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

Authors: Wael S. Abu El Azm; Ehab M. Almetwally; Abdulaziz S. Alghamdi; Hassan M. Aljohani; Abdisalam Hassan Muse; O. E. Abo-Kasem
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

For the first time and by using an entire sample, we discussed the estimation of the unknown parametersθ1,θ2, and β and the system of stress-strength reliability R=PY<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.


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## Body

## 1. 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=PY<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 stress-strength 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.Recently, many authors investigated the estimation ofR=PY<X for different life testing schemes based on different distributions by using the maximum likelihood and Bayesian estimation methods; see, for example, Mokhlis [6], Kundu and Raqab [7], Asgharzadeh et al. [8], Asgharzadeh 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=PY<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. They 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 parametersθ1,θ2, 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=PY<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 θ1,θ2, and β and stress-strength reliability R=PY<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. The physical interpretation of the exponentiated inverted Weibull distribution is also available.The EIW with scale parameterθ and shape parameter β, denoted by EIW θ,β, has the following probability density function (PDF):(1)fx;θ,β=θβx−β+1exp−x−βθ,x>0,α,β>0,and the corresponding cumulative distribution function (CDF) is given by(2)Fx;θ,β=exp−x−βθ,x>0,θ,β>0.LetX and Y be independent exponentiated inverted Weibull random variables and follow EIW θ1,β and EIW θ2,β, respectively; then R=PY<X can be written as follows (see Hassan et al. [24]): (3)R=θ1θ1+θ2.Figure1 shows different values for R when θ1 and θ2 change.Figure 1
3 dimensions of stress-strength reliability for EIW distribution with different parameters.The main objective of this study is to estimate unknown parametersθ1,θ2, and β and stress-strength reliability R=PY<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. The rest of this paper is organized as follows: In Section 2, we proposed the different estimation methods. Percentile bootstrap and bias-corrected 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.


## 2. 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. These 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].


### 2.1. Maximum Likelihood Estimation

Letx1,x2,…,xn and y1,y2,…,yk be random samples from EIW θ1,β and EIW θ2,β, respectively, and the likelihood function of the observed sample can be expressed as(4)Lθ1,θ2,β=∏i=1nθ1βxi−β+1exp−xi−βθ1∏j=1kθ2βyj−β+1exp−yj−βθ2.We obtainl=logLθ1,θ2,β by taking the natural logarithm likelihood function as(5)l=n+klogβ+nlogθ1+klogθ2−β+1∑i=1nlogxi+∑j=1klogyj−θ1∑i=1nxi−β−θ2∑j=1kyj−β.The MLEs ofθ1, θ2, and β denoted by θ^1MLE, θ^2MLE, and β^MLE, respectively, can be obtained by solving the subsequent equations:(6)∂l∂θ1=nθ1−∑i=1nxi−β,(7)∂l∂θ2=kθ2−∑j=1kyi−β,(8)∂l∂β=n+kβ−∑i=1nlogxi−∑j=1klogyj+θ1∑i=1nxi−βlogxi+θ2∑j=1kyj−βlogyj.θ^1MLE and θ^2MLE can be obtained as a function of the unknown parameter β from (6) and (7), respectively, as follows:(9)θ^1MLEβ=n∑i=1nxi−β,θ^2MLEβ=k∑j=1nyi−β.Substituting the estimatorsθ1MLEβ and θ2MLEβ obtained from (9) in (5), the profile log-likelihood function of parameter β can then be obtained as follows:(10)l=n+klogβ−β∑i=1nlogxi+∑j=1klogyj−nlog∑i=1nxi−β−klog∑j=1kyj−β.To obtainβMLE, as a result of differentiating (10) with respect to β and equating the result by zero,(11)ψβ=∑i=1nlogxi+∑j=1nlogyin+k−n∑i=1nxi−βlogxin+k∑i=1nxi−β−k∑j=1nyi−βlogyjn+k∑j=1nyi−β−1.After obtainingβ^MLE from (11) by using any iteration procedure, we can obtain θ^1MLE and θ^2MLE from (9). Now, the MLE of a system R can be obtained as(12)R^MLE=θ^1MLEθ^1MLE+θ^2MLE⋅gλj∝λjaj−1e−λjbj,aj,bj>0,j=1,2.


### 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 x1:n,x2:n,…,xn:n denote the order statistics of a random sample n from EIW θ1,β and let y1:k,y2:k,…,yk:k denote the order statistics of a random sample k from EIW θ2,β; the uniform spacings of the two samples can therefore be defined as follows:Letx1:n,x2:n,…,xn:n denote the order statistics of a random sample from EIW.(13)Δ1i=Fxi:nθ1,β−Fxi−1:nθ1,β,Δ2j=Fyj:kθ2,β−Fyj−1:kθ2,β.The MPSEs of the unknown parameters are produced by maximization of the following function, as in the work of Cheng and Amin [30].(14)MPθ1,θ2,β=1n+1∑i=1n+1logΔ1i+1k+1∑j=1k+1logΔ2j.From (2) and (14), the MPSEs of the unknown parameters θ1,θ2, and β denoted by θ^1MPSE, θ^2MPSE, and β^MPSE can be obtained by maximizing, with respect to θ1,θ2, and β, the following function:(15)MPθ1,θ2,β=1n+1∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1+1k+1∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2.These estimates can be obtained equivalently by solving the following equations simultaneously:(16)∂MPθ1,θ2,β∂θ1=1n+1∑i=1n+1xi:n−βe−xi:n−βθ1−xi−1:n−βe−xi−1:n−βθ1e−xi:n−βθ1−e−xi−1:n−βθ1=0,∂MPθ1,θ2,β∂θ2=1k+1∑j=1k+1yj:k−βe−yj:k−βθ2−yj−1:k−βe−yj−1:k−βθ2e−yj:k−βθ2−e−yj−1:k−βθ2=0,∂MPθ1,θ2,β∂β=θ1n+1∑i=1n+1Ζi:n−Ζi−1:ne−xi:n−βθ1−e−xi−1:n−βθ1+θ2k+1∑j=1k+1Ψj:k−Ψj−1:ke−yj:k−βθ2−e−yj−1:k−βθ2=0,where Ζi:n=xi:n−βe−xi:n−βθ1logxi:n and Ψj:k=yj:k−βe−yj:k−βθ2logyj:k. Using the obtained estimates, we can obtain the MPSE of a system R as(17)R^MPSE=θ^1MPSEθ^1MPSE+θ^2MPSE.


### 2.3. Minimum Spacing Distance Estimation

Torabi [31] was the first to propose the minimum spacing distance estimating method. The minimum spacing distance estimators (MSADEs) are obtained by minimizing the following function, using the same notations as in the previous subsections:(18)MDθ1,θ2,β=∑i=1n+1ψΔ1i,ϕ1n+∑j=1k+1ψΔ2j,ϕ2m,where ϕ1n=1/n+1, ϕ2m=1/m+1, and ψa,b is an appropriate distance. The most common selections of ψa,b in (18) are called absolute distance a−b and absolute-log distance loga−logb. The MSADEs of the unknown parameters denoted by θ^1MSADE, θ^2MSADE, and β^MSADE can be determined by minimizing the the next function in terms of θ1,θ2, and β.(19)MDθ1,θ2,β=∑i=1n+1e−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1n+∑j=1k+1e−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2m.Simultaneously, the three following equations are solved:(20)∂MDθ1,θ2,β∂θ1=∑i=1n+1e−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1nxi:n−βe−xi:n−βθ1−xi−1:n−βe−xi−1:n−βθ1=0,∂MDθ1,θ2,β∂θ2=∑j=1m+1e−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2me−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2myj:k−βe−yj:k−βθ2−yj−1:k−βe−yj−1:k−βθ2=0,∂MDθ1,θ2,β∂β=θ1∑i=1n+1e−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1nZi:n−Zi−1:n+θ2∑j=1m+1e−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2me−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2mΨj:k−Ψj−1:k=0.Similarly, the MSALDEs of the unknown parametersθ1,θ2, and β denoted by θ^1MSALDE, θ^2MSALDE, and β^MSALDE can be obtained by minimizing the function that follows:(21)Mdθ1,θ2,β=∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1ϕ1n+∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2ϕ2k.The three following equations are solved:(22)∂Mdθ1,θ2,β∂θ1=∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1nloge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1×xi:n−βe−xi:n−βθ1−xi−1:n−βe−xi−1:n−βθ1=0,∂Mdθ1,θ2,β∂θ2=∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2kloge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2ke−yj:k−βθ2−e−yj−1:k−βθ2×yj:k−βe−yj:k−βθ2−yj−1:k−βe−yj−1:k−βθ2=0,∂Mdθ1,θ2,β∂β=θ1∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1nZi:n−Zi−1:nloge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1+θ2∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2kΨj:k−Ψj−1:kloge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2ke−yj:k−βθ2−e−yj−1:k−βθ2=0.Now, the MSADE and MSALDE of a systemR can be obtained, respectively:(23)R^MSADE=θ^1MSADEθ^1MSADE+θ^2MSADE,R^MSADE=θ^1MSADEθ^1MSADE+θ^2MSADE.


