Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers

For the first time and by using an entire sample, we discussed the estimation of the unknown parameters θ1, θ2, and β and the system of stress-strength reliability R=P(Y < X) for exponentiated inverted Weibull (EIW) distributions with an equivalent scale parameter supported eight methods. We will use maximum likelihood method, maximum product of spacing estimation (MPSE), minimum spacing absolute-log distance estimation (MSALDE), least square estimation (LSE), weighted least square estimation (WLSE), method of Cramér-von Mises estimation (CME), and Anderson-Darling estimation (ADE) when X and Y are two independent a scaled exponentiated inverted Weibull (EIW) distribution. Percentile bootstrap and bias-corrected percentile bootstrap confidence intervals are introduced. To pick the better method of estimation, we used the Monte Carlo simulation study for comparing the efficiency of the various estimators suggested using mean square error and interval length criterion. From cases of samples, we discovered that the results of the maximum product of spacing method are more competitive than those of the other methods. A two real‐life data sets are represented demonstrating how the applicability of the methodologies proposed in real phenomena.


Introduction
Since Birnbaum's [1] pioneering research, statistical inference of a system stress-strength parameter has received increased attention and is widely used in a variety of engineering applications. If X and Y are two independent random variables that represent the strength and the stress, then R � P(Y < X) is a measure of system performance that naturally arises in mechanical dependability. In this case, the system fails if and only if the applied stress is greater than its strength at any time. Several studies in the statistical literature have investigated the problem of estimating the stressstrength parameters of a system. Kundu and Gupta [2,3] introduced the estimation of stress-strength reliability for generalized exponential and Weibull random variables, respectively. um likelihood method, max Raqab et al. [4] discussed the estimation of R, where X and Y are distributed as two independent three-parameter generalized exponential random variables. Rezaei et al. [5] considered the estimation of R, where X and Y are two independent generalized Pareto distributions. et al. [9], Asgharzadeh et al. [10], Valiollahi et al. [11], Rao et al. [12], Rao et al. [13], Mirjalili et al. [14], Jia et al. [15], Nadeb et al. [16], Alshenawy et al. [17], El-Sherpieny et al. [18], Nassr et al. [19], and Muhammad et al. [20]. As frequentist methods, the maximum likelihood method and the Bayesian estimation method were used in these studies. However, little thought was given to estimate R � P(Y < X) using other methods, although, in some cases, they can provide better estimates than the maximum likelihood approach. Almetwally and Almongy [21] examined classical and Bayesian estimation methods for the stress-strength model of the power Lomax distribution.
Aside from the maximal likelihood estimation (MLE) approach, Almarashi et al. [22] offered nine other frequentist estimate methods to estimate the stress-strength reliability of the Weibull distribution, namely, least square, weighted least square, percentile, maximum product of spacing, minimum spacing absolute distance, minimum spacing absolute-log distance, method of Cramér-von Mises, and Anderson-Darling and right-tail Anderson-Darling. ey compared the efficiency of the different proposed estimators by using Monte Carlo simulation study. In terms of relative biases and relative mean squared errors, the performance and finite sample properties of the various estimators are compared.
To the authors' knowledge, the MLE and the maximum product of spacing method which were used to estimate the parameters θ 1 , θ 2 , and β of life of an EIW under a finite sample (MPSE), minimum spacing absolute-log distance estimation (MSALDE) method, least square estimation (LSE) method, weighted least square estimation (WLSE) method of Cramér-von Mises estimation (CME), and Anderson-Darling estimation (ADE) method have not yet been investigated. In this paper, our main purpose is to use eight approaches to derive estimates of the unknown parameters and stress-strength reliability R � P(Y < X), which we think if applied by statisticians/reliability engineers would be very interesting in a scaled exponentiated inverted Weibull distribution. Furthermore, the simulation study and real data analysis demonstrate that there are classical methods, rather than MLE methods, which can provide desirable estimates, justifying their use in applied areas. It should also be noted that this is the first time that eight estimation methods have been considered to estimate the unknown parameters θ 1 , θ 2 , and β and stress-strength reliability R � P(Y < X) of the EIW distribution.
Flaih et al. [23] proposed the exponentiated inverted Weibull distribution as a generalization of the standard parent distribution known as the exponentiated-parent distribution and the standard inverted Weibull distribution. More research on the EIW distribution is needed, both theoretically (estimation methods) and practically (analysis further data). According to this study, the EIW can be used as an alternative to the inverted Weibull distribution and may perform better than the inverted Weibull distribution. For θ � 1, it represents the standard inverted Weibull distribution, and for β � 1, it represents the exponentiated standard inverted exponential distribution. As a result, the exponentiated inverted Weibull distribution is a generalization of both the exponentiated inverted exponential and inverted Weibull distributions. e physical interpretation of the exponentiated inverted Weibull distribution is also available.
e main objective of this study is to estimate unknown parameters θ 1 , θ 2 , and β and stress-strength reliability R � P(Y < X) when X and Y are independent of a scaled EIW distribution using the eight estimation methods listed above. Furthermore, for the stress-strength parameter, we use percentile bootstrap and bias-corrected percentile bootstrap confidence intervals. To compare the efficiency of the various estimates, we conduct an extensive Monte Carlo numerical simulation study, as well as an analysis of two real-life data sets, the applicability of the methodologies proposed in real phenomena. e rest of this paper is organized as follows: In Section 2, we proposed the different estimation methods. Percentile bootstrap and biascorrected percentile bootstrap confidence intervals are discussed in Section 3. In Section 4, a Monte Carlo numerical simulation research is carried out. In Section 5, two real-life data sets are examined. Finally, Section 6 concludes the paper.

