Inferences for Stress-Strength Reliability Model in the Presence of Partially Accelerated Life Test to Its Strength Variable

We focus on estimating the stress-strength reliability model when the strength variable is subjected to the step-stress partially accelerated life test. Based on the assumption that both stress and strength random variables follow Weibull distribution with a common first shape parameter, the inferences for this reliability system are constructed. The maximum likelihood, two parametric bootstraps, and Bayes estimates are obtained. Moreover, approximate confidence intervals, asymptotic variance-covariance matrix, and highest posterior density credible intervals are derived. A simulation study and application to real-life data are conducted to compare the proposed estimation methods developed here and also check the accuracy of the results.


Introduction
Stress-strength models have attracted many statisticians for many years due to their applicability in different and diverse areas such as engineering, economics, and quality control, and, in the last years, there have been numerous applications to medical and engineering problems.
In the last ten years, many authors have been interested in studying the application of the simple stress-strength reliability model, which is more handled theoretically and at the same time is more simple and applicable to implement in practice.
is model of reliability contains a strength variable X and a stress variable Y, which is exposed to it. Such a system will properly function when X exceeds Y; namely, R � P(X > Y) denotes the reliability system. Many estimation studies of reliability system are considered by several statistician researchers under both complete and censored samples from different models, for example, exponential distribution under progressive type-II censoring by Saraçoglu et al. [1], Weibull distribution under complete samples by Kundu and Gupta [2], Kumaraswamy distribution under upper record values by Nadar and Kızılaslan [3], Kumaraswamy distribution under progressive type-II censoring by Nadar et al. [4], Lomax distribution under record values by Mahmoud et al. [5], Burr X distribution under complete samples by Surles and Padgett [6], inverse Lindley distribution under complete samples by Sharma et al. [7], exponential and Weibull by Kumar and Siju [8], Weibull-Gamma distribution under progressively type-II censored samples by Mahmoud et al. [9], general exponential form distribution under complete samples by Mokhlis et al. [10], Rayleigh distribution under complete samples by Afshin [11], modified Weibull model under progressively type-II censored samples by Soliman et al. [12], Lindley distribution using progressively firstfailure censoring by Kumar et al. [13], generalized inverted exponential distribution under progressively first-failure censoring by Krishna et al. [14], Kumaraswamy exponential distribution under progressively type-II censored samples by El-Sagheer and Mansour [15], Burr XII distribution under progressively first-failure censored samples by Saini et al. [16], and generalized Maxwell failure distribution under progressive first-failure censoring by Saini et al. [17].
In previous reliability studies, it is evident that it is difficult to observe the lifetime of highly reliable components because few failures occur in a limited test time due to the very long lifetimes under normal test conditions. erefore, to overcome this problem, we are looking for a catalyst for early failure of the components. Since testing under normal conditions takes a long time, then the development of accelerated life testing (ALT) or partially accelerated life test (PALT) is needed, where units are subjected to a more severe environment (increased or decreased stress levels) than the normal operating environment so that failures can be induced quickly. In this case, ALT or PALT allows experimenters to control higher stress levels to be used in the test.
In PALT, only part of the test components run under a higher stress level than the normal level, while all the test components run under a higher stress level in ALT. We use PALT when the acceleration factor is unknown, where items are examined at both normal and accelerated conditions. According to Nelson [18], there are three types of stress in PALT: constant stress, step stress, and progressive stress. In step-stress partially accelerated life test (S-SPALT), items are tested at a normal level; if it does not fail, then the stress will be changed at a certain time. is type allows the experimenter to select multiple stress factors, for instance, temperature, voltage stress, thermal and electrical cycling and shock, vibration, mechanical stress, and radiation in life testing.
Recently, in parallel with progress in engineering, technology, and manufacturing, the experimenters may want to investigate the stress-strength reliability in case the strength component of the reliability system is exposed to an ALT. In this paper, we study a simple stress-strength model R � P(X > Y) when the component strength exposes to S-SPALT. is system can be described as follows: such a system starts with the normal use condition of the strength variable X and stress variable Y. If the system does not fail before the prespecified time τ, then the strength variable X runs at an acceleration factor (λ). is model will help us to evaluate R when induced early failures to X. Moreover, force to failure on strength may help us to see the effect of change on R due to not only stress variable Y but also exposed stress by accelerating on strength variable X. For this reason, we consider the S-SPALT model introduced by DeGroot and Goel [29] for strength variable X. e outline of the paper is as follows. In Section 2, assumptions of S-SPALT for the stress-strength reliability model are provided. Section 3 deals with the maximum likelihood estimate and asymptotic confidence intervals.
Two parametric bootstrap methods are proposed in Section 4. In a Bayes paradigm, estimation techniques have been assayed in Section 5. In Section 6, a simulation study is conducted to compare the proposed procedures. In Section 7, a real-life data example is presented to illustrate the application of the proposed inference procedures. Finally, a conclusion is furnished in Section 8.