### 2.4. 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 x1:n,x2:n,…,xn:n be the order statistics of a random sample of size n from EIW θ1,β and let y1:k,y2:k,…,yk:k be the order statistics of a random sample of size k from EIW θ2,β. The least square estimations (LEs) of the unknown parameters θ1,θ2, and β denoted by θ^1LSE, θ^2LSE, and β^LSE can be obtained by minimizing the following function with respect to θ1,θ2, and β as follows:(24)LSθ1,θ2,β=∑i=1nFxi:n−in+12+∑j=1kFyj:k−jm+12=∑i=1ne−xi:n−βθ1−in+12+∑j=1ke−yj:k−βθ2−jm+12.Instead of minimizing (18), the estimates θ^1LSE, θ^2LSE, and β^LSE can be obtained by simultaneously solving the three following equations:(25)∂LSθ1,θ2,β∂θ1=∑i=1nxi:n−βe−xi:n−βθ1e−xi:n−βθ1−in+1=0,∂LSθ1,θ2,β∂θ2=∑j=1kyj:k−βe−yj:k−βθ2e−yj:k−βθ2−jk+1=0,∂LSθ1,θ2,β∂β=θ1∑i=1nxi:n−βlogxi:ne−xi:n−βθ1e−xi:n−βθ1−in+1+θ2∑j=1kyj:klogyj:ke−yj:k−βθ2e−yj:k−βθ2−jm+12=0.Upon obtaining the estimatesθ^1LSE, θ^2LSE, and β^LSE, the LSE of R can be obtained as follows:(26)R^LSE=θ^1LSEθ^1LSE+θ^2LSE.Similarly, the unknown parameters’ WLSEsθ1,θ2, and β denoted by θ^1WLSE, θ^2WLSE, and β^WLSE can be obtained by minimizing the following function:(27)WLSθ1,θ2,β=∑i=1nω1i,ne−xi:n−βθ1−in+12+∑j=1kω2j,ke−yj:k−βθ2−jm+12,where ω1i,n=n+12n+2/in−i+1 and ω2j,k=k+12k+2/jk−j+1. These estimates can also be obtained by simultaneously solving the three following equations:(28)∂WLSθ1,θ2,βθ1=∑i=1nω1i,nxi:n−βe−xi:n−βθ1e−xi:n−βθ1−in+1=0,∂WLSθ1,θ2,β∂θ2=∑j=1kω2j,kyj:k−βe−yj:k−βθ2e−yj:k−βθ2−jm+1=0,∂WLSθ1,θ2,β∂β=θ1∑i=1nω1i,nxi:n−βlogxi:ne−xi:n−βθ1e−xi:n−βθ1−in+1+θ2∑j=1kω2j,kyj:klogyj:ke−yj:k−βθ2e−yj:k−βθ2−jm+1=0.The WLSE ofR can be obtained as(29)R^WLSE=θ^1WLSEθ^1WLSE+θ^2WLSE.


### 2.5. Cramér-von Mises Estimation

Cramér [33] and von Mises [34] introduced the Cramér-von Mises method of estimation to estimate the unknown parameters θ1,θ2, and β denoted by θ^1CME, θ^2CME, and β^CME which are obtained by minimizing the following goodness-of-fit statistic:(30)CMθ1,θ2,β=112n+112k+∑i=1ne−xi:n−βθ1−φ1i,n2+∑j=1ke−yj:k−βθ2−φ2j,k2,with respect to θ1,θ2, and β, where φ1i,n=2n−i+1/2n and φ2j,k=2k−j+1/2k. These estimates can also be obtained by solving the three following equations simultaneously:(31)∂CMθ1,θ2,β∂θ1=∑i=1nxi:n−βe−xi:n−βθ1e−xi:n−βθ1−φ1i,n=0,∂CMθ1,θ2,β∂θ2=∑j−1kyj:k−βe−yj:k−βθ2e−yj:k−βθ2−φ2j,k=0,∂CMθ1,θ2,β∂β=θ1∑i=1nxi:n−βlogxi:ne−xi:n−βθ1e−xi:n−βθ1−φ1i,n+θ2∑j−1kyj:k−βlogyj:k×e−yj:k−βθ2e−yj:k−βθ2−φ2j,k.The CME ofR can be obtained as follows:(32)R^CME=θ^1CMEθ^1CME+θ^2CME.


### 2.6. Anderson-Darling Estimation

Another type of minimum distance estimator is the Anderson-Darling estimation, which is obtained by minimizing Anderson-Darling statistics. Right-tail Anderson-Darling estimation (ADEs) statistics were introduced by Luceño [35] as a modification to the Anderson-Darling estimation (ADEs) statistics (RADEs). The unknown parameters of ADEs θ1,θ2, and β denoted by θ^1ADE, θ^2ADE, and β^ADE are obtained by minimizing the following function:(33)ADEθ1,θ2,β=−n−k−1n∑i=1n2i−1loge−θ1xi:n−β−θ1xn+1−i:n−β−1k∑j=1k2j−1loge−θ2yj:k−β−θ2yk+1−j:k−β,with respect to θ1,θ2, and β. These estimates can also be obtained by solving the three following equations simultaneously:(34)∂ADEθ1,θ2,β∂θ1=1n∑i=1n2i−1xi:n−βe−θ1xi:n−βe−θ1xi:n−β+xn+1−i:n−β=0,∂ADEθ1,θ2,β∂θ2=1k∑j=1k2j−1yj:k−βe−θ2yj:k−βe−θ2yj:k−β+yk+1−j:k−β=0,∂ADEθ1,θ2,β∂β=θ1n∑i=1n2i−1xi:n−βlogxi:ne−θ1xi:n−βe−θ1xi:n−β−xn+1−i:n−βlogxn+1−i:n+θ2k∑j=1k2j−1yj:k−βlogyj:ke−θ2yj:k−βe−θ2yj:k−β−yk+1−j:k−βlogyk+1−j:k=0.The ADE ofR can be obtained, respectively, by(35)R^ADE=θ^1ADEθ^1ADE+θ^2ADE.


## 2.1. Maximum Likelihood Estimation

Letx1,x2,…,xn and y1,y2,…,yk be random samples from EIW θ1,β and EIW θ2,β, respectively, and the likelihood function of the observed sample can be expressed as(4)Lθ1,θ2,β=∏i=1nθ1βxi−β+1exp−xi−βθ1∏j=1kθ2βyj−β+1exp−yj−βθ2.We obtainl=logLθ1,θ2,β by taking the natural logarithm likelihood function as(5)l=n+klogβ+nlogθ1+klogθ2−β+1∑i=1nlogxi+∑j=1klogyj−θ1∑i=1nxi−β−θ2∑j=1kyj−β.The MLEs ofθ1, θ2, and β denoted by θ^1MLE, θ^2MLE, and β^MLE, respectively, can be obtained by solving the subsequent equations:(6)∂l∂θ1=nθ1−∑i=1nxi−β,(7)∂l∂θ2=kθ2−∑j=1kyi−β,(8)∂l∂β=n+kβ−∑i=1nlogxi−∑j=1klogyj+θ1∑i=1nxi−βlogxi+θ2∑j=1kyj−βlogyj.θ^1MLE and θ^2MLE can be obtained as a function of the unknown parameter β from (6) and (7), respectively, as follows:(9)θ^1MLEβ=n∑i=1nxi−β,θ^2MLEβ=k∑j=1nyi−β.Substituting the estimatorsθ1MLEβ and θ2MLEβ obtained from (9) in (5), the profile log-likelihood function of parameter β can then be obtained as follows:(10)l=n+klogβ−β∑i=1nlogxi+∑j=1klogyj−nlog∑i=1nxi−β−klog∑j=1kyj−β.To obtainβMLE, as a result of differentiating (10) with respect to β and equating the result by zero,(11)ψβ=∑i=1nlogxi+∑j=1nlogyin+k−n∑i=1nxi−βlogxin+k∑i=1nxi−β−k∑j=1nyi−βlogyjn+k∑j=1nyi−β−1.After obtainingβ^MLE from (11) by using any iteration procedure, we can obtain θ^1MLE and θ^2MLE from (9). Now, the MLE of a system R can be obtained as(12)R^MLE=θ^1MLEθ^1MLE+θ^2MLE⋅gλj∝λjaj−1e−λjbj,aj,bj>0,j=1,2.