Different Estimation Methods
In this section, the eight recurrent estimation methods considered in this paper to obtain the unknown parameters and different estimates of the stress-strength parameter will be discussed. ese estimation methods would be of particular interest when comparing them with other maximum likelihood estimation procedures. For more examples of classical estimation method, see the works of Almetwally [25], El-Morshedy et al. [26], Almetwally et al. [27], and Sabry et al. [28]. EIW (θ 2 , β), respectively, and the likelihood function of the observed sample can be expressed as We obtain l � log L(θ 1 , θ 2 , β) by taking the natural logarithm likelihood function as l � (n + k)log β + n log θ 1 + k log θ 2 − (β + 1) e MLEs of θ 1 , θ 2 , and β denoted by θ MLE 1 , θ MLE 2 , and β MLE , respectively, can be obtained by solving the subsequent equations: θ MLE 1 and θ MLE 2 can be obtained as a function of the unknown parameter β from (6) and (7), respectively, as follows: Computational Intelligence and Neuroscience Substituting the estimators θ MLE 1 (β) and θ MLE 2 (β) obtained from (9) in (5), the profile log-likelihood function of parameter β can then be obtained as follows: To obtain β MLE , as a result of differentiating (10) with respect to β and equating the result by zero, After obtaining β MLE from (11) by using any iteration procedure, we can obtain θ MLE 1 and θ MLE 2 from (9). Now, the MLE of a system R can be obtained as 2.2. Maximum Product of Estimation. Cheng and Amin [29] introduced the method of maximum product of spacing as an alternative to the maximum likelihood method to estimate the parameters of the lognormal distribution. Let x 1: n , x 2: n , . . . , x n: n denote the order statistics of a random sample n from EIW (θ 1 , β) and let y 1: k , y 2: k , . . . , y k: k denote the order statistics of a random sample k from EIW (θ 2 , β); the uniform spacings of the two samples can therefore be defined as follows: Let x 1: n , x 2: n , . . . , x n: n denote the order statistics of a random sample from EIW.
e MPSEs of the unknown parameters are produced by maximization of the following function, as in the work of Cheng and Amin [30].
From (2) and (14), the MPSEs of the unknown parameters θ 1 , θ 2 , and β denoted by θ MPSE 1 , θ MPSE 2 , and β MPSE can be obtained by maximizing, with respect to θ 1 , θ 2 , and β, the following function: ese estimates can be obtained equivalently by solving the following equations simultaneously: 4 Computational Intelligence and Neuroscience where Using the obtained estimates, we can obtain the MPSE of a system R as

