# Reliability Estimation for the Remained Stress-Strength Model under the Generalized Exponential Lifetime Distribution

Authors: Mohammad Mehdi Saber; Marwa M. Mohie El-Din; Haitham M. Yousof
Journal: Journal of Probability and Statistics
 (2021)

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## Abstract

A stress-strength reliability model compares the strength and stresses on a certain system; it is used not only primarily in reliability engineering and quality control but also in economics, psychology, and medicine. In this paper, a novel extension of stress-strength models is presented. The mew model is applied under the generalized exponential distribution. The maximum likelihood estimator, asymptotic distribution, and Bayesian estimation are obtained. A comprehensive simulation study along with real data analysis is performed for illustrating the importance of the new stress-strength model.


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## Body

## 1. Introduction

Stress-strength reliability analysis is a statistical analysis of the interference of of the strength of the component and the stresses placed on the component. The stress-strength reliability analysis is a statistical tool used in reliability engineering.In a stress-strength reliability model, both strength and stresses are considered as separate random variables. Stress experienced by a certain component is usually presented by the random variableX and the strength of the same component is presented by the ramdom variable Y. A situation in which X>Y is one in which the stress is greater than the strength, and then, the component fails.The abovementionedprobability model can be expressed asR=PrX>Y and then called as the stress-strength quantity. The stress-strength reliability model has various applications in many fields such as reliability, quality control, and engineering. For more details in this matter, see the work of Kotz et al. [1] and Ventura and Racugno [2]. Rezaei et al. [3] presented a list of probability distributions used under the stress-strength reliability model. Recently, Rasekhi et al. [4] presented a Bayesian and the classical inference of reliability in multicomponent stress-strength under the generalized logistic model. Saber and Yousof [5] investigated the Bayesian and the classical inference for the generalized stress-strength parameter under generalized logistic distribution.We suppose that we know these two components have been worked till a known time, and then, we are going to have some inferences onR. For this case, Saber and Khorshidian [6] introduce the conditional stress-strength model R|a,b:(1)R|a,b=PX>Y|X>a,Y>b.When independent random variablesX and Y are continues, R|a,b can be computed by the following equation:(2)R|a,b=1−FYa−∫a+∞FXyfYydy1−FXa1−FYb|a=b,1−FYb−∫b+∞FXyfYydy1−FXa1−FYb|a<b,∫a+∞FYxfXxdx−FYb1−FXa1−FXa1−FYb|a>b.Saber and Khorshidian [6] studied this model for the case of exponential distributed components under the nonparametric case, and the Bayesian estimation is ignored. However, in this work, we study the conditional stress-strength model for two-parameter generalized exponential (GE) distributed components, and the Bayesian estimation is considered. The GE distribution can be used as an alternative to gamma, Weibull, and log-normal distributions. For more related studies, see the work of Gupta and Kundu [7] and Kundu (2002) and Rao [8].The probability density function and cumulative distribution function of GE distribution are, respectively,(3)fXx=αλe−λx1−e−λxα−1,FXx=1−e−λxα,where x≥0 and α>0 are the shape parameters and λ>0 is the scale parameter.The rest of the paper is organized as follows. We devote Section2 to study R|a,b in case of GE distribution. In Section 3, the ML estimator of quantity R|a,b and its corresponding asymptotic distribution and confidence interval are provided. A simulation study is presented in Section 4, and Section 5 has been devoted to applying a real dataset to the recommended model.


