    		$\beta$	2.59797	6.80969 × 10−6	2.59987	2.59494	0.00493
    60	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19895	6.83017 × 10−6	1.20155	1.19741	0.00237
    		$b$	1.30141	2.29929 × 10−6	1.30209	1.29961	0.00248
    		$\theta$	0.79873	2.42454 × 10−6	0.80455	0.79652	0.00373
    		$\beta$	2.59747	1.54899 × 10−6	2.59961	2.59588	0.00372
    100	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19978	9.04491 × 10−7	1.20385	1.19874	0.00232
    		$b$	1.29938	6.17997 × 10−7	1.30029	1.29823	0.00207
    		$\theta$	0.80011	3.53507 × 10−7	0.80152	0.79869	0.00238
    		$\beta$	2.60014	7.84607 × 10−7	2.60169	2.59868	0.00302
    LINEX loss function, $\mathit{v}$ = −2
    n	$\mathit{\pi }$	Parameters	Averages	ERs	UL	LL	Length
    30	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.20281	9.93651 × 10−6	1.20468	1.19989	0.00479
    		$b$	1.29755	7.47020 × 10−6	1.29947	1.29535	0.00412
    		$\theta$	0.79765	7.07696 × 10−6	0.79953	0.79464	0.00489
    		$\beta$	2.59786	6.42881 × 10−6	2.60004	2.59530	0.00474
    60	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.20204	5.29223 × 10−6	1.20396	1.19995	0.00400
    		$b$	1.30106	3.84336 × 10−6	1.30271	1.29928	0.00342
    		$\theta$	0.800042	1.20481 × 10−6	0.80167	0.79827	0.00339
    		$\beta$	2.59937	1.32895 × 10−6	2.60064	2.59761	0.00303
    100	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19955	7.07557 × 10−7	1.20037	1.19807	0.00229
    		$b$	1.30048	8.02209 × 10−7	1.30177	1.29892	0.00285
    		$\theta$	0.79959	6.33673 × 10−7	0.80066	0.79788	0.00278
    		$\beta$	2.60026	8.53778 × 10−7	2.60141	2.59892	0.00249

    and

    Table 4. The averages, ERs, and 95% CIs for the parameters $a, b, \theta , \beta ,$ using informative prior under the SEL and LINEX loss function based on Type II censoring (N = 10,000$,{\pi }_1=20\%,{\pi }_2=50\%$) (Case 3, $a=1.2, b=1.3, \theta =0.8,\beta =3.5)$.

    n	$\mathit{\pi }$	Parameters	Averages	ERs	UL	LL	length
    30	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19879	1.87579 × 10−6	1.20123	1.19749	0.00224
    		$b$	1.29836	3.28453 × 10−6	1.29968	1.29637	0.00331
    		$\theta$	0.80073	1.47123 × 10−6	0.80244	0.79928	0.00271
    		$\beta$	3.50141	2.42485 × 10−6	3.50241	3.49966	0.00275
    60	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.20064	8.61574 × 10−7	1.20003	1.19948	0.00222
    		$b$	1.29960	9.91519 × 10−7	1.30025	1.29826	0.00199
    		$\theta$	0.80065	8.81992 × 10−7	0.80024	0.79939	0.00249
    		$\beta$	3.50076	8.72280 × 10−7	3.50163	3.49904	0.00259
    100	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19966	2.65474 × 10−7	1.20055	1.19859	0.00183
    		$b$	1.29981	1.87694 × 10−7	1.30035	1.29869	0.00166
    		$\theta$	0.80031	2.28577 × 10−7	0.80157	0.79946	0.00146
    		$\beta$	3.49979	1.73696 × 10−7	3.50029	3.49892	0.00137
    LINEX loss function, $\mathit{v}$ = −2
    n	$\mathit{\pi }$	Parameters	Averages	ERs	UL	LL	Length
    30	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19949	1.10353 × 10−6	1.20086	1.19747	0.00339
    		$b$	1.29860	3.16168 × 10−6	1.300072	1.29644	0.00363
    		$\theta$	0.80002	1.18626 × 10−6	0.80129	0.79730	0.00399
    		$\beta$	3.50085	1.11262 × 10−6	3.50179	3.49942	0.00238
    60	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19948	8.79795 × 10−7	1.20077	1.19803	0.00274
    		$b$	1.29895	7.51404 × 10−7	1.30044	1.29761	0.00284
    		$\theta$	0.80057	9.47693 × 10−7	0.80207	0.79924	0.00283
    		$\beta$	3.49931	8.60677 × 10−7	3.50039	3.49810	0.00229
    100	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	1.19946	3.55234 × 10−7	1.20013	1.19833	0.00180
    		$b$	1.30024	2.89304 × 10−7	1.30092	1.29893	0.00199
    		$\theta$	0.80033	3.49186 × 10−7	0.80106	0.79896	0.00209
    		$\beta$	3.50004	1.85641 × 10−7	3.50078	3.49909	0.00168