Assumptions of S-SPALT for Reliability System
Suppose that X denote the lifetime of a test item as strength under S-SPALT can be determined, according to DeGroot and Goel [29], by the relation with probability density function (PDF) where T is the lifetime under normal use condition, τ is the time when stress is changed, and λ is the acceleration factor as λ > 1. Suppose X and Y are independent random variables following Weibull distribution (WD) with parameters (α, β 1 ) and (α, β 2 ), respectively, that is, X ∼ W D(α, β 1 ) and Y ∼ W D(α, β 2 ), where the parameter α is common and known, considering strength X under S-SPALT with the PDF f(x) and CDF F(x) and primary stress Y with PDF g(y) and CDF G(y). According to Çetinkaya [30], a partially accelerated life test implemented stress-strength reliability estimation can be written as e PDF of S-SPALT implemented strength variable X as suggested by DeGroot and Goel [29], which is given as follows: and CDF is given as follows: Also, the PDF and CDF of primary stress Y are given by en, by using equations (4) and (6) in equation (3), the reliability of such a system can be obtained as 2 Computational Intelligence and Neuroscience en, if we put α � 1 and β i � 1/θ i , i � 1, 2, the reliability of such a system, R � P(X > Y), devolves to one-parameter exponential distribution. If λ � 1, equation (7) becomes the reliability for a simple stress-strength system without any acceleration. From Figures 1 and 2, we notice the following: (i) e reliability of the system increases with increasing stress change time τ, when the acceleration factor λ is fixed. (ii) Increasing on acceleration factor λ reduces the reliability quickly.
In equation (7), α is common and known, τ is the predetermined stress change time, and β 1 , β 2 , and λ are unknown and need to be estimated; then

Maximum Likelihood Inference
Let X 1 , X 2 , . . . , X n be a random sample of strength from W D(α, β 1 ) and Y 1 , Y 2 , . . . , Y m be a random sample of stress from W D(α, β 2 ). en, by considering equations (4) and (6), the likelihood function of observed samples in this reliability system is given by (see Çetinkaya [30]) and equally Hence, the logarithm of the likelihood function may then be written as Taking the first partial derivatives of the log-likelihood in (11) with respect to β 1 , β 2 , and λ, we get Computational Intelligence and Neuroscience where r ≠ 0 and r ≠ n. To get the MLEs of the unknown parameters, denoted by β 1 , β 2 , and λ, we should equate zℓ/zβ 1 , zℓ/zβ 2 , and zℓ/zλ to zero; thus,  Computational Intelligence and Neuroscience en, we use the Newton-Raphson iteration method to solve (15). erefore, the MLE of R, denoted by R ML , can be obtained by considering the invariance property of the MLEs by replacing the parameters with their estimates as follows: where the parameter α is common and known.

Asymptotic Confidence Interval.
In this subsection, we construct an asymptotic confidence interval (ACI) for R based on the asymptotic normal property of MLEs. Let δ � (β 1 , β 2 , λ) be the MLEs of δ � (β 1 , β 2 , λ); according to Cohen [31], the observed Fisher information matrix, denoted by I(δ), is defined by where Also, the variance of R is obtained by using the delta method as follows: where the first partial derivatives included in (20) can be easily obtained and I − 1 ij is the ij − th element of the inverse of the information matrix I(δ) as given by where 2 13 , and Λ 33 � I 11 I 22 . erefore, the 100(1 − c)% ACI of R is constructed as where z c is 100(1 − c)th upper percentile of standard normal variate N(0, 1).

Parametric Bootstrap
In this section, we propose a resampling technique, the bootstrap procedure, to obtain a more widely used confidence interval. DiCiccio and Efron [32] introduced the bootstrap method and showed that the bootstrap method can improve the accuracy of the confidence intervals, especially when the sample is small such that the normal approximation is inappropriate. Besseris [33] showed that the bootstrap method can provide tighter confidence intervals. Reiser et al. [34] compared difference bootstrap confidence intervals by applying Monte Carlo simulation.
Here, we study two bootstrap methods: bootstrap-p and bootstrap-t. ese bootstrap confidence intervals work as follows.

Bootstrap-t
(1) e same as the bootstrap-p.
(2) e same as the bootstrap-p.

Computational Intelligence and Neuroscience
(3) e same as the bootstrap-p.

