Optimal Plan of Multi-Stress–Strength Reliability Bayesian and Non-Bayesian Methods for the Alpha Power Exponential Model Using Progressive First Failure

Systems failing to perform in their harsh working settings is a common occurrence in real-life scenarios. When crossing their lower, upper, or both extreme operating conditions, systems frequently fail to perform their intended roles. Stress–strength reliability, often known as $\mathrm{R}=\mathrm{p}\left(\mathrm{X}<\mathrm{Y}\right)$, has been extensively investigated in the literature. When the applied stress exceeds the system’s strength, a system working under such stress–strength conditions fail to function. Refs. [1,2,3,4,5,6,7,8,9,10,11] are only a few of the significant efforts in this direction. Moreover, the study of stress–strength models has been expanded to multi-component systems, which are systems with several components. Despite the fact that Ref. [12] developed the multi-component stress–strength model decades ago, it has garnered a lot of attention in recent years and has been explored by numerous scholars for both complete and filtered data [13,14,15,16,17]. Stress–strength reliability, $\mathrm{R}=\mathrm{P}(\mathrm{X}<\mathrm{Y})$, has been extensively investigated as a stress–strength model, and the research has also been extended to multi-component systems. However, an equally important practical scenario in which equipment fails in extreme lower and upper working environments receives significantly less attention. When electrical equipment is placed below or above a specified power supply, for example, it will fail. A person’s systolic and diastolic pressure limits should not be exceeded at the same time. There are a plethora of such applications, many of them are straightforward and natural, reflecting sound correlations between diverse real-world events. It is a valuable relationship in a variety of subfields of genetics and psychology, where strength Y should not only be more than stress X, but also less than stress Z. For numerous statistical models, several scholars have examined the estimation of the stress–strength parameter. Refs. [18,19,20] investigated the estimation of $\mathrm{R}=\mathrm{P}(\mathrm{X}<\mathrm{Y}<\mathrm{Z})$ based on independent samples. Ref. [21] obtained the estimation in the stress–strength model with the assumption that a component’s strength lies in an interval and the probability $\mathrm{R}=\mathrm{P}({\mathrm{X}}_1<\mathrm{Y}<{\mathrm{X}}_2),$ where ${\mathrm{X}}_1$ and ${\mathrm{X}}_2$ are random stress variables and $\mathrm{Y}$ is a random strength variable. When $\left({\mathrm{Y}}_1,{\mathrm{Y}}_2,\dots ,{\mathrm{Y}}_{\mathrm{k}}\right)$ are normal random variables, $\mathrm{X}$ is another independent normal random variable, and the estimation of $\mathrm{R}=\mathrm{P}\left[\max \left({\mathrm{Y}}_1,{\mathrm{Y}}_2,\dots ,{\mathrm{Y}}_{\mathrm{k}}\right)<\mathrm{X}\right]$ is considered. Ref. [22] calculated the reliability of a component that was subjected to two separate stresses that were unrelated to the component’s strength. Ref. [23] used a multi-component series stress–strength model to predict system reliability. Using current U-statistics, Ref. [24] proposed a straightforward computation procedure for $\mathrm{P}(\mathrm{X}<\mathrm{Y}<\mathrm{Z})$ and its variance. $\mathrm{P}(\mathrm{X}<\mathrm{Y}<\mathrm{Z})$ was used by Ref. [24] to study the cascade system. Nonparametric statistical inference for $\mathrm{P}(\mathrm{X}<\mathrm{Y}<\mathrm{Z})$ was studied by Ref. [25]. Ref. [26] achieved inference of $\mathrm{R}=\mathrm{P}(\mathrm{X}<\mathrm{Y}<\mathrm{Z})$ for the n-Standby System: a Monte-Carlo simulation approach. Ref. [27] discussed $\mathrm{R}=\mathrm{P}(\mathrm{X}<\mathrm{Y}<\mathrm{Z})$ for the progressive first failure of the Kumaraswamy model.

Many articles appeared in the censored sample, including a multi-component stress–strength model with adaptive hybrid progressive censored data. Ref. [28] discussed Bayesian and maximum likelihood estimation methods of reliability. Weibull distribution is a type of probability distribution. Using progressively first-failure censored data, Ref. [29] determined the reliability of a multi-component stress–strength system based on the Burr XII distribution. Under adaptive hybrid progressive censoring, Ref. [30] proposed multi-component stress–strength estimation of a non-identical component strengths system. Ref. [31] used progressive Type-II censoring data to estimate the reliability of multi-component stress–strength with a generalized linear failure rate distribution. Ref. [32] studied the estimation of multi-component reliability based on progressively Type-II censored data from unit Weibull distribution.

When dealing with reliability features in statistical analysis, even when it is known that some efficiency loss may occur, different censoring strategies, or early deletions of active units, are frequently utilized to save time and money. The Type-II censoring scheme, progressive Type-II censoring system, and progressive first failure censoring method, for example, are all well-known censoring schemes. Ref. [33] presented the progressive first failure censoring scheme, which combines progressive Type-II censoring and first failure censoring strategies to create a new life-test plan.