## 2. Conditional Stress-Strength Model for GE Distribution

In this section, quantity (2) is computed when distribution of components is GE.Theorem 1.
We suppose random variablesX and Y are independent and X∼GEα1,λ and Y∼GEα2,λ; then,(4)R|a,b=1−1−e−λbα2−α2/α1+α21−1−e−λaα11−1−e−λbα2|a=b,1−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λaα11−1−e−λbα2|a<b,α2/α1+α21−1−e−λaα1+α2−1−e−λbα21−1−e−λaα11−1−e−λaα11−1−e−λbα2|a>b.Proof.
Let(5)fXy=α1λe−λy1−e−λyα1−1,fYy=α2λe−λy1−e−λyα2−1,FXy=1−e−λyα1,FYy=1−e−λyα2,FXa=1−e−λaα1,FYb=1−e−λbα2.
Substitute the last six equations in equation (2). Let a=b; then,(6)R|a,b=1−FYa−∫a+∞FXyfYydy1−FXa1−FYb.
Then,(7)⇒R|a,b=1−FYa−∫a+∞1−e−λyα1α2λe−λy1−e−λyα2−1dy1−1−e−λbα11−1−e−λbα2,⇒R|a,b=1−FYa−∫a+∞α2λe−λy1−e−λyα1+α2−1dy1−1−e−λbα11−1−e−λbα2,⇒R|a,b=1−FYa−α2/α1+α2∫a+∞α1+α2λe−λy1−e−λyα1+α2−1dy1−1−e−λbα11−1−e−λbα2,⇒R|a,b=1−FYa−α2/α1+α21−FWa1−1−e−λbα11−1−e−λbα2.
In the abovementioned result, it is noted thatW∼GEα1+α2,λ; thus,(8)FWa=1−e−λaα1+α2.
Sincea=b, the proof of Theorem 1 is completed.
Ifa<b, then(9)R|a,b=1−FYb−∫b+∞FXyfYydy1−FXa1−FYb.
The dominator ofR|a,b in this case is exactly the same as in the first case with substitution of b instead of a. Therefore,(10)R|a,b=1−FYb−α2/α1+α21−FWb1−1−e−λaα11−1−e−λbα2.
Then,(11)R|a,b=1−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λaα11−1−e−λbα2.
Eventually, ifa>b, we can calculate R|a,b like previous ones.(12)R|a,b=∫a+∞FYxfXxdx−FYb1−FXa1−FXa1−FYb.
Then,(13)R|a,b=∫a+∞1−e−λyα2α1λe−λy1−e−λyα1−1dy−FYb1−FXa1−FXa1−FYb,which can be expressed as(14)R|a,b=α1/α1+α21−FWa−FYb1−FXa1−FXa1−FYb.
Below, we derive the maximum likelihood estimation (MLE) of theR|a,b model; and hence, the asymptotic distribution of those is presented in order to constructing the corresponding confidence interval.
LetX1,X2,…,Xm be a random sample of size m of GEα1,λ and Y1,Y2,…,Yn be a random sample of size n of GEα2,λ such that X and Y are independent. Then, the likelihood function can be expressed as(15)L=α1mα2nλm+ne−λ∑i=1mxi+∑j=1nyj×∏i=1m1−e−λxiα1−1∏j=1n1−e−λyjα2−1.
Then, the log-likelihood function is given by(16)Li,jm,n=mlnα1+nlnα2+m+nlnλ−λ∑i=1mxi+∑j=1nyj+α1−1∑i=1mln1−e−λxi+α2−1∑j=1nln1−e−λyj.
Therefore, the maximum likelihood estimator of parameters can be obtained by solving∂Li,jm,n/∂λ=0, ∂Li,jm,n/∂α1=0 and ∂Li,jm,n/∂α2=0. A simple computation shows(17)∂Li,jm,n∂α1=mα1+∑i=1mln1−e−λxi,∂L∂α2=nα2+∑i=1mln1−e−λxi,∂Li,jm,n∂λ=m+nλ−∑i=1mxi+∑j=1nyj+α1−1∑i=1mxie−λxi1−e−λxi+α2−1∑j=1nyje−λyj1−e−λyj.
Therefore, we have(18)α1=−m∑i=1mln1−e−λxi,(19)α2=−n∑j=1nln1−e−λyj.
Equations (18) and (19) depend on unknown parameter λ. We substitute equations (18) and (19) in ∂Li,jm,n/∂λ, and we can find λ^ by solving the following nonlinear equation:(20)∂Li,jm,n∂λ=m+nλ−∑i=1mxi+∑j=1nyj−m∑i=1mln1−e−λxi+1∑i=1mxie−λxi1−e−λxi−n∑j=1nln1−e−λyj+1∑j=1nyje−λyj1−e−λyj=0.