Minimum Spacing Distance Estimation.
Torabi [31] was the first to propose the minimum spacing distance estimating method. e minimum spacing distance estimators (MSADEs) are obtained by minimizing the following function, using the same notations as in the previous subsections: where ϕ 1 (n) � (1/(n + 1)), ϕ 2 (m) � (1/(m + 1)), and ψ(a, b) is an appropriate distance. e most common selections of ψ(a, b) in (18) are called absolute distance |a − b| and absolute-log distance |log(a) − log(b)|. e MSADEs of the unknown parameters denoted by θ MSADE 1 , θ MSADE 2 , and β MSADE can be determined by minimizing the the next function in terms of θ 1 , θ 2 , and β.
Computational Intelligence and Neuroscience Simultaneously, the three following equations are solved: Similarly, the MSALDEs of the unknown parameters θ 1 , θ 2 , and β denoted by θ MSALDE 1 , θ MSALDE 2 , and β MSALDE can be obtained by minimizing the function that follows: 6 Computational Intelligence and Neuroscience e three following equations are solved: Now, the MSADE and MSALDE of a system R can be obtained, respectively:

Least Square and Weighted Least Square Estimation.
Swain et al. [32] proposed the least squares and weighted least squares estimation methods for estimating the Beta distribution parameters. Let x 1: n , x 2: n , . . . , x n: n be the order statistics of a random sample of size n from EIW (θ 1 , β) and let y 1: k , y 2: k , . . . , y k: k be the order statistics of a random sample of size k from EIW (θ 2 , β). e least square estimations (LEs) of the unknown parameters θ 1 , θ 2 , and β denoted by θ LSE 1 , θ LSE 2 , and β LSE can be obtained by minimizing the following function with respect to θ 1 , θ 2 , and β as follows: Computational Intelligence and Neuroscience Instead of minimizing (18), the estimates θ LSE 1 , θ LSE 2 , and β LSE can be obtained by simultaneously solving the three following equations: Upon obtaining the estimates θ LSE 1 , θ LSE 2 , and β LSE , the LSE of R can be obtained as follows: Similarly, the unknown parameters' WLSEs θ 1 , θ 2 , and β denoted by θ WLSE 1 , θ WLSE 2 , and β WLSE can be obtained by minimizing the following function: where ω 1 (i, n) � ((n + 1) 2 (n + 2)/i(n − i + 1)) and ese estimates can also be obtained by simultaneously solving the three following equations: e WLSE of R can be obtained as

Cramér-von Mises Estimation. Cramér [33] and von
Mises [34] introduced the Cramér-von Mises method of estimation to estimate the unknown parameters θ 1 , θ 2 , and β denoted by θ CME 1 , θ CME 2 , and β CME which are obtained by minimizing the following goodness-of-fit statistic: 8 Computational Intelligence and Neuroscience with respect to θ 1 , θ 2 , and β, where φ 1 (i, n) � (2(n − i) + 1/2n) and φ 2 (j, k) � (2(k − j) + 1/2k). ese estimates can also be obtained by solving the three following equations simultaneously: e CME of R can be obtained as follows: , and β ADE are obtained by minimizing the following function: with respect to θ 1 , θ 2 , and β. ese estimates can also be obtained by solving the three following equations simultaneously: Computational Intelligence and Neuroscience e ADE of R can be obtained, respectively, by

Bootstrap Confidence Intervals
ere are two confidence intervals for parameters θ 1 , θ 2 , and β in this section, and parametric bootstrap methods will be proposed. Percentile bootstrap (Boot-P) and bias-corrected percentile bootstrap (Boot-BCP) confidence intervals are shown as two distinct parametric confidence intervals. e steps below will show how to estimate the confidence intervals of R.