Bayes Estimation
Bayes estimation is quite different from MLE and bootstrap methods because it takes into consideration both the information from observed sample data and the prior information. It can characterize the problems more rationally and reasonably. Assume that both parameters β 1 and β 2 have independent Gamma priors, while the parameter λ has usual noninformative prior; see Carlin and Louis [35]: Here, a 1 , a 2 , b 1 , and b 2 are the hyperparameters that reflect the prior knowledge about the unknown parameters. e joint prior of the unknown parameters β 1 , β 2 , and λ is then given by Via Bayes' theorem, based on the considered joint prior (27) and the likelihood (10), the posterior distribution of β 1 , β 2 , and λ given data takes the form It is clear that the conditional posterior densities of β 1 , β 2 , and λ can be written as us, under the squared error loss function, the Bayes estimate of R, denoted by R MC , can be obtained as the mean of the posterior function as given in the following: From (29) and (30), the full conditional posterior densities of β 1 and β 2 are Gamma(n + a 1 , r i�1 x α (i) + n i�r+1 [τ + λ(x (i) − τ)] α +b 1 ) and Gamma(m + a 2 , m i�1 y α (i) + b 2 ), respectively. us, the samples of β 1 and β 2 can be generated by using any Gamma routine. On the other hand, the expression of π * 3 (λ|β 1 , β 2 ; x, y) cannot be written as any wellknown distribution. One can use the method proposed by Devroye [36] to generate sample data from this distribution.
However, the Metropolis-Hastings (MH) with the Gibbs sampling scheme by using normal proposal N(., .) can be effectively used to simulate random samples from (25)- (27).
where N 0 is burn-in to guarantee the convergence and to remove the affection of the selection of initial values. erefore, the 100(1 − c)% highest posterior density (HPD) Bayes credible interval is given by

Simulation Study
In this section, we apply a Monte Carlo simulation to assess the performance of MLEs, bootstrap, and Bayes estimator methods for the stress-strength reliability model with component strength under PALT, along with their ACIs, bootstrap CIs, and HPD credible intervals. Furthermore, we study the variations on reliability with the different cases for both acceleration factor λ and stress change time τ. e performance of estimators is evaluated in terms of mean square error (MSE) for the point estimates, also coverage probability (CP), and average lengths (ALs) for interval estimates (asymptotic, bootstrap, and HPD). We consider five sample sizes such as (n, m) � (10, 15), (20,25), (40, 45), (60, 60), and (90, 90) for eight cases of the true values of the parameters, stress change times, acceleration factor, and corresponding actual values of R, when the common parameter α � 2. ese cases are as follows:

Application to Real-Life Data
In this section, for illustrative purposes, real-life data are presented to inspect the inference procedures. We used two sets of data introduced by Badar and Priest [37] and used by Kundu and Gupta [2]. Set 1, denoted by (X) given in Table 9, is strength measured in GPA for single carbon fibers tested under tension at gauge lengths of 20 mm. Set 2, denoted by (Y) given in Table 10, is strength measured in GPA for single carbon fibers tested under tension at gauge lengths of 10 mm. Both Set 1 and Set 2 are of size 63. For the purpose of the goodness of fit test, the Kolmogorov-Smirnov (KS) distance between the empirical and the fitted distribution functions has been computed. It is 0.087588 and the associated p value is 0.7192 for Set 1, while it is 0.050055 and the associated p value is 0.9975 for Set 2. Hence, the p value for KS has the highest value for Sets 1 and 2. is leads us to conclude that WD is the best fit for the two real data sets. Empirical, Q − Q, and P − P plots are shown in Figures 3 and  4, which make it clear that the WD fits the data very well.
Under the S-SPALT implementation on the strength variable X, the corresponding estimates of parameters change depending on the stress change time τ. For example, at τ � 2.25 and α � 2, MLEs of β 1 � 6.0934, β 2 � 3.0594, and λ � 16.6242. us, using (16), the MLE of R and its corresponding ACI is calculated as R ML � 0.48062 and (0.38595, 0.57531) with length 0.18936, respectively. Also, by implementing the iterative algorithms mentioned in Section 4, the boot-p and boot-t CIs are computed as             Figure 6. Also, we obtained more results with different values of τ as shown in Table 11.

Conclusions
In this paper, we consider the estimation of the stressstrength reliability model when the strength variable is subjected to the S-SPALT. By exposing the strength variable to the acceleration factor, a normal stress-strength model is forced to early failure; this design of the reliability has an effect on the level of reliability. We can illustrate it as follows: (i) In a specific stress change time, the increasing on acceleration loadings decreases the stress-strength reliability.
(ii) Delaying stress change time helps maintain a level of reliability. e main aim of this paper is to study the effect of external stress loading on the strength variable. is external stress includes, for instance, temperature, voltage stress, thermal and electrical cycling and shock, vibration, mechanical stress, and radiation. us, the estimations of the stress-strength reliability model with the corresponding ACIs using the maximum likelihood, two parametric bootstrap, and Bayesian estimation methods are obtained. For illustrative purposes, we have applied a real-life example. A simulation study is computerized to inspect and compare the rendition of the proposed methods for different sample sizes (n, m), different acceleration factor λ, and different stress change time τ. From the results, we observe the following: (1) It is clear that, from Tables 1-8, as sample  ACIs of the bootstrap-t have the largest ALs. At the same time, the Bayesian HPD intervals have the highest CPs. (6) Finally, we can conclude that the proposed inference methods give consistent results. (7) Sometimes, it is worth noting that the available data may be affected by uncertainties and/or inaccuracies. en, strictly speaking, it would be necessary to carry out a fuzzy preprocessing of the data; see [38] as a future work.
Data Availability e article contains data generated from the statistical models used in the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.