Then, two other parameters are earned by substitutionλ^ in equations (18) and (19) as(21)α^1=−m∑i=1mln1−e−λ^xi,α^2=−n∑j=1nln1−e−λ^yj.
Therefore, the maximum likelihood estimator ofR|a,b becomes(22)R^|a,b=1−1−e−λ^bα^2−α^2/α^1+α^21−1−e−λ^bα^2+α^11−1−e−λ^aα^11−1−e−λ^bα^2|a=b,1−1−e−λ^bα^2−α^2/α^1+α^21−1−e−λ^bα^2+α^11−1−e−λ^aα^11−1−e−λ^bα^2|a<b,α^1/α^1+α^21−1−e−λ^aα^1+α^2−1−e−λ^bα^21−1−e−λ^aα^11−1−e−λ^aα^11−1−e−λ^bα^2|a>b.
In continuation with this section, the asymptotic distribution ofθ^=λ^,α^1,α^2 and the asymptotic distribution of R|a,b are obtained. The Fisher information matrix of θ=λ,α1,α2 denoted by Jθ=EIθ, where Iθ=Ii,ji,j=1,2,3 is the observed information matrix; i.e.,(23)Iθ=−∂2Li,jm,nλ,α1,α2∂λ2∂2Li,jm,nλ,α1,α2∂λ∂α1∂2Li,jm,nλ,α1,α2∂λ∂α2∂2Li,jm,nλ,α1,α2∂α1∂λ∂2Li,jm,nλ,α1,α2∂α12∂2Li,jm,nλ,α1,α2∂α1∂α2∂2Li,jm,nλ,α1,α2∂α2∂λ∂2Li,jm,nλ,α1,α2∂α2∂α1∂2Li,jm,nλ,α1,α2∂α22.
Now, the elements ofJθ follow by(24)J11=−E∂2Li,jm,n∂λ2,and then,(25)J11=−E−m+nλ2+α1−1∑i=1m−xi2e−λxi1−e−λxi2+α2−1∑j=1n−yj2e−λyj1−e−λyj2.
It can finally be derived as(26)J11=m+nλ2+mα1−1α1λ2∂2Betat+1,α1−2∂2t|t=2+nα2−1α2λ2∂2Betat+1,α2−2∂2t|t=2,J12=J21=−E∂2L∂λ∂α1=−E∑i=1mxie−λxi1−e−λxi,=mα1λ∂Betat+1,α1−1∂t|t=1,J13=J31=−E∂2L∂λ∂α2=−E∑j=1nyje−λyj1−e−λyj=nα2λ∂Betat+1,α2−1∂t|t=1,J22=−E∂2Li,jm,n∂α12=−E−mα12=mα12,J23=J32=−E∂2Li,jm,n∂α1∂α2=0,J33=−E∂2Li,jm,n∂α22=−E−nα22=nα22.
By characteristics of MLEs, we haveθ^≈N3θ,Σ for a large number of n and m, where Σ is the inverse of the Fisher information matrix Iθ.(27)Σ=1detIθJ22J33−J23J32J13J32−J12J33J12J23−J13J22J23J31−J21J33J11J33−J13J31J13J21−J11J23J21J32−J22J31J12J31−J11J32J11J22−J12J21,Σ=1detIθJ22J33−J12J33−J13J22−J21J33J11J33−J13J31J13J21−J22J31J12J31J11J22−J12J21.
In Lemma1, we recall the multivariate Delta method in order to find an asymptotic distribution of R|a,b.Lemma 1.
LetXnn=1∞ be a sequence of random vector, where Xn⟶Nμ,Σ in distribution. If the function g is continuous in the first partial derivatives and τ=∇gμTΣ∇gμ>0, then gXn⟶Ngμ,τ in distribution. Lemma 1 is related to the the multivariate Delta method.Here, with the help of a theorem well known as the multivariate Delta method, we find the asymptotic distribution ofR|a,b. As n⟶∞ and m⟶∞; then,(28)1σ1R^|a,b−R|a,b⟶N0,1|a=b,1σ2R^|a,b−R|a,b⟶N0,1||a<b,1σ3R^|a,b−R|a,b⟶N0,1|a>b,where σ12,σ22, and σ32 are obtained from the following equations. In all three cases,(29)σ2λ,α1,α2=∇R|a,bI−1λ,α1,α2∇R|a,bT,∇R|a,b=∂R|a,b∂λ,∂R|a,b∂α1,∂R|a,b∂α2.For all cases,I−1λ,α1,α2 is equal and the only difference is in ∇R|a,b.In the following, this quantity has been computed for casesa=b, a<b, and a>b, respectively.Case 1.