## 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 x1:n,x2:n,…,xn:n denote the order statistics of a random sample n from EIW θ1,β and let y1:k,y2:k,…,yk:k denote the order statistics of a random sample k from EIW θ2,β; the uniform spacings of the two samples can therefore be defined as follows:Letx1:n,x2:n,…,xn:n denote the order statistics of a random sample from EIW.(13)Δ1i=Fxi:nθ1,β−Fxi−1:nθ1,β,Δ2j=Fyj:kθ2,β−Fyj−1:kθ2,β.The MPSEs of the unknown parameters are produced by maximization of the following function, as in the work of Cheng and Amin [30].(14)MPθ1,θ2,β=1n+1∑i=1n+1logΔ1i+1k+1∑j=1k+1logΔ2j.From (2) and (14), the MPSEs of the unknown parameters θ1,θ2, and β denoted by θ^1MPSE, θ^2MPSE, and β^MPSE can be obtained by maximizing, with respect to θ1,θ2, and β, the following function:(15)MPθ1,θ2,β=1n+1∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1+1k+1∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2.These estimates can be obtained equivalently by solving the following equations simultaneously:(16)∂MPθ1,θ2,β∂θ1=1n+1∑i=1n+1xi:n−βe−xi:n−βθ1−xi−1:n−βe−xi−1:n−βθ1e−xi:n−βθ1−e−xi−1:n−βθ1=0,∂MPθ1,θ2,β∂θ2=1k+1∑j=1k+1yj:k−βe−yj:k−βθ2−yj−1:k−βe−yj−1:k−βθ2e−yj:k−βθ2−e−yj−1:k−βθ2=0,∂MPθ1,θ2,β∂β=θ1n+1∑i=1n+1Ζi:n−Ζi−1:ne−xi:n−βθ1−e−xi−1:n−βθ1+θ2k+1∑j=1k+1Ψj:k−Ψj−1:ke−yj:k−βθ2−e−yj−1:k−βθ2=0,where Ζi:n=xi:n−βe−xi:n−βθ1logxi:n and Ψj:k=yj:k−βe−yj:k−βθ2logyj:k. Using the obtained estimates, we can obtain the MPSE of a system R as(17)R^MPSE=θ^1MPSEθ^1MPSE+θ^2MPSE.


## 2.3. Minimum Spacing Distance Estimation

Torabi [31] was the first to propose the minimum spacing distance estimating method. The minimum spacing distance estimators (MSADEs) are obtained by minimizing the following function, using the same notations as in the previous subsections:(18)MDθ1,θ2,β=∑i=1n+1ψΔ1i,ϕ1n+∑j=1k+1ψΔ2j,ϕ2m,where ϕ1n=1/n+1, ϕ2m=1/m+1, and ψa,b is an appropriate distance. The most common selections of ψa,b in (18) are called absolute distance a−b and absolute-log distance loga−logb. The MSADEs of the unknown parameters denoted by θ^1MSADE, θ^2MSADE, and β^MSADE can be determined by minimizing the the next function in terms of θ1,θ2, and β.(19)MDθ1,θ2,β=∑i=1n+1e−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1n+∑j=1k+1e−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2m.Simultaneously, the three following equations are solved:(20)∂MDθ1,θ2,β∂θ1=∑i=1n+1e−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1nxi:n−βe−xi:n−βθ1−xi−1:n−βe−xi−1:n−βθ1=0,∂MDθ1,θ2,β∂θ2=∑j=1m+1e−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2me−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2myj:k−βe−yj:k−βθ2−yj−1:k−βe−yj−1:k−βθ2=0,∂MDθ1,θ2,β∂β=θ1∑i=1n+1e−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1−ϕ1nZi:n−Zi−1:n+θ2∑j=1m+1e−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2me−yj:k−βθ2−e−yj−1:k−βθ2−ϕ2mΨj:k−Ψj−1:k=0.Similarly, the MSALDEs of the unknown parametersθ1,θ2, and β denoted by θ^1MSALDE, θ^2MSALDE, and β^MSALDE can be obtained by minimizing the function that follows:(21)Mdθ1,θ2,β=∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1ϕ1n+∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2ϕ2k.The three following equations are solved:(22)∂Mdθ1,θ2,β∂θ1=∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1nloge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1×xi:n−βe−xi:n−βθ1−xi−1:n−βe−xi−1:n−βθ1=0,∂Mdθ1,θ2,β∂θ2=∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2kloge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2ke−yj:k−βθ2−e−yj−1:k−βθ2×yj:k−βe−yj:k−βθ2−yj−1:k−βe−yj−1:k−βθ2=0,∂Mdθ1,θ2,β∂β=θ1∑i=1n+1loge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1nZi:n−Zi−1:nloge−xi:n−βθ1−e−xi−1:n−βθ1−logϕ1ne−xi:n−βθ1−e−xi−1:n−βθ1+θ2∑j=1k+1loge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2kΨj:k−Ψj−1:kloge−yj:k−βθ2−e−yj−1:k−βθ2−logϕ2ke−yj:k−βθ2−e−yj−1:k−βθ2=0.Now, the MSADE and MSALDE of a systemR can be obtained, respectively:(23)R^MSADE=θ^1MSADEθ^1MSADE+θ^2MSADE,R^MSADE=θ^1MSADEθ^1MSADE+θ^2MSADE.


## 2.4. 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 x1:n,x2:n,…,xn:n be the order statistics of a random sample of size n from EIW θ1,β and let y1:k,y2:k,…,yk:k be the order statistics of a random sample of size k from EIW θ2,β. The least square estimations (LEs) of the unknown parameters θ1,θ2, and β denoted by θ^1LSE, θ^2LSE, and β^LSE can be obtained by minimizing the following function with respect to θ1,θ2, and β as follows:(24)LSθ1,θ2,β=∑i=1nFxi:n−in+12+∑j=1kFyj:k−jm+12=∑i=1ne−xi:n−βθ1−in+12+∑j=1ke−yj:k−βθ2−jm+12.Instead of minimizing (18), the estimates θ^1LSE, θ^2LSE, and β^LSE can be obtained by simultaneously solving the three following equations:(25)∂LSθ1,θ2,β∂θ1=∑i=1nxi:n−βe−xi:n−βθ1e−xi:n−βθ1−in+1=0,∂LSθ1,θ2,β∂θ2=∑j=1kyj:k−βe−yj:k−βθ2e−yj:k−βθ2−jk+1=0,∂LSθ1,θ2,β∂β=θ1∑i=1nxi:n−βlogxi:ne−xi:n−βθ1e−xi:n−βθ1−in+1+θ2∑j=1kyj:klogyj:ke−yj:k−βθ2e−yj:k−βθ2−jm+12=0.Upon obtaining the estimatesθ^1LSE, θ^2LSE, and β^LSE, the LSE of R can be obtained as follows:(26)R^LSE=θ^1LSEθ^1LSE+θ^2LSE.Similarly, the unknown parameters’ WLSEsθ1,θ2, and β denoted by θ^1WLSE, θ^2WLSE, and β^WLSE can be obtained by minimizing the following function:(27)WLSθ1,θ2,β=∑i=1nω1i,ne−xi:n−βθ1−in+12+∑j=1kω2j,ke−yj:k−βθ2−jm+12,where ω1i,n=n+12n+2/in−i+1 and ω2j,k=k+12k+2/jk−j+1. These estimates can also be obtained by simultaneously solving the three following equations:(28)∂WLSθ1,θ2,βθ1=∑i=1nω1i,nxi:n−βe−xi:n−βθ1e−xi:n−βθ1−in+1=0,∂WLSθ1,θ2,β∂θ2=∑j=1kω2j,kyj:k−βe−yj:k−βθ2e−yj:k−βθ2−jm+1=0,∂WLSθ1,θ2,β∂β=θ1∑i=1nω1i,nxi:n−βlogxi:ne−xi:n−βθ1e−xi:n−βθ1−in+1+θ2∑j=1kω2j,kyj:klogyj:ke−yj:k−βθ2e−yj:k−βθ2−jm+1=0.The WLSE ofR can be obtained as(29)R^WLSE=θ^1WLSEθ^1WLSE+θ^2WLSE.