The progressive first failure censoring system can be summarized as follows. Assume that a life test is administered to $n$ independent groups, each having $\mathrm{k}$ items. The ${\mathrm{R}}_1$ units and the group in which the first failure is identified are randomly withdrawn from the experiment once the first failure ${\mathrm{Y}}_{1;\mathrm{m},\mathrm{n},\mathrm{k}}$ has occurred. The ${\mathrm{R}}_2$ units and the group in which the second failure is observed are randomly withdrawn from the remaining live $\left(\mathrm{n}-{\mathrm{R}}_1-2\right)$ groups at the moment of the second failure ${\mathrm{Y}}_{2;\mathrm{m},\mathrm{n},\mathrm{k}}$. When the $\mathrm{m}$-th observation ${\mathrm{Y}}_{\mathrm{m};\mathrm{m},\mathrm{n},\mathrm{k}}$ fails at the end, the remaining living units ${\mathrm{R}}_{\mathrm{m}}$are removed from the test. The resultant ordered observations ${\mathrm{Y}}_{1;\mathrm{m},\mathrm{n},\mathrm{k}},\dots ,{\mathrm{Y}}_{\mathrm{m};\mathrm{m},\mathrm{n},\mathrm{k}}$are then referred to as progressive first-failure censored with a progressive censored scheme described by $\mathrm{R}=\left({\mathrm{R}}_1,{\mathrm{R}}_2,\dots ,{\mathrm{R}}_{\mathrm{m}}\right),$ where $\mathrm{m}$ are failures and the sum of all removals equals $\mathrm{n}$, that is, $\mathrm{n}=\mathrm{m}+{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{m}}{\mathrm{R}}_{\mathrm{i}}$. The progressive first-failure censoring scheme is reduced to a first-failure censoring scheme when ${\mathrm{R}}_1={\mathrm{R}}_2=\dots ={\mathrm{R}}_{\mathrm{m}}=0$. Similarly, first-failure Type-II censoring is a special instance of this censoring technique when ${\mathrm{R}}_1={\mathrm{R}}_2=\dots ={\mathrm{R}}_{\mathrm{m}-1}=0$ and ${\mathrm{R}}_{\mathrm{m}}=\mathrm{n}-\mathrm{m}$. The progressive first-failure censoring scheme is simplified to the progressive Type-II censoring scheme with the premise that each group contains precisely one unit, $\mathrm{k}=1$. Progressive first-failure censoring is a generalization of progressive censoring.

Letting ${\mathrm{Y}}_{1;\mathrm{m},\mathrm{n},\mathrm{k}},\dots ,{\mathrm{Y}}_{\mathrm{m};\mathrm{m},\mathrm{n},\mathrm{k}}$ denote a progressive first-failure Type-II censored population sample with probability density function (PDF)${\mathrm{f}}_{\mathrm{X}}(.)$ and cumulative distribution function (CDF)${\mathrm{F}}_{\mathrm{X}}(.)$ and the progressive censoring scheme of R, on the basis of considering progressive first-failure, the likelihood function is based on Ref. [34]. The following is a censored sample: where $\mathrm{A}=\mathrm{n}\left(\mathrm{n}-{\mathrm{R}}_1-1\right)\left(\mathrm{n}-{\mathrm{R}}_1-{\mathrm{R}}_2-2\right)\dots \left(\mathrm{n}-{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{m}-1}{\mathrm{R}}_{\mathrm{i}}-\mathrm{m}+1\right)$.

Ref. [35] constructed the APE distribution from the exponential baseline distribution and explored its essential aspects as well as parameter estimation. Ref. [36] developed the alpha power Weibull distribution and demonstrated that it outperforms certain other variants of the Weibull distribution using two real data sets. Ref. [37] used the generalized exponential baseline distribution and the APE approach to introduce the alpha power generalized exponential (APGE) distribution. Closed-form formulas for the APGE distribution’s moment properties were established by Ref. [38]. Because of APE flexibility, recently, many studies gave been conducted, such as in Refs. [39,40]. The PDF and hazard functions of the APE distribution are similar to the Weibull, gamma, and GE distributions. As a result, it can be used to replace the popular Weibull, gamma, and GE distributions. Because the APE distribution’s CDF may be precisely defined, it can also be used to evaluate censored data. The PDF, CDF, and hazard rate function of the APE with parameters $\mathsf{\alpha }$and $\mathsf{\beta }$ are described by and

To the best of our knowledge, statistical inference and optimality on multi-component stress–strength models have been derived for some well-known models using progressively censored sample conditions; this subject has not received much attention under censored data. As a result, we plan to introduce multi-component reliability inference where stress–strength variables follow unit APE based on progressive first-failure. This work addresses the problem of predicting the stress–strength function R, where X, Y, and Z are three independent APE. The moments, skewness, and kurtosis measures of APE are computed. The assessment of likelihood based on increasing first-failure point estimation filtered, asymptotic confidence interval, boot-p, and boot-t approaches are also covered. Using Markov chain Monte Carlo (MCMC), Bayesian estimate methods based on progressive first-failure censoring are produced. A Bayesian estimate has made use of both symmetric and asymmetric loss functions. Based on progressive first-failure censored samples, the balanced and unbalanced loss functions were utilized to assess the reliability of the multi-stress–strength APE distribution. The different optimal schemes of the progressively censored samples are obtained. Monte Carlo simulations and real-world application examples are utilized to assess and compare the performance of the various proposed estimators.