a=b.(30)∂R|a,b∂λ=α2be−λb1−e−λbα2−1−α2be−λb1−e−λbα1+α2−1×1−1−e−λbα11−1−e−λbα21−1−e−λbα11−1−e−λbα22−α1be−λb1−e−λbα1−11−1−e−λbα2+α2be−λb1−e−λbα2−11−1−e−λbα11−1−e−λbα11−1−e−λbα22×1−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λbα11−1−e−λbα22,∂R|a,b∂α1=−α2/α1+α21−1−e−λbα2+α1+1−e−λbα2+α1ln1−e−λbα2/α1+α21−1−e−λbα11−1−e−λbα22×1−1−e−λbα11−1−e−λbα2−1−e−λbα1ln1−e−λb1−1−e−λbα21−1−e−λbα11−1−e−λbα22×1−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λbα11−1−e−λbα22,∂R|a,b∂α2=1−e−λbα2ln1−e−λb+α2/α1+α221−e−λbα2+α1+α2/α1+α22ln1−e−λb1−e−λbα2+α11−1−e−λbα11−1−e−λbα221−1−e−λaα11−1−e−λbα2−1−e−λbα2ln1−e−λb1−1−e−λaα11−1−e−λbα11−1−e−λbα221−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λbα11−1−e−λbα22.Case 2.
a<b.(31)∂R|a,b∂λ=α2be−λb1−e−λbα2−1−α2be−λb1−e−λbα1+α2−11−1−e−λaα11−1−e−λbα21−1−e−λaα11−1−e−λbα22−α1ae−λa1−e−λaα1−11−1−e−λbα2+α2be−λb1−e−λbα2−11−1−e−λaα11−1−e−λaα11−1−e−λbα221−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λaα11−1−e−λbα22,∂R|a,b∂α1=−α2/α1+α21−1−e−λbα2+α1+1−e−λbα2+α1ln1−e−λbα2/α1+α21−1−e−λaα11−1−e−λbα221−1−e−λaα11−1−e−λbα2−1−e−λaα1ln1−e−λa1−1−e−λbα21−1−e−λaα11−1−e−λbα221−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λaα11−1−e−λbα22,∂R|a,b∂α2=1−e−λbα2ln1−e−λb+α2/α1+α221−e−λbα2+α1+α2/α1+α22ln1−e−λb1−e−λbα2+α11−1−e−λbα11−1−e−λbα22×1−1−e−λaα11−1−e−λbα2−1−e−λbα2ln1−e−λb1−1−e−λaα11−1−e−λbα11−1−e−λbα22×1−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λbα11−1−e−λbα22×1−1−e−λbα2−α2/α1+α21−1−e−λbα2+α11−1−e−λbα11−1−e−λbα22.Case 3.
a>b.(32)∂R|a,b∂λ=α1ae−λa1−e−λaα1+α2−1−α2be−λb1−e−λbα2−11−1−e−λaα1+α1ae−λa1−e−λaα1−11−e−λbα21−1−e−λaα11−1−e−λbα221−1−e−λaα11−1−e−λbα2−α1ae−λa1−e−λaα1−11−1−e−λbα2+α2be−λb1−e−λbα2−11−1−e−λaα11−1−e−λaα11−1−e−λbα22α1/α1+α21−1−e−λaα1+α2−1−e−λbα21−1−e−λaα11−1−e−λaα11−1−e−λbα22,∂R|a,b∂α1=α2/α1+α221−1−e−λaα1+α2+α1/α1+α21−e−λaα1+α2ln1−e−λa−1−e−λaα1ln1−e−λa1−e−λbα21−1−e−λaα11−1−e−λbα221−1−e−λaα11−1−e−λbα2−1−e−λaα1ln1−e−λa1−1−e−λbα21−1−e−λaα11−1−e−λbα22α1/α1+α21−1−e−λaα1+α2−1−e−λbα21−1−e−λaα11−1−e−λaα11−1−e−λbα22,∂R|a,b∂α2=−α1/α1+α221−1−e−λaα1+α2+α1/α1+α21−e−λaα1+α2ln1−e−λa−1−e−λbα2ln1−e−λb1−1−e−λaα11−1−e−λbα11−1−e−λbα221−1−e−λaα11−1−e−λbα2−1−e−λbα2ln1−e−λb1−1−e−λaα11−1−e−λbα11−1−e−λbα221−1−e−λbα2−α1/α1+α21−1−e−λbα2+α11−1−e−λbα11−1−e−λbα22.
Equation (28) can be used for finding the confidence interval of R|a,b, however, by using of estimates of asymptotic variance. From this theorem, the 1−α×100 percentage confidence interval of R|a,b is given by(33)R|a,b∈R^|a,b−z1−α/2σ^1,R^|a,b+z1−α/2σ^1|a=b,R|a,b∈R^|a,b−z1−α/2σ^2,R^|a,b+z1−α/2σ^2|a<b,R|a,b∈R^|a,b−z1−α/2σ^3,R^|a,b+z1−α/2σ^3|a>b.
In the abovementioned equations,σ^i2i=1,2,3 are similar to σ^i2i=1,2,3 in equation (28) with substitution λ^, α^1, and α^2 instead of λ, α1, and α2.