Simulation Study
In the simulation section, a Monte Carlo simulation is done to estimate the unknown parameters of EIW distribution to get stress-strength reliability for MLE, MPSE, MSADE, MSALDE, LSE, WLSE, CME, and ADE methods using R-program are described as follows: Step 1: Generate 10000 random samples, in strength variable (X), the sample size is n � 30, 35, 50 , and 70 from the EIW distribution, and in stress variable the sample size is m � 40, 45, 60 , and 80 from the EIW distribution.
Step 3: e MLE, MPSE, MSADE, MSALDE, LSE, WLSE, CME, and ADE of the model parameters are obtained by solving the nonlinear equations for the stress-strength model.
Step 4: e mean and mean square errors (MSE) of the parameters are obtained.
Step 5: e length of CI by using bootstrapping of the stress-strength reliability is obtained in Tables 4-6.
Step 6: e numerical results of parameters estimation of EIW distribution are listed in Tables 1-3.

Application of Real Data
In this section, we consider two applications of the stressstrength reliability model by using breaking strengths of jute fiber and carbon fibers data to describe all the details for illustrative purposes. We used Kolmogorov-Smirnov statistics (KSS) with P value to check the fit of the model and standard errors (SE) of estimators.  Table 7, we can see that although the EIW distribution fits the data because the difference between the values of KSS is very small and the P value is more than 0.05, for more illustration, Figures 2 and 3 show the fitted CDF with empirical CDF, fitted PDF with histogram, and P-P plot for strength and stress, respectively, computed at the estimated parameters of EIW distribution. e estimates of the parameters model of stress-strength reliability for EIW distribution are obtained in Table 8. MSADE has the smallest SE and the largest reliability.

Carbon Fibers Data.
In this subsection, we look at two data sets and discuss all of the specifics for the sake of illustration. e two data sets were first published by Bader and Priest [37]; and they reflected the GPA strength of single carbon fibers with lengths of 10 mm (Data Set I) and 10 mm (Data Set II), respectively, with sample sizes of n � 63 and m � 69. ese data were analyzed previously by Hassan et al. [38]. e following are the data sets:         Table 9, we can see that although the EIW distribution fits the carbon fibers data because the difference between the values of KSS is very small and the P value is more than 0.05, for more illustration, Figures 4 and 5 show the fitted CDF with empirical CDF, fitted PDF with histogram, and P-P plot for strength and stress, respectively, computed at the estimated parameters of EIW distribution. e estimates of the parameters model of stressstrength reliability for EIW distribution are obtained in Table 10. MSADE has the smallest SE and the largest reliability.     Emprical EIW Figure 5: Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution for Data Set II of carbon fibers data.

Conclusion
In this paper, we assumed that X and Y are two independent EIW distributions with the same scale parameter, and by using eight methods of estimation, we could propose the estimations of the unknown parameters θ 1 , θ 2 , and β and the system of stress-strength parameter R � P(Y < X). e eight methods of estimations are MLEs, MPSEs, MSADEs, MSALDEs, LSEs, WLSEs, CMEs, and ADEs. e percentile bootstrap and bias-corrected percentile bootstrap confidence intervals which are two parametric bootstrap confidence intervals of R were introduced. Breaking strengths of jute fiber and carbon fibers data were used as two real data sets to demonstrate the performance of the unknown parameters θ 1 , θ 2 , and β and the system of stress-strength reliability R � P(Y < X) in practical applications, the goodness of fit of the methods estimators for each real data set was examined using the KSS, and the results were sufficient and satisfactory. We investigated the proposed point and interval estimates using simulation studies, and they performed admirably for a variety of sample sizes, as evidenced by their MSE and confidence intervals. In both techniques, the MSE decreases as the sample size increases; however, the method of maximum product of spacing outperforms other estimation methods. By comparing the estimators using an extensive Monte Carlo numerical simulation study and analyzing a real-world data set, in all sample cases, the MPSEs method outperformed the MLEs. Overall, simulation results show that the maximum product of spacing methods outperforms the other methods in terms of minimum MSE and confidence interval length in the majority of cases. In terms of minimum confidence interval lengths, Boot-PCP outperforms Boot-P confidence intervals.

Data Availability
All the data are included in the manuscript.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.