    , with sample sizes of (n = 30, 60, 100), where (Case 1,$a=1.2, b=1.3, \theta =0.8, \beta =1.1,{\pi }_1=20\%, {\pi }_2=20\% \mathit{and} {\pi }_1=20\%, {\pi }_2=50\%)$, (Case 2, $a=1.2, b=1.3, \theta =0.8, \beta =2.6,{\pi }_1=20\%,{\pi }_2=20\%)$, and (Case 3, $a=1.2, b=1.1, \theta =0.8, \beta =3.5, {\pi }_1=20\%, {\pi }_2=20\%)$.
  • The set of hyperparameters (${\alpha }_i=10, 0.5, 0.1, 0.5,{ c}_i=10, 10, 20, 15$), for $i=1,\dots ,4,$ respectively.
  • An informative prior distribution was specified using the gamma distribution.
• Step 3: To ensure robustness and obtain sufficient data, the previous steps were repeated $N=10,000$, where $N$ denotes the predetermined number of simulated samples.
• Step 4: Accuracy measures are used to evaluate the estimates’ performance. The average and the ER will be utilized to study their precision and variation,
• where $\overline{{\widehat{\mathsf{\Phi }}}_i}={\displaystyle \frac{\sum _{j=1}^N{\widehat{\mathsf{\Phi }}}_i^j}{N}}$, ${\mathsf{\Phi }}_i=a,b,\theta$ and $\beta$ and $ER={\displaystyle \frac{\sum _{j=1}^N{\left(estimate-true value\right)}^2}{N}}.$
• Step 5: Simulation yielded the Bayes estimates for parameters and the acceleration factor under BSEL and BLL loss functions, based on (23) and (24), according to the MCMC method, as displayed in

  Table 5. Estimates and ERs of the parameters under BSEL and BLL function based on informative prior (N = 10,000$, n=30, {\pi }_1=20\%,{\pi }_2=20\%$) ($a=1.2, b=1.3, \theta =0.8, \beta =1.1,v=-2)$.

  Estimate	$\mathit{\omega }$ = 0“BSEL”	$\mathit{\omega }$ = 0.2	$\mathit{\omega }$ = 0.4	$\mathit{\omega }$ = 0.6	$\mathit{\omega }$ = 0.8	$\mathit{\omega }$ = 1“MLE”
  $\widehat{a}$	1.19965	1.23948	1.27983	1.31948	1.35956	1.40000
  ER ($\widehat{a}$)	2.08351 × 10−7	4.56824 × 10−7	6.52945 × 10−7	8.18127 × 10−7	1.91091 × 10−6	1.18964 × 10−6
  $\widehat{b}$	1.29981	1.33871	1.37956	1.42063	1.45994	1.50000
  ER ($\widehat{b}$)	1.41115 × 10−7	5.01444 × 10−7	6.01049 × 10−7	9.05320 × 10−7	1.17660 × 10−6	9.03869 × 10−6
  $\widehat{\theta }$	0.79997	0.88024	0.96082	1.04028	1.11995	1.20000
  ER ($\widehat{\theta }$)	1.81662 × 10−7	4.71937 × 10−7	6.75914 × 10−7	9.22254 × 10−7	1.73144 × 10−6	9.28580 × 10−6
  $\widehat{\beta }$	1.09979	1.14045	1.18013	1.22001	1.26001	1.30000
  ER ($\widehat{\beta }$)	1.73757 × 10−7	3.49318 × 10−7	5.99530 × 10−7	6.60153 × 10−7	2.18880 × 10−6	5.68955 × 10−6
  BLL function, $\mathit{v}$ = −2
  Estimate	$\mathit{\omega }$ = 0“BSEL”	$\mathit{\omega }$ = 0.2ER	$\mathit{\omega }$ = 0.4	$\mathit{\omega }$ = 0.6	$\mathit{\omega }$ = 0.8	$\mathit{\omega }$ = 1“MLE”
  $\widehat{a}$	1.20026	1.23854	1.28095	1.32057	1.36018	1.40000
  ER ($\widehat{a}$)	2.66801 × 10−7	3.41489 × 10−7	6.64653 × 10−7	8.13729 × 10−7	1.18056 × 10−6	8.92956 × 10−6
  $\widehat{b}$	1.30006	1.33887	1.37957	1.42026	1.45996	1.50000
  ER ($\widehat{b}$)	2.03979 × 10−7	3.23921 × 10−7	6.76975 × 10−7	9.94193 × 10−7	2.81884 × 10−6	8.65829 × 10−6
  $\widehat{\theta }$	0.80006	0.88105	0.96043	1.04013	1.12024	1.20000
  ER ($\widehat{\theta }$)	2.06671 × 10−7	3.62945 × 10−7	6.22699 × 10−7	9.81924 × 10−7	2.72779 × 10−6	1.08953 × 10−5
  $\widehat{\beta }$	1.09995	1.13881	1.18079	1.22004	1.25974	1.30000
  ER ($\widehat{\beta }$)	1.66267 × 10−7	5.75918 × 10−7	6.08285 × 10−7	8.30178 × 10−7	1.18664 × 10−6	5.67469 × 10−6