## 2.5. Cramér-von Mises Estimation

Cramér [33] and von Mises [34] introduced the Cramér-von Mises method of estimation to estimate the unknown parameters θ1,θ2, and β denoted by θ^1CME, θ^2CME, and β^CME which are obtained by minimizing the following goodness-of-fit statistic:(30)CMθ1,θ2,β=112n+112k+∑i=1ne−xi:n−βθ1−φ1i,n2+∑j=1ke−yj:k−βθ2−φ2j,k2,with respect to θ1,θ2, and β, where φ1i,n=2n−i+1/2n and φ2j,k=2k−j+1/2k. These estimates can also be obtained by solving the three following equations simultaneously:(31)∂CMθ1,θ2,β∂θ1=∑i=1nxi:n−βe−xi:n−βθ1e−xi:n−βθ1−φ1i,n=0,∂CMθ1,θ2,β∂θ2=∑j−1kyj:k−βe−yj:k−βθ2e−yj:k−βθ2−φ2j,k=0,∂CMθ1,θ2,β∂β=θ1∑i=1nxi:n−βlogxi:ne−xi:n−βθ1e−xi:n−βθ1−φ1i,n+θ2∑j−1kyj:k−βlogyj:k×e−yj:k−βθ2e−yj:k−βθ2−φ2j,k.The CME ofR can be obtained as follows:(32)R^CME=θ^1CMEθ^1CME+θ^2CME.


## 2.6. Anderson-Darling Estimation

Another type of minimum distance estimator is the Anderson-Darling estimation, which is obtained by minimizing Anderson-Darling statistics. Right-tail Anderson-Darling estimation (ADEs) statistics were introduced by Luceño [35] as a modification to the Anderson-Darling estimation (ADEs) statistics (RADEs). The unknown parameters of ADEs θ1,θ2, and β denoted by θ^1ADE, θ^2ADE, and β^ADE are obtained by minimizing the following function:(33)ADEθ1,θ2,β=−n−k−1n∑i=1n2i−1loge−θ1xi:n−β−θ1xn+1−i:n−β−1k∑j=1k2j−1loge−θ2yj:k−β−θ2yk+1−j:k−β,with respect to θ1,θ2, and β. These estimates can also be obtained by solving the three following equations simultaneously:(34)∂ADEθ1,θ2,β∂θ1=1n∑i=1n2i−1xi:n−βe−θ1xi:n−βe−θ1xi:n−β+xn+1−i:n−β=0,∂ADEθ1,θ2,β∂θ2=1k∑j=1k2j−1yj:k−βe−θ2yj:k−βe−θ2yj:k−β+yk+1−j:k−β=0,∂ADEθ1,θ2,β∂β=θ1n∑i=1n2i−1xi:n−βlogxi:ne−θ1xi:n−βe−θ1xi:n−β−xn+1−i:n−βlogxn+1−i:n+θ2k∑j=1k2j−1yj:k−βlogyj:ke−θ2yj:k−βe−θ2yj:k−β−yk+1−j:k−βlogyk+1−j:k=0.The ADE ofR can be obtained, respectively, by(35)R^ADE=θ^1ADEθ^1ADE+θ^2ADE.


## 3. Bootstrap Confidence Intervals

There are two confidence intervals for parametersθ1,θ2, 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. The steps below will show how to estimate the confidence intervals of R.


### 3.1. Boot-P Confidence

(1)
Generate samplesx1:n,x2:n,…,xn:n and yi:k,y2:k,…,yk:k to obtain the bootstrap estimates of θ^1∗,θ^2∗,β^∗, and R^∗ from the original data, where θ^1∗,θ^2∗,β^∗, and R^∗ are the estimates obtained from the different estimation.(2)
Useθ^1∗ and β^∗ to generate a bootstrap sample x1:nBoot,x2:nBoot,…,xn:nBoot and θ^2∗ and β^∗ to generate a bootstrap sample y1:kBoot,y2:kBoot,…,yk:kBoot.(3)
Based onx1:nBoot,x2:nBoot,…,xn:nBoot and y1:kBoot,y2:kBoot,…,yk:kBoot, obtain the bootstrap estimate of a system R, say R^Boot.(4)
Repeat steps 1–3B times to have R^Boot1,R^Boot2,…,R^BootB.(5)
Arrange the bootstrap estimates in step 4 in ascending order asR^Boot1,R^Boot2,…,R^BootB.(6)
A two-side1001−γ% Boot-P confidence interval of R is given by R^BootBγ/2,R^BootB1−γ/2.


### 3.2. Boot-BCP Confidence Interval

(1)
The same steps as (1–4) in Boot-P(2)
A two-side1001−γ% Boot-BCP confidence interval for the unknown parameters is given by(36)R^BootBδ1,R^BootBδ2,where(37)δ1=Φ2z0+zγ/2,δ2=Φ2z0+z1−γ/2,whereΦ. is the CDF of the standard normal distribution, zγ=Φ−1γ, and z0 can be obtained as follows:(38)z0=Φ−1R^BootiB,i=1,2,…,B.


## 3.1. Boot-P Confidence

(1)
Generate samplesx1:n,x2:n,…,xn:n and yi:k,y2:k,…,yk:k to obtain the bootstrap estimates of θ^1∗,θ^2∗,β^∗, and R^∗ from the original data, where θ^1∗,θ^2∗,β^∗, and R^∗ are the estimates obtained from the different estimation.(2)
Useθ^1∗ and β^∗ to generate a bootstrap sample x1:nBoot,x2:nBoot,…,xn:nBoot and θ^2∗ and β^∗ to generate a bootstrap sample y1:kBoot,y2:kBoot,…,yk:kBoot.(3)
Based onx1:nBoot,x2:nBoot,…,xn:nBoot and y1:kBoot,y2:kBoot,…,yk:kBoot, obtain the bootstrap estimate of a system R, say R^Boot.(4)
Repeat steps 1–3B times to have R^Boot1,R^Boot2,…,R^BootB.(5)
Arrange the bootstrap estimates in step 4 in ascending order asR^Boot1,R^Boot2,…,R^BootB.(6)
A two-side1001−γ% Boot-P confidence interval of R is given by R^BootBγ/2,R^BootB1−γ/2.


## 3.2. Boot-BCP Confidence Interval

(1)
The same steps as (1–4) in Boot-P(2)
A two-side1001−γ% Boot-BCP confidence interval for the unknown parameters is given by(36)R^BootBδ1,R^BootBδ2,where(37)δ1=Φ2z0+zγ/2,δ2=Φ2z0+z1−γ/2,whereΦ. is the CDF of the standard normal distribution, zγ=Φ−1γ, and z0 can be obtained as follows:(38)z0=Φ−1R^BootiB,i=1,2,…,B.