## 3. Bayesian Method for Estimation

In this section, we provide a Bayes estimator forR|a,b. All priors for α1, α2, and λ are considered Gamma distribution. We more exactly denoted(34)πα1|ξ1,d1=d1ξ1Γξ1α1ξ1−1e−d1α1,πα2|ξ2,d2=d2ξ2Γξ2α2ξ2−1e−d2α2,πλ|ξ3,d3=d3ξ3Γξ3λξ3−1e−d3λ.Now, the joint posterior density ofθ is given by(35)Lθ|x,y∝α1m+ξ1−1α2n+ξ2−1λm+n+ξ3−1×e−λ∑i=1mxi+∑j=1nyj+d3−d1α1−d2α2×∏i=1m1−e−λxiα1−1∏j=1n1−e−λyjα2−1.Equation (35) is complicated, and it does not belong to a known distrinution. Therefore, we use Gibs sampler to generate samples from (35). By (35), the full posterior density functions are(36)Lα1|α2,λ,x,y∝α1m+ξ1−1e−α1d1+∑i=1mln1−e−λxi,Lα2|α1,λ,x,y∝α2n+ξ2−1e−α2d2+∑j=1nln1−e−λyj,(37)Lλ|α1,α2,x,y∝λm+n+ξ3−1e−λ∑i=1mxi+∑j=1nyj+d3×∏i=1m1−e−λxiα1−1∏j=1n1−e−λyjα2−1.It is clear that the posteriors ofα1 and α2 are Gamma distribution while (37) does not have a known distribution. Henceforth, we use Metropolis–Hastigs (M–H) algorithm to generate data from (37). The proposal distribution for M–H algorithm is considered Gamma with the shape parameter m+n+ξ3 and scale parameter ∑i=1mxi+∑j=1nyj+d3.