  , using samples of size (n = 30, $\pi =0.20$ and $r=0.1$). Also, the Bayes estimates for the parameters and their standard error for the real datasets under Type II censoring using the SE and LINEX loss functions (${\pi }_1=20\%,{\pi }_2=20\%$) are proposed in

  Table 6. Bayes estimates for the parameters and their standard errors for the real datasets based on Type II censoring (${\pi }_1=20\%,{\pi }_2=20\%$).

  Application I
  ${\pi }_1=20\%,$ ${\pi }_2=20\%$	n	Parameters	Estimates	SE		Parameters	Estimates	SE
  	30	$a$	1.19967	1.12506 × 10−4	${\pi }_1=20\%,$ ${\pi }_2=50\%$	$a$	1.20142	0.00010
  		$b$	1.19753	2.41407 × 10−4		$b$	1.20072	0.00012
  		$\theta$	2.39801	1.29049 × 10−4		$\theta$	2.40077	0.00013
  		$\beta$	1.49843	1.11329 × 10−4		$\beta$	1.50037	0.00012
  LINEX loss function, $\mathit{v}$ = −2
  ${\pi }_1=20\%,$ ${\pi }_2=20\%$	n	Parameters	Estimates	SE		Parameters	estimates	SE
  	30	$a$	1.19913	1.15734 × 10−4	${\pi }_1=20\%,$ ${\pi }_2=50\%$	$a$	1.20183	0.00022
  		$b$	1.19957	1.25401 × 10−4		$b$	1.20160	0.00012
  		$\theta$	2.40181	2.32409 × 10−4		$\theta$	2.40277	0.00021
  		$\beta$	1.50245	2.89954 × 10−4		$\beta$	1.49979	0.00011
  Application II
  ${\pi }_1=20\%,$ ${\pi }_2=20\%$	n	Parameters	Estimates	SE		Parameters	estimates	SE
  	20	$a$	0.89922	1.27238 × 10−4	${\pi }_1=20\%$ ${\pi }_2=50\%$	$a$	0.89919	0.00017
  		$b$	1.39874	2.32324 × 10−4		$b$	1.39958	0.00022
  		$\theta$	1.20049	1.29285 × 10−4		$\theta$	1.20240	0.00028
  		$\beta$	2.20092	7.80205 × 10−5		$\beta$	2.19958	0.00014
  LINEX loss function, $\mathit{v}$ = −2
  ${\pi }_1=20\%,$ ${\pi }_2=20\%$	n	Parameters	Estimates	SE		Parameters	estimates	SE
  	20	$a$	0.89934	7.68515 × 10−5	${\pi }_1=20\%,$ ${\pi }_2=50\%$	$a$	0.90092	0.00015
  		$b$	1.39995	8.56565 × 10−5		$b$	1.39995	0.00015
  		$\theta$	1.19875	1.79654 × 10−4		$\theta$	1.20177	0.00014
  		$\beta$	2.19924	1.71894 × 10−4		$\beta$	2.20045	0.00016

  .

4.2. Some Applications

This subsection aims to show the practical applicability of the proposed method. This is achieved by employing two real sets of lifetime data. The R programming language is utilized to conduct a Kolmogorov–Smirnov goodness of fit test, which helped to determine how well the TL-IK $\left(a, b, \theta \right)$ distribution aligns with the two real datasets.

The validity of the fitted model was checked by applying the Kolmogorov–Smirnov goodness-of-fit test. The high p-values obtained, 0.5860 and 0.6465, indicate and confirm a very good representation of the data by the model in both cases.