## 4. 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 variableX, 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 2: Use the quantilexi=−lnqi1/θ1−1/β,yi=−lnqi1/θ2−1/β;0<qi<1, where x and y are distributed as EIW for different parameters θ1,θ2,β and different cases of actual parameters values are selected; see Tables 1–3.Case 1:θ1=0.1 and 3, θ2=0.1, and β=0.5.Case 2:θ2=0.4 and 2, θ1=3, and β=0.9.Case 3:θ2=0.4 and 1.5, θ1=0.5, and β=0.5.Step 3: The 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: The mean and mean square errors (MSE) of the parameters are obtained.Step 5: The length of CI by using bootstrapping of the stress-strength reliability is obtained in Tables4–6.Step 6: The numerical results of parameters estimation of EIW distribution are listed in Tables1–3.Table 1
Mean and MSE for parameter and stress-strength reliability for EIW distribution with different sample size: Case 1.
θ2=0.1,β=0.5n = 30, m = 35n = 40, m = 45n = 30, m = 35n = 40, m = 45θ1θ1θ2βRθ1θ2βRθ1θ2βRθ1θ2βR0.1MLEMean0.09870.09810.51470.50030.10020.09840.51350.50390.09940.09860.50960.50210.09910.09940.50640.4993MSE0.00100.00090.00260.00360.00090.00080.00230.00330.00060.00060.00170.00230.00040.00040.00110.0016MPSEMean0.09260.09370.50980.49580.09520.09480.50840.50030.09610.09610.50490.50020.09680.09750.50240.4983MSE0.00100.00090.00250.00360.00080.00080.00230.00330.00060.00060.00160.00230.00040.00040.00110.0016MSADEMean0.17920.17960.43230.49770.17220.16880.43650.50350.14300.14100.45410.50200.12650.12660.46700.4990MSE0.02370.02370.01270.00470.01840.01680.01090.00460.00680.00600.00630.00320.00270.00260.00380.0025MSALDEMean0.11040.11200.48540.49500.11120.11050.48660.50050.10860.10860.48740.49970.10670.10760.48810.4976MSE0.00180.00170.00340.00440.00160.00140.00310.00400.00110.00110.00220.00300.00070.00070.00150.0021LSEMean0.10940.10870.49070.49980.10960.10790.49330.50380.10640.10530.49510.50230.10360.10410.49670.4987MSE0.00140.00130.00310.00440.00130.00120.00300.00410.00080.00080.00220.00290.00060.00060.00160.0021WLSEMean0.10500.10450.50010.50000.10540.10380.50220.50380.10280.10190.50260.50230.10090.10140.50260.4987MSE0.00120.00110.00270.00420.00110.00100.00270.00390.00070.00070.00200.00270.00050.00050.00140.0019CMEMean0.10090.10000.51260.50020.10210.10030.51260.50420.10090.09980.50890.50250.09960.10010.50680.4987MSE0.00130.00110.00360.00480.00110.00110.00340.00450.00080.00070.00250.00310.00050.00050.00170.0022ADEMean0.12990.14080.45020.47940.14070.15110.43380.48280.15270.16380.41020.48290.15650.16830.40110.4824MSE0.00240.00330.00470.00380.00310.00430.00640.00320.00380.00540.00940.00210.00400.00560.01080.00163MLEMean3.21100.09730.51600.96893.15970.09800.51440.96863.12360.09810.51290.96853.07890.09930.50690.9681MSE0.59460.00080.00260.00010.38230.00080.00230.00010.27780.00060.00170.00010.15840.00040.00120.0001MPSEMean2.91560.09270.51150.96742.89990.09420.50970.96722.92470.09550.50830.96732.92810.09740.50290.9671MSE0.45480.00080.00250.00020.31020.00070.00230.00010.23560.00060.00160.00010.14070.00040.00110.0001MSADEMean2.91700.18240.42950.93932.90250.17260.43530.94222.91800.14590.45490.95072.92140.12830.46600.9568MSE0.43100.02360.01380.00250.36530.01870.01170.00200.28400.00950.00720.00110.20350.00340.00410.0004MSALDEMean2.91980.11110.48650.96122.89720.10970.48800.96182.92260.10850.49100.96282.92130.10700.48960.9638MSE0.47660.00160.00330.00030.34370.00150.00310.00030.28010.00110.00230.00020.17480.00070.00160.0001LSEMean3.02430.10850.49200.96253.00180.10730.49420.96343.01580.10530.49770.96452.99850.10440.49550.9652MSE0.68120.00130.00350.00030.43860.00110.00290.00020.35240.00090.00230.00020.21800.00060.00160.0001WLSEMean3.10260.10420.50150.96493.07300.10330.50310.96563.06830.10200.50470.96643.04580.10160.50170.9668MSE0.68650.00110.00320.00020.44900.00090.00270.00020.34100.00070.00200.00010.20100.00050.00140.0001CMEMean3.23070.09980.51400.96723.17670.09970.51350.96763.13990.09980.51160.96753.08660.10040.50560.9674MSE0.93030.00110.00400.00020.57310.00100.00340.00020.42960.00080.00260.00020.25020.00050.00170.0001ADEMean2.73520.14030.45160.94842.59830.15040.43480.94322.45300.16350.41280.93572.38750.16810.40120.9329MSE0.44640.00340.00490.00080.37470.00410.00620.00100.43850.00540.00900.00140.46260.00560.01080.0015Table 2
Mean and MSE for parameter and stress-strength reliability for EIW distribution with different sample size: Case 2.
θ1=3,β=0.9n=30,m=40n=35,m=45n=50,m=60n=70,m=80θ2θ1θ2βRθ1θ2βRθ1θ2βRθ1θ2βR0.4MLEMean3.20530.40140.92530.88513.17860.39530.92870.88623.12870.40090.91970.88423.07160.40260.91330.8826MSE0.47490.00680.00860.00100.42060.00590.00760.00090.26370.00410.00530.00060.17240.00360.00400.0005MPSEMean2.90790.37790.91670.88132.91430.37490.92000.88282.92770.38580.91070.88132.92030.39090.90570.8804MSE0.36450.00640.00840.00100.32920.00590.00730.00090.21880.00400.00500.00060.15520.00340.00380.0005MSADEMean2.92140.55920.75870.83752.90880.51810.78940.84632.92560.48860.81040.85452.90950.45150.84240.8635MSE0.38400.08470.05320.00600.39380.05920.04120.00470.27880.03300.02690.00300.20620.01230.01290.0015MSALDEMean2.92120.41600.87120.87072.91210.40730.88240.87332.91870.41290.87970.87292.93080.41130.88280.8749MSE0.44030.01010.01160.00160.38690.00850.01030.00130.28250.00630.00730.00110.19810.00510.00540.0008LSEMean3.02730.41580.88580.87413.03340.40780.89240.87663.02650.40880.89330.87773.00050.40820.89530.8779MSE0.51150.00790.01150.00140.50470.00670.00930.00130.30670.00500.00720.00090.22220.00420.00530.0007WLSEMean3.09980.41030.90210.87823.09220.40280.90710.88033.07860.40470.90620.88103.03960.40490.90510.8805MSE0.52860.00750.01020.00130.49030.00640.00840.00110.29120.00470.00610.00080.20460.00390.00460.0006CMEMean3.23320.40540.92550.88313.21200.39860.92720.88443.15150.40220.91830.88343.08910.40330.91350.8821MSE0.70920.00800.01330.00130.66860.00690.01090.00120.37890.00500.00800.00090.25620.00420.00580.0006ADEMean2.74290.48210.81310.84622.62220.48790.78650.83922.46370.50530.74170.82742.38500.50790.72390.8227MSE0.36280.01530.01570.00290.38660.01510.01940.00330.41310.01620.02960.00410.46490.01570.03440.00442MLEMean3.20202.08340.92550.60233.19162.06160.92650.60453.12972.06090.91600.60103.08992.04680.91290.6008MSE0.54300.14550.00860.00380.43910.10660.00720.00320.30000.08880.00500.00230.15360.06350.00380.0016MPSEMean2.90781.93050.91700.59742.92841.92520.91850.60042.93131.95220.90800.59832.93751.96070.90550.5990MSE0.43070.12080.00840.00380.33880.09440.00700.00310.25090.07700.00480.00230.13310.05680.00370.0016MSADEMean2.94052.04300.75480.58522.93422.02280.77480.58902.90562.00050.81340.59022.90832.00420.83750.5908MSE0.58340.14330.05810.00520.38190.12790.04580.00420.28730.09980.02490.00320.21270.08590.01330.0026MSALDEMean2.91861.95570.87150.59452.91741.94400.88050.59652.91851.96450.87810.59522.92531.97740.88000.5956MSE0.50770.13540.01140.00460.41180.11270.00900.00390.29980.08880.00700.00280.16930.06990.00510.0021LSEMean3.02292.00930.88550.59623.03412.00460.89290.59833.01832.00740.88990.59853.01302.01810.89470.5975MSE0.57680.15310.01070.00450.48720.12320.00880.00380.31910.10650.00660.00270.21200.07880.00510.0021WLSEMean3.09512.03750.90170.59843.09772.02760.90680.60053.07202.03130.90240.59983.05492.03510.90480.5990MSE0.59130.14550.00960.00440.49180.11210.00800.00360.31770.09790.00570.00260.19330.07190.00440.0019CMEMean3.22902.10250.92510.60063.21232.08510.92770.60223.14282.06470.91480.60123.10192.06000.91300.5995MSE0.79320.19730.01230.00490.64670.15380.01040.00400.39020.12480.00710.00290.24650.08950.00550.0021ADEMean2.73632.04520.81230.56922.63011.98000.78640.56792.46331.89180.73890.56422.39331.84810.72340.5635MSE0.40530.11750.01560.00470.38770.07850.01920.00400.43100.06750.03030.00320.44880.06040.03420.0027Table 3