The remainder of the paper is structured as follows: Moments of APE are calculated in Section 2. Section 3 considers the traditional point estimates, maximum likelihood estimation of R, and the parameter model under progressive first failure. Fisher information matrix of the parameter model is obtained in Section 4, while confidence intervals, namely asymptotic intervals, boot-p, and boot-t, are computed in Section 5. In Section 6, the Bayesian approach is considered. Optimization criterion is used to choose the appropriate progressive censoring approach in Section 7. Simulation research is carried out to demonstrate the relative effectiveness of multi-stress–strength reliability under progressive first failure based on different censoring methods in Section 8. Section 9 provides real-world data application examples. Finally, Section 10 has the concluding remarks of this paper.

Let $\mathrm{X}\sim \mathrm{APE}\left(\mathsf{\alpha },\mathsf{\beta }\right)$ and $\mathrm{Y}\sim \mathrm{APE}\left(\mathsf{\alpha },1\right)$. Then, we have $\mathrm{X}=\mathrm{Y}/\mathsf{\beta }$. Thus, we have $\mathrm{E}\left(\mathrm{X}\right)=\frac{1}{\mathsf{\beta }}\mathrm{E}\left(\mathrm{Y}\right)\mathrm{and}\mathrm{E}\left({\mathrm{X}}^2\right)=\frac{1}{{\mathsf{\beta }}^2}\mathrm{E}\left({\mathrm{Y}}^2\right).$

Thus, we have

Then, using Lemmas 1 and 2, we have

Using $\mathrm{E}\left(\mathrm{X}\right)$ and $\mathrm{E}\left({\mathrm{X}}^2\right)$ in the above, we can obtain the method-of-moments estimate as follows. We can set

Thus, by solving the above for $\mathsf{\alpha }$, we can estimate $\mathsf{\alpha }$. We denote this estimate as $\hat{\mathsf{\alpha }}$. Then, the estimate of $\mathsf{\beta }$ can be explicitly obtained by setting $\mathrm{E}\left(\mathrm{X}\right)=\frac{1}{\mathrm{n}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}$ with (11), which is given by $\hat{\mathsf{\beta }}=\frac{1}{\frac{1}{\mathrm{n}}{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\mathrm{X}}_{\mathrm{i}}}\cdot \frac{\hat{\mathsf{\alpha }}}{\hat{\mathsf{\alpha }}-1}\left\{\mathrm{loglog}\left(\hat{\mathsf{\alpha }}\right)+\mathsf{\Gamma }\left(0,\log \left(\hat{\mathsf{\alpha }}\right)\right)+\mathsf{\gamma }\right\}.$

Figure 1 shows the skewness (SK) and kurtosis (KT) by using moment measures of quartile with different values of parameters. Table 1 discusses the first quartile, median, and third quartile and well as SK and KT of the APE distribution with different values.

In this part, the classical point and interval estimation methods are discussed, namely maximum likelihood estimation for finding point estimates of R and asymptotic, boot-p, and boot-t intervals for R for obtaining interval estimates.

The Fisher information matrix of the $\mathsf{\phi }=\left({\mathsf{\beta }}_1,{\mathsf{\beta }}_2,{\mathsf{\beta }}_3\right)$ is expressed as where $A_{11}=E\left(\frac{{\partial }^2\ell }{\partial {\mathsf{\beta }}_1^2}\right)$, $A_{12}=A_{21}=E\left(\frac{{\partial }^2\ell }{\partial {\mathsf{\beta }}_2\partial {\mathsf{\beta }}_1}\right)$, $A_{13}=A_{31}=E\left(\frac{{\partial }^2\ell }{\partial {\mathsf{\beta }}_1\partial {\mathsf{\beta }}_3}\right),$ $A_{22}=E\left(\frac{{\partial }^2\ell }{\partial {\mathsf{\beta }}_2^2}\right)$, $A_{23}=A_{32}=E\left(\frac{{\partial }^2\ell }{\partial {\mathsf{\beta }}_2\partial {\mathsf{\beta }}_3}\right)$ $A_{33}=E\left(\frac{{\partial }^2\ell }{\partial {\mathsf{\beta }}_3^2}\right)$,

In this section, the parameters’ confidence intervals (CIs) are computed. Because our point estimate is the most likely value for the parameter, we should build the confidence intervals on it. CIs are a set of values (intervals) that serve as good approximations of an unknown population parameter. In this investigation, two types of CIs were computed.

Bayesian inference has gained appeal in a variety of sectors in recent years, including engineering, clinical medicine, biology, and so on. Its capacity to analyze data using prior knowledge makes it valuable in dependability studies, where data availability is a major issue. The model parameters ${\mathsf{\beta }}_1,{\mathsf{\beta }}_2,{\mathsf{\beta }}_3$ and $\mathrm{R}$ Bayesian estimates, as well as the corresponding credible intervals, are derived in this section.