A comparative analysis is included using standard benchmark distributions such as the Weibull, TL-log-logistic (TL-Fisk), and TL-Lomax (TL-Pareto Type II). Specifically, we evaluated the performance of these models on two real lifetime datasets using widely accepted model selection criteria: the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Consistent AIC (CAIC). The results, presented in Table 7 and Table 8, show that the TL-IK distribution consistently yields the lowest values across these criteria and exhibits higher p-values, indicating a better fit. These findings empirically support the claim of the TL-IK distribution’s superior flexibility in capturing diverse failure patterns compared to traditional alternatives.

4.2.1. Application 1

The data given by ref. [33] refer to the time between successive failures of a repairable item, a crucial aspect for understanding the reliability and maintainability of systems that can be restored after failure. Analyzing these inter-failure times, especially under the varying stress levels of SS-PALT, offers valuable insights into how the failure process evolves. The data are 1.43, 0.11, 0.71, 0.77, 2.63, 1.49, 3.46, 2.46, 0.59, 0.74, 1.23, 0.94, 4.36, 0.40, 1.74, 4.73, 2.23, 0.45, 0.70, 1.06, 1.46, 0.30, 1.82, 2.37, 0.63, 1.23, 1.24, 1.97, 1.86, and 1.17.

The dataset provided by ref. [33], which consists of inter-failure times of a repairable system, is utilized in this paper. These inter-failure times represent the durations between successive failures and are crucial for analyzing the reliability and maintainability of systems that are subject to repair. In the analysis, the data are treated within the framework of an SS-PALT model. The observed failure times were aligned with a hypothetical step-stress testing structure to reflect varying stress levels over time. This alignment allowed the proposed estimation procedures to be applied directly, and the model parameters were estimated accordingly.

By using this dataset, the practical applicability of the proposed methodology is demonstrated. The structure of the data, featuring variable failure intervals under conditions that can be interpreted as sequentially elevated stress levels, makes it particularly suitable for the SS-PALT framework. As such, the dataset serves as a meaningful validation case for illustrating how the model performs under real-world conditions.

Table 7. AIC, BIC, CAIC, and p-value for the data of Application I.

Model	AIC	BIC	CAIC	p-Value
TL-IK	66.795	70.997	67.718	0.5860
Weibull	460.322	464.526	461.245	0.1324
TL-log-logistic	67.013	71.216	67.936	0.2391
TL-Lomax	97.041	101.244	97.964	0.1350

Table 7. AIC, BIC, CAIC, and p-value for the data of Application I.

Model	AIC	BIC	CAIC	p-Value
TL-IK	66.795	70.997	67.718	0.5860
Weibull	460.322	464.526	461.245	0.1324
TL-log-logistic	67.013	71.216	67.936	0.2391
TL-Lomax	97.041	101.244	97.964	0.1350

4.2.2. Application 2

The second application, presented by ref. [34], utilizes data on the maximum flood level of the Susquehanna River at Harrisburg, Pennsylvania, measured in millions of cubic feet per second. Each data point corresponds to the peak flood level recorded over a four-year interval, spanning from 0.654 million cubic feet per second for the years 1890–1893 to 0.265 million cubic feet per second for the period 1966–1969.

The data are 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.3235 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, and 0.265.

Table 8. AIC, BIC, CAIC, and p-value the data of Application II.

Model	AIC	BIC	CAIC	p-Value
TL-IK	13.212	17.415	14.135	0.6465
Weibull	55.950	58.937	57.450	0.2007
TL-log-logistic	14.333	18.536	15.256	0.1178
TL-Lomax	18.639	21.627	20.139	0.1190

Table 8. AIC, BIC, CAIC, and p-value the data of Application II.

Model	AIC	BIC	CAIC	p-Value
TL-IK	13.212	17.415	14.135	0.6465
Weibull	55.950	58.937	57.450	0.2007
TL-log-logistic	14.333	18.536	15.256	0.1178
TL-Lomax	18.639	21.627	20.139	0.1190

4.3. Concluding Remarks

(a)

Table 1, Table 2, Table 3 and Table 4 show that the accuracy of the ERs improves as the sample size increases. This suggests the Bayesian estimation procedure employed is likely stable and converges to more reliable risk assessments with increased data availability, which indicates the suitability of the TL-IK distribution for modeling the failure data under accelerated testing conditions. This improved accuracy and parameter precision are essential for making informed decisions about product reliability and for extrapolating performance to normal operating conditions.