Mean and MSE for parameter and stress-strength reliability for EIW distribution with different sample size: Case 3.
θ1=0.5,β=0.5n=30,m=40n=35,m=45n=50,m=60n=70,m=80θ2θ1θ2βRθ1θ2βRθ1θ2βRθ1θ2βR0.4MLEMean0.50860.40380.51270.55600.50540.40080.51360.55650.50510.39970.50810.55770.50190.39960.50580.5564MSE0.01200.00670.00260.00350.01080.00620.00240.00350.00720.00460.00160.00250.00490.00320.00100.0017MPSEMean0.46900.38000.50820.55120.47150.38010.50870.55260.48050.38390.50390.55540.48430.38810.50170.5548MSE0.01100.00630.00260.00350.00990.00590.00230.00340.00680.00450.00160.00250.00470.00310.00100.0017MSADEMean0.65710.54860.42370.54550.62030.50880.43720.54930.58130.47140.45320.55190.55010.44750.46590.5514MSE0.08380.06980.01490.00450.05520.04270.01070.00440.03180.02210.00710.00310.01500.01150.00400.0024MSALDEMean0.51420.42200.48250.54810.51380.41490.48600.55180.51130.41090.48700.55400.50600.40680.48930.5540MSE0.01490.00990.00350.00440.01430.00870.00300.00420.00940.00650.00240.00290.00650.00440.00140.0022LSEMean0.52000.41710.48990.55270.51420.41110.49460.55450.51140.40810.49260.55540.50860.40550.49480.5558MSE0.01450.00750.00340.00430.01200.00700.00290.00410.00810.00510.00220.00280.00580.00350.00150.0021WLSEMean0.51500.41180.49910.55360.51030.40680.50270.55520.50770.40410.50000.55620.50540.40190.50080.5565MSE0.01380.00720.00300.00410.01160.00670.00270.00390.00770.00490.00190.00270.00540.00340.00130.0020CMEMean0.51310.40680.51180.55530.50780.40190.51400.55680.50650.40150.50640.55710.50500.40060.50490.5570MSE0.01530.00760.00380.00470.01260.00710.00330.00440.00840.00510.00230.00300.00590.00360.00160.0022ADEMean0.54420.48420.44980.52810.55040.49140.43590.52800.56430.50560.40820.52740.56700.50450.40090.5292MSE0.01390.01530.00490.00420.01220.01590.00610.00380.01040.01650.00980.00270.00880.01460.01080.00211.5MLEMean0.51041.54040.51400.25070.50251.54050.51400.24770.50571.52770.50890.24980.50221.51360.50640.2499MSE0.01300.07460.00270.00260.01050.06380.00270.00230.00690.04110.00150.00150.00540.02980.00110.0011MPSEMean0.47111.43220.50900.24930.46891.44210.50900.24710.48121.45100.50450.25020.48461.45410.50220.2507MSE0.01170.06700.00270.00260.01000.05690.00260.00230.00650.03820.00150.00150.00520.02910.00110.0011MSADEMean0.67101.59190.41930.29290.62881.57980.43480.28160.59031.54540.45020.27510.55121.51150.46610.2672MSE0.09330.11250.01560.00680.06520.09640.01350.00510.03560.05920.00750.00360.01660.04210.00410.0023MSALDEMean0.51911.47830.48250.26180.50671.48100.48660.25650.51241.48150.48790.25820.50471.47170.49050.2562MSE0.01680.07960.00360.00360.01390.06610.00370.00300.00960.04750.00220.00200.00720.03600.00160.0015LSEMean0.52171.50470.49220.25990.51001.51060.49460.25500.51201.50660.49450.25540.50841.49370.49710.2552MSE0.01550.07860.00350.00350.01260.07640.00330.00300.00800.04890.00210.00200.00630.03630.00160.0016WLSEMean0.51691.51990.50130.25580.50641.52470.50350.25150.50831.51640.50150.25260.50571.50290.50250.2527MSE0.01470.07490.00310.00310.01190.07090.00300.00270.00740.04560.00180.00180.00580.03270.00130.0013CMEMean0.51481.55400.51430.25170.50341.55400.51390.24760.50721.53710.50840.25000.50481.51540.50720.2512MSE0.01640.09600.00400.00350.01320.09190.00380.00320.00820.05570.00230.00200.00650.03940.00180.0016ADEMean0.54641.56780.45070.26010.54811.54080.43500.26390.56401.48980.41070.27550.56741.44900.40160.2820MSE0.01480.06760.00490.00260.01240.05300.00660.00240.01020.02890.00940.00200.00920.02210.01060.0020Table 4
The length of bootstrap CI for stress-strength reliability with different level of intervals: Case 1.
θ10.13θ2=0.1,β=0.590%95%99%90%95%99%n,mL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCP30, 40MLE0.006570.006580.007640.007640.010290.010300.001290.001280.001510.001520.002010.00200MPSE0.006230.006280.007480.007440.010010.009910.001230.001240.001470.001470.002040.00204MSADE0.007040.007080.008440.008460.011150.011300.004070.004030.004950.004940.006810.00667MSALDE0.007070.007070.008500.008450.010700.010730.001660.001670.002010.002010.002620.00273LSE0.006680.006760.007940.007810.010630.010850.001650.001640.002020.002030.002560.00258WLSE0.006740.006800.008120.008110.010340.010340.001640.001630.001910.001910.002490.00250CME0.007440.007440.009000.009000.011540.011580.001580.001580.001950.001960.002550.00256ADE0.006200.006220.007640.007580.009870.009780.002180.002170.002550.002590.003320.0034435, 45MLE0.005860.005920.007240.007180.009480.009430.001180.001170.001420.001410.001840.00185MPSE0.006290.006070.007350.007440.009740.009790.001220.001220.001460.001460.001850.00188MSADE0.006900.006970.008320.008380.011650.011350.003900.003940.004730.004800.006100.00653MSALDE0.006740.006690.007860.007820.009820.009930.001670.001690.002020.002030.002520.00252LSE0.006680.006680.007900.007900.010610.010610.001510.001500.001770.001780.002330.00232WLSE0.006210.006260.007700.007720.010540.010810.001470.001450.001700.001720.002150.00216CME0.006760.006800.008360.008390.011400.011330.001430.001410.001700.001690.002240.00226ADE0.005740.005750.006740.006740.008750.008770.002060.002090.002450.002440.003410.0034750, 60MLE0.005050.005000.005980.005980.008140.008210.001080.001080.001250.001250.001610.00162MPSEE0.004840.004720.005850.005860.008150.007960.001090.001090.001260.001260.001630.00163MSADE0.006180.006200.007400.007370.009310.009330.002970.003020.003690.003650.004650.00482MSALDE0.005690.005680.006770.006810.009100.009080.001360.001340.001610.001630.002160.00220LSE0.005520.005470.006500.006520.008870.008840.001420.001420.001660.001660.002110.00211WLSE0.005580.005580.006810.006850.009470.009360.001240.001230.001480.001490.002110.00207CME0.005970.006000.006940.006970.009080.009010.001310.001330.001630.001620.002110.00209ADE0.004420.004380.005340.005170.006860.006890.002060.002030.002350.002360.003220.0032270, 80MLE0.004050.004160.005010.004970.006460.006590.000870.000870.001050.001060.001460.00146MPSE0.004070.004120.004880.004880.006690.006620.000880.000910.001080.001080.001430.00153MSADE0.005120.005130.006230.006280.008070.008070.001910.001920.002300.002300.003100.00311MSALDE0.004800.004830.005980.005960.007870.007980.001080.001100.001300.001290.001680.00168LSE0.004650.004640.005570.005550.007970.007910.001080.001070.001280.001280.001810.00178WLSE0.004630.004610.005510.005510.007620.007740.001000.001010.001230.001200.001530.00153CME0.004920.004840.005850.005830.007820.007810.001100.001110.001290.001300.001600.00160ADE0.003820.003780.004490.004490.005790.005750.001770.001780.002170.002190.002840.00280Table 5
The length of bootstrap CI for stress-strength reliability with different level of intervals: Case 2.