## 4. Simulation Study

In this section, we conduct a simulation study in order to survey quality and efficiency of the introduced model and its estimator. All results are the mean of 10000 iteration. To put it more clearly, note that we have iterated our simulation 10000 times. In theith iteration, two random samples with size n and m are generated and R^|a,b is computed. The values of R^|a,b demonstrated in the tables are the mean of these 10000 computed estimates as follows:(38)R^|a,b=∑i=110000R^i|a,b10000.Four criteria containing bias, Mean Square Error (MSE), Coverage Probability (CP), and Length of Confidence Interval (LCI) are used in order to investigate the effectiveness and potentialities of the method. The results are demonstrated in Table1 for values of parameters α1=2, α2=1.6, λ=2, a=2, and b=2. Also, the results are represented in Table 2 for values of parameters α1=3, α2=3, λ=1.5, a=1, and b=3.Table 1
Results forR|2,2=0.50061 and some values of sample sizes n and m.
n152510040253020m152510010456080R^|a,b0.5006560.50061940.50061280.5008390.5005520.500560.500506Bias−0.00117−6×10−5−0.00078750.000149−0.000670.000820.000148MSE1.37E − 063.6×10−96.2×10−72.2E − 084.4×10−76.7×10−72×10−8CP0.004720.00335170.00160440.0045630.0027770.0025060.002723LCI0.9240.920.94780.92540.9360.94390.9577Table 2
Results forR|1,3=0.5 and different values of sample sizes n and m.
n15251001525100m15251001525100R^|a,b0.5006560.50061940.50061280.5006560.5006190.5006128Bias−0.00117−6×10−5−0.0007875−0.00117−6×10−5−0.0007875MSE1.4×10−63.6×10−96.2×10−71.4×10−63.6×10−96.20×10−7CP0.004720.00335180.001604370.004720.00335180.00160437LCI0.9240.920.94780.9240.920.9478As these tables show two criteria MSE and bias are very small; therefore, our estimation method is appropriate. Also, values of CP and LCI as cover of probability and length of confidence interval forR|2,2 represent the same findings.


## 5. Application

In this section, two real datasets reported by Lawless [9] (data A) and Linhardt and Zucchini [10] (data B), respectively, are analyzed. We fit the GE distribution to the two datasets separately. The first dataset arose in tests on endurance of deep groove ball bearings and is the number of million revolutions before failure for each of the 23 ball bearings in the life test. The other dataset denotes the failure times of the air-conditioning system of an airplane (in hours).Gupta and Kundu [7] and Gupta and Kundu [11] studied the validity of GE distribution for these two datasets, respectively. In Table 3, the Kolmogorov–Smirnov distance and its corresponding P value are provided for these data which confirm that the generalized exponential model fits quite well to both the datasets where data I: 17.88, 28.92, 33, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.8, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, and 173.40 and data II: 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, and 52, 95.Table 3
Results of goodness-of-fit tests for fitting GE distribution to two datasets.
DataK–SP valueI0.10580.9592II0.17440.2926Using the results previously presented in Section2, we can obtain the MLE of parameters α1, α2, and λ. Our computations show that α^1=2.8082, α^2=1.0067, and λ^=0.5036. For these estimated parameters, R^|a,b and its corresponding confidence interval have been computed for some values of a and b, as in Table 4.Table 4
R^|a,b and its corresponding C.I for real datasets.
R^|1,3=0.5371R^|2,2=0.5662R^|1,3=0.5126R^|0.1,0.4=0.6843(0.5125, 0.5616)(0.5231, 0.609)(0.5037, 0.5215)(0.5856, 0.7828)


## 6. Conclusions

In this paper, a new extension of the stress-strength model is defined, studied, and applied under many particular cases. The novel stress-strength model is applied under the generalized exponential distribution. The maximum likelihood estimator, asymptotic distribution, and Bayesian estimation are obtained with details. A simulation study along with analysis of two real datasets is also performed for illustrative purposes. We hope that the proposed method would enable engineers and system designers to design better products.



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