(b)

Table 1, Table 2, Table 3 and Table 4 show that the width of the parameter intervals decreases with larger sample sizes, reflecting improved precision in the estimation of the TL-IK distribution parameters within the proposed Bayesian framework. This is because smaller intervals indicate reduced uncertainty about the true parameter values.

(c)

The numerical results in Table 5 reveal that the Bayes estimates (using both the BSEL and BLL loss functions) yield the smallest ERs compared to the ML estimates. This implies that the Bayes estimators, when used with these specific loss functions, perform better in minimizing the estimated risk. This underscores the importance and potential benefits of carefully choosing an appropriate loss function within a Bayesian framework. Also, Table 6 displays the Bayes estimates for the parameters and their standard errors for the real datasets based on Type II censoring under the SE and LINEX loss functions (${\pi }_1=20\%,{\pi }_2=20\%$).

(d)

Table 1, Table 2, Table 3 and Table 4 indicate an inverse relationship between the acceleration factor and the ERs; higher acceleration factors lead to lower ERs. This implies that the TL-IK model or the analysis approach might be capturing a non-linear effect, suggesting that within the tested range, very high acceleration levels could be interpreted by the model as indicating greater apparent robustness during the short, high-stress period, which is not generally expected in standard degradation scenarios.

(e)

The results in Table 1 and Table 2 demonstrate that allocating a smaller proportion $\pi$ of the sample to accelerated conditions enhances the accuracy of the model parameters’ ERs and the acceleration factor. This implies that allocating a smaller proportion of the sample to accelerated conditions in SS-PALT with a TL-IK distribution enhances model accuracy by providing richer data at usual use conditions, leading to a more stable baseline for parameter estimation, a clearer understanding of the dominant failure mechanism under normal stress, and reduced uncertainty from extrapolating high-stress data. This also yields a more accurate acceleration factor by improving the understanding of the stress–life relationship, mitigating over-acceleration effects that might introduce non-representative failures, and resulting in a more robust model fit grounded in real-world behavior, ultimately leading to more reliable predictions of product life.

5. Overall Summary

Assessing the lifespan of highly reliable products under usual operating conditions can be time-consuming. To speed up reliability estimation, ALT or PALT are employed. While ALT exposes items only to elevated stress levels, which may not always reflect real-world conditions, PALT incorporates both normal and accelerated environments to provide a more realistic assessment. This paper focuses on SS-PALT with Type II censoring, assuming product lifetimes have a TL-IK distribution. The Bayes estimators are developed for the shape parameters and acceleration factor of the TL-IK distribution, using symmetric (BSEL), asymmetric (BLL) functions, and informative priors. Numerical studies and real-world data applications demonstrate the performance of our proposed methodology. It is generally noted that lower weighting $\omega$ leads to increased accuracy in estimated ERs. Moreover, utilizing informative priors appears to reduce the ERs of the estimated parameters, the acceleration factor, and the interval length as the sample size grows.

6. Future Research

This manuscript opens several avenues for future research, such as exploring and comparing ML prediction methods alongside the established Bayesian prediction approaches based on SS-PALT, which would be valuable. Also, investigating the design of optimal constant-stress PALT and optimal SS-PALT specifically fitted for the TL-IK distribution deserves further attention. Extending the parameter, rf, and hrf estimation to accommodate other censoring schemes, such as Type I, progressive, joint, and hybrid censoring, could provide a more comprehensive understanding. Moreover, examining the impact of different stress types, including progressive, cyclic, random, and combined stress, on the proposed models would expand their applicability. Employing and comparing various alternative estimation methodologies like least squares, modified moments, modified ML, E-Bayesian estimation, and one-sample prediction could offer valuable insights.

Future work may explore the integration of semi-parametric models, such as the Cox proportional hazards model, to enhance modeling flexibility—particularly in scenarios where the underlying hazard function is complex or not explicitly defined. This direction could offer a robust alternative to the fully parametric framework by relaxing assumptions on the baseline hazard, thereby broadening the applicability of the proposed methodology to more diverse reliability data.

Additionally, investigating the properties of the TL-IK distribution within the framework of generalized order statistics or dual generalized order statistics, and subsequently specializing in record statistics, presents an interesting theoretical extension. Considering a wider range of symmetric and asymmetric loss functions for Bayesian estimation, such as balanced absolute, balanced binary, balanced modified LINEX, and balanced general entropy loss functions, could provide more robust inference.

Furthermore, exploring empirical Bayesian inference based on Type II censored samples, particularly when hyperparameters are unknown, offers a practical extension. Finally, investigating stress-strength reliability models based on the TL-IK distribution could provide valuable tools for reliability assessment.