θ20.42θ1=3,β=0.990%95%99%90%95%99%n,mL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCP30, 40MLE0.003210.003200.003930.003960.004930.004930.006520.006530.007650.007670.010110.01049MPSE0.003260.003270.003920.003930.005370.005300.006440.006420.007630.007580.010480.01050MSADE0.006890.006840.008050.008070.010320.010120.007260.007230.008750.008830.011610.01170MSALDE0.004060.004080.004970.004970.006550.006420.006860.006870.008310.008330.010640.01064LSE0.003730.003760.004430.004420.005750.005750.007060.007110.008670.008580.011250.01131WLSE0.003610.003620.004270.004290.005630.005620.006790.006790.007930.007940.010740.01076CME0.003690.003690.004430.004640.006070.006050.007210.007150.008340.008370.011010.01104ADE0.004040.003960.004820.004830.006060.006220.005850.005850.007350.007330.009750.0096635, 45MLE0.003120.003110.003700.003730.004440.004430.005640.005750.006890.006880.008390.00836MPSE0.003200.003110.003740.003700.004870.004920.005790.005920.007160.007100.009550.00976MSADE0.006170.006160.007230.007190.009520.009410.006680.006690.008510.008420.010740.01069MSALDE0.003610.003590.004060.004190.005830.006010.006460.006390.007450.007520.009450.00949LSE0.003540.003540.004280.004280.005780.005780.006640.006440.007870.007810.010640.01079WLSE0.003580.003560.004250.004280.005380.005590.006380.006210.007340.007400.010030.01014CME0.003630.003640.004240.004250.005700.005730.007000.007070.008350.008270.011220.01164ADE0.003840.003830.004860.004870.005820.005950.005750.005750.007010.006990.009320.0091950, 60MLE0.002500.002480.003020.003050.004460.004460.004960.004950.005980.005990.008100.00813MPSE0.002510.002510.003060.003060.004180.004180.005150.005140.006260.006240.008240.00822MSADE0.004890.004900.005830.005840.007810.007780.005520.005500.006600.006580.008780.00891MSALDE0.003320.003300.003880.003940.005250.005330.005490.005490.006480.006490.008520.00853LSE0.003210.003130.003890.003920.005090.005150.005200.005170.006120.006090.007710.00775WLSE0.002860.002850.003420.003410.004810.004740.005140.005160.006330.006200.008010.00799CME0.003110.003120.003700.003690.004840.004800.005740.005640.006870.006870.008790.00883ADE0.003500.003530.004210.004100.005470.005500.004830.004780.005820.005840.007610.0075670, 80MLE0.002270.002250.002670.002680.003630.003820.004070.004030.005100.005080.006580.00660MPSE0.002250.002250.002660.002650.003400.003400.003910.003840.004790.004890.006380.00633MSADE0.003530.003510.004270.004230.005830.005720.005160.005150.006440.006320.008430.00835MSALDE0.002620.002650.003170.003170.004240.004270.004890.004890.005580.005690.008200.00817LSE0.002740.002770.003350.003280.004410.004330.004720.004720.005650.005590.007400.00727WLSE0.002630.002620.003140.003220.003940.003990.004820.004820.005810.005760.006890.00697CME0.002810.002840.003200.003240.004390.004220.004620.004650.005580.005640.007810.00803ADE0.003090.002990.003670.003660.004990.005160.003820.003780.004450.004400.005730.00585Table 6
The length of bootstrap CI for stress-strength reliability with different level of intervals: Case 3.
θ20.41.5θ1=0.5,β=0.590%95%99%90%95%99%n, mL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCPL.BPL.BCP30, 40MLE0.005880.005870.007120.007130.010180.010170.005290.005380.006200.006250.008850.00870MPSE0.006190.006300.007480.007510.009250.009270.005350.005460.006470.006440.008350.00801MSADE0.006930.007010.008280.008380.011890.010700.006970.006880.008250.008320.010670.01074MSALDE0.006980.006980.008020.008020.010580.010570.006130.006100.007070.007200.010450.01032LSE0.006640.006640.008070.008070.010660.010650.005820.005810.007120.006970.009550.00984WLSE0.006750.006740.008010.008050.010580.010320.006280.006280.007200.007170.009240.00916CME0.006900.006860.008840.008790.011850.011490.006070.006060.007270.007270.010310.01032ADE0.006510.006430.007920.007800.010310.010400.005300.005300.006250.006280.008660.0087135, 45MLE0.006160.006070.007250.007120.009760.009710.004890.004890.005940.005940.008380.00838MPSE0.005850.005890.007130.007150.010390.010520.004880.004890.005820.005820.007810.00781MSADE0.006860.006860.008260.008210.010910.011020.006710.006650.007870.007820.010180.00997MSALDE0.006760.006890.008400.008310.010600.010780.005570.005550.006480.006490.008980.00903LSE0.006350.006370.007830.007820.010190.010210.005560.005610.006740.006730.009360.00934WLSE0.006300.006410.007680.007690.009690.009680.005380.005400.006370.006410.008600.00850CME0.006760.006700.007820.007930.010560.010750.006010.005930.006950.007020.009690.00873ADE0.005790.005790.006750.006810.009750.009750.004720.004740.005740.005720.007470.0075150, 60MLE0.005400.005470.006580.006320.008620.008480.004120.004110.005120.005110.006760.00683MPSE0.005330.005350.006190.006130.007750.007680.004150.004100.004950.004970.006640.00656MSADE0.005930.005930.007020.007040.009040.009010.005610.005500.006700.006680.009300.00906MSALDE0.005830.005750.006660.006650.009470.009450.004620.004750.005690.005620.007970.00786LSE0.005440.005430.006620.006660.008340.008740.004710.004680.005580.005570.007160.00709WLSE0.005230.005240.006100.006110.008480.008470.004260.004220.005070.005080.006980.00698CME0.005390.005400.006420.006300.008690.008660.004740.004770.005760.005810.007260.00745ADE0.004620.004640.005380.005380.007730.007780.003850.003830.004720.004720.005920.0059570, 80MLE0.004390.004390.005170.005170.006160.006160.003510.003600.004210.004230.005440.00593MPSE0.004230.004250.005160.005120.006840.006920.003520.003480.004090.004100.005510.00555MSADE0.004980.004810.005960.005850.007980.007970.004710.004800.005750.005690.007480.00749MSALDE0.004780.004770.005830.005740.007730.007760.004180.004150.004990.004970.006640.00655LSE0.004870.004790.005920.005890.007470.007370.004020.004020.004750.004750.006350.00634WLSE0.004770.004780.005830.005920.007510.007500.003780.003780.004610.004610.005970.00597CME0.004790.004780.005580.005590.007600.007600.004200.004230.005060.004960.006370.00636ADE0.003780.003870.004610.004590.005890.005820.003350.003370.004000.004010.005050.00504The simulation outcomes of point estimation are recorded in Tables1–3. The following concluding remakes are noticed based on these tables:(i)
In some cases, as the sample size of strength increases and for a fixed sample size of stress, the MSEs associated with the parameter estimates decrease for all methods of estimation.(ii)
In some cases, as the sample size of stress increases and for a fixed sample size of strength, the MSEs associated with the parameter estimates decrease for all methods of estimation.(iii)
In Case 1, asθ1 increases and for a fixed sample size of strength and stress, the MSEs of parameters of EIW are increasing for all methods of estimation.(iv)
In Case 2, asθ2 increases and for a fixed sample size of strength and stress, the MSEs for most of the parameters of EIW are increasing for all methods of estimation.The simulation outcomes of interval estimation of stress-strength reliability are recorded in Tables4–6. The following concluding remakes are noticed based on these tables:(i)
In some cases, as the sample size of strength increases and for a fixed sample size of stress, the length of CI of stress-strength reliability estimates decreases for all methods of estimation.(ii)
In some cases, as the sample size of stress increases and for a fixed sample size of strength, the length of CI of stress-strength reliability estimates decreases for all methods of estimation.(iii)
In some cases, as a level of interval increases and for a fixed sample size of stress and strength, the length of CI of stress-strength reliability estimates increases for all methods of estimation.


## 5. Application of Real Data

In this section, we consider two applications of the stress-strength 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) withP value to check the fit of the model and standard errors (SE) of estimators.


### 5.1. Breaking Strengths of Jute Fiber Data

A pair of real data sets are studied for demonstration purposes. The breaking strengths of jute fiber at two different gauge lengths are depicted in these data. Xia et al. [36] used these two data sets in their study, where X represents the breaking strength of jute fiber with a diameter of 10 mm and Y represents the breaking strength of a 20 mm diameter jute fiber.The breaking strengths of jute fiber with a gauge length of 10 mm are “X = 693.73, 704.66, 323.83, 778.17, 123.06, 637.66, 383.43, 151.48, 108.94, 50.16, 671.49, 183.16, 257.44, 727.23, 291.27, 101.15, 376.42, 163.40, 141.38, 700.74, 262.90, 353.24, 422.11, 43.93, 590.48, 212.13, 303.90, 506.60, 530.55, and 177.25.”The breaking strengths of jute fibers with a gauge length of 20 mm are “Y = 71.46, 419.02, 284.64, 585.57, 456.60, 113.85, 187.85, 688.16, 662.66, 45.58, 578.62, 756.70, 594.29, 166.49, 99.72, 707.36, 765.14, 187.13, 145.96, 350.70, 547.44, 116.99, 375.81, 581.60, 119.86, 48.01, 200.16, 36.75, 244.53, and 83.55.”From Table7, 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.Table 7
MLEs, SEs, and KSS test withP value for breaking strengths of jute fiber data.
XYθ1β1θ2β2Estimate483.98331.1803228.95001.0849SE363.08400.1515158.69150.1466KSS0.17080.1566P value0.30920.4116Figure 2
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution forX of breaking strengths of jute fiber data.Figure 3
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution forY of breaking strengths of jute fiber data.The estimates of the parameters model of stress-strength reliability for EIW distribution are obtained in Table8. MSADE has the smallest SE and the largest reliability.Table 8
Estimation with SE and stress-strength reliability for breaking strengths of jute fiber data.
MLEMPSEMSADEMSALDEEstimateSEEstimateSEEstimateSEEstimateSEθ1441.8773262.2884451.5945231.2679483.9824201.5166483.8995198.1652θ2315.0805171.8451274.3958112.3324228.9514106.1569229.003886.2184β1.15690.11471.16360.95571.00470.01541.12840.0955R0.58380.62200.67890.6788LSEWLSECMEADEθ1431.0775316.5598851.4172175.4259461.7079319.4659408.3478465.6265θ2328.7274716.6545653.3498130.7536368.3411416.8481315.7379341.9234β1.13590.40721.27520.03581.14970.49901.13390.2050R0.56740.56580.55620.5639


### 5.2. Carbon Fibers Data

In this subsection, we look at two data sets and discuss all of the specifics for the sake of illustration. The 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. These data were analyzed previously by Hassan et al. [38]. The following are the data sets:Data Set I (length of 10 mm):X (n = 63): “1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020.”Data Set II (length of 20 mm):Y (m = 69): “1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128, 3.233, 3.433, 3.585, 3.585.”From Table9, 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.Table 9
MLEs, SEs, and KSS test withP value for carbon fibers data.
XYθ1β1θ2β2Estimates23.26754.1271230.47635.4338SE5.71330.3382110.96230.5081KSS0.10010.1336P value0.55310.1700Figure 4
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution for Data Set I of carbon fibers data.Figure 5
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution for Data Set II of carbon fibers data.The estimates of the parameters model of stress-strength reliability for EIW distribution are obtained in Table10. MSADE has the smallest SE and the largest reliability.Table 10
Estimation with SE and stress-strength reliability for carbon fibers data.
MLEMSADELSEstimatesSEREstimatesSEREstimatesSERθ1103.946030.24230.7722223.483221.28610.7108225.1419434.18150.7456θ230.66876.343590.93061.311276.8348123.6565β4.57450.27925.27340.00645.35191.7957WLSCVMADθ1325.818729.04740.7626252.1115490.79540.749386.187451.93260.7119θ2101.45127.673984.3294137.073634.874917.2440β5.73520.08295.45691.81284.44010.5642


## 5.1. Breaking Strengths of Jute Fiber Data

A pair of real data sets are studied for demonstration purposes. The breaking strengths of jute fiber at two different gauge lengths are depicted in these data. Xia et al. [36] used these two data sets in their study, where X represents the breaking strength of jute fiber with a diameter of 10 mm and Y represents the breaking strength of a 20 mm diameter jute fiber.The breaking strengths of jute fiber with a gauge length of 10 mm are “X = 693.73, 704.66, 323.83, 778.17, 123.06, 637.66, 383.43, 151.48, 108.94, 50.16, 671.49, 183.16, 257.44, 727.23, 291.27, 101.15, 376.42, 163.40, 141.38, 700.74, 262.90, 353.24, 422.11, 43.93, 590.48, 212.13, 303.90, 506.60, 530.55, and 177.25.”The breaking strengths of jute fibers with a gauge length of 20 mm are “Y = 71.46, 419.02, 284.64, 585.57, 456.60, 113.85, 187.85, 688.16, 662.66, 45.58, 578.62, 756.70, 594.29, 166.49, 99.72, 707.36, 765.14, 187.13, 145.96, 350.70, 547.44, 116.99, 375.81, 581.60, 119.86, 48.01, 200.16, 36.75, 244.53, and 83.55.”From Table7, 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.Table 7
MLEs, SEs, and KSS test withP value for breaking strengths of jute fiber data.
XYθ1β1θ2β2Estimate483.98331.1803228.95001.0849SE363.08400.1515158.69150.1466KSS0.17080.1566P value0.30920.4116Figure 2
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution forX of breaking strengths of jute fiber data.Figure 3
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution forY of breaking strengths of jute fiber data.The estimates of the parameters model of stress-strength reliability for EIW distribution are obtained in Table8. MSADE has the smallest SE and the largest reliability.Table 8
Estimation with SE and stress-strength reliability for breaking strengths of jute fiber data.
MLEMPSEMSADEMSALDEEstimateSEEstimateSEEstimateSEEstimateSEθ1441.8773262.2884451.5945231.2679483.9824201.5166483.8995198.1652θ2315.0805171.8451274.3958112.3324228.9514106.1569229.003886.2184β1.15690.11471.16360.95571.00470.01541.12840.0955R0.58380.62200.67890.6788LSEWLSECMEADEθ1431.0775316.5598851.4172175.4259461.7079319.4659408.3478465.6265θ2328.7274716.6545653.3498130.7536368.3411416.8481315.7379341.9234β1.13590.40721.27520.03581.14970.49901.13390.2050R0.56740.56580.55620.5639


## 5.2. Carbon Fibers Data

In this subsection, we look at two data sets and discuss all of the specifics for the sake of illustration. The 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. These data were analyzed previously by Hassan et al. [38]. The following are the data sets:Data Set I (length of 10 mm):X (n = 63): “1.901, 2.132, 2.203, 2.228, 2.257, 2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886, 3.971, 4.024, 4.027, 4.225, 4.395, 5.020.”Data Set II (length of 20 mm):Y (m = 69): “1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090, 3.096, 3.128, 3.233, 3.433, 3.585, 3.585.”From Table9, 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.Table 9
MLEs, SEs, and KSS test withP value for carbon fibers data.
XYθ1β1θ2β2Estimates23.26754.1271230.47635.4338SE5.71330.3382110.96230.5081KSS0.10010.1336P value0.55310.1700Figure 4
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution for Data Set I of carbon fibers data.Figure 5
Cumulative function and empirical CDF, histogram, and P-P plot for the EIW distribution for Data Set II of carbon fibers data.The estimates of the parameters model of stress-strength reliability for EIW distribution are obtained in Table10. MSADE has the smallest SE and the largest reliability.Table 10
Estimation with SE and stress-strength reliability for carbon fibers data.
MLEMSADELSEstimatesSEREstimatesSEREstimatesSERθ1103.946030.24230.7722223.483221.28610.7108225.1419434.18150.7456θ230.66876.343590.93061.311276.8348123.6565β4.57450.27925.27340.00645.35191.7957WLSCVMADθ1325.818729.04740.7626252.1115490.79540.749386.187451.93260.7119θ2101.45127.673984.3294137.073634.874917.2440β5.73520.08295.45691.81284.44010.5642


## 6. Conclusion

In this paper, we assumed thatX 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 θ1,θ2, and β and the system of stress-strength parameter R=PY<X. The eight methods of estimations are MLEs, MPSEs, MSADEs, MSALDEs, LSEs, WLSEs, CMEs, and ADEs. The 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 θ1,θ2, and β and the system of stress-strength reliability R=PY<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.



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*Source: 4227346-2021-09-22.xml*