# Classical and Bayesian Inference of Conditional Stress-Strength Model under Kumaraswamy Distribution

Authors: Fathy H. Riad; Mohammad Mehdi Saber; Mehrdad Taghipour; M. M. Abd El-Raouf
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

Stress-strength models have been frequently studied in recent years. An applicable extension of these models is conditional stress-strength models. The maximum likelihood estimator of conditional stress-strength models, asymptotic distribution of this estimator, and its confidence intervals are obtained for Kumaraswamy distribution. In addition, Bayesian estimation and bootstrap method are applied to the model.


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## Body

## 1. Introduction

In reliability,R=PX>Y is named as the “stress-strength model.” This model has application in some literatures in addition to reliability, such as biostatistics, quality control, engineering, psychology, stochastic precedence, medicine, and probabilistic mechanical design. For a comprehensive review with details, refer to the study by Kotz et al. [1]. Regarding this reference, a sample in a clinical study can be considered in such a way that Y and X are assumed to be the results of a treatment group and a control group, respectively. Therefore, the expression (1 − R) measures the effect of treatment.For some other applications, refer to the study by Ventura and Racugno (2011). In terms of reliability,Y is considered the strength of a component, which is under X stress. Therefore, R and (1 − R) indicate the system performance and the probability of system failure, respectively.Some authors have extensively studied quantityR in parametric case for different distributions and in nonparametric case. Refer the study by Rezaei et al. [2] for a list of distributions used in this matter.In usual situations, it is known thatX and Y are bigger than two fixed values. Especially when X and Y are lifetime of two components of a system, we may know that these components had been alive for a known time until we are going to have some inferences about R=PX>Y. Therefore, Saber and Khorshidian [3] introduced the conditional stress-strength model as follows:(1)R|a,b=PX>Y|X>a,Y>b.The quantityR is special case of this quantity if we set a=b=−∞. In the study by Saber and Khorshidian [3], if independent random variables X and Y are continues, then(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.They worked on exponential distribution in the first paper on conditional stress-strength models.In this paper, the Kumaraswamy distribution is applied to conditional stress-strength models. The Kumaraswamy distribution is a continuous distribution whose values are similar to the beta distribution in the distance [0.1]. In this respect, it is very similar to the beta distribution. However, it has significant advantages in terms of performance, and the cumulative irreversible distribution of booklets has advantages.In the study by Kumaraswamy [4]; Kumaraswamy [5], hydrological data were shown, for example, daily rainfall and daily flow are not compatible with known and widely used distribution distributions such as normal, log-normal, and beta distributions. In addition, distributions such as Johnson and natural transformed polynomials have this problem too. Therefore, he defined a new probability density function called the sine probability density function. It seems that researchers for such data have considered this type of distribution (see Sundar and Subbiah [6]; Fletcher and Ponnambalam [7]; Seifi et al. [8]; Ponnambalam et al. [9]; and Ganji et al. (2006)).The Kumaraswamy (Kw) distribution has a probability density function (pdf) and a cumulative distribution function as follows:(3)fXx=αβxα−11−xαβ−1,FXx=1−1−xαβ,respectively, where x∈0,1 and α,β∈0,∞are two positive shape parameters. Some works on stress-strength models by using of this distribution have been listed as follows. Nadar et al. [10] studied classical and Bayesian estimation of R, and then Nadar and Kizilaslan [11] performed the same study but by using upper record values. Estimation of reliability of multicomponent stress-strength models (introduced by Bhattacharyya and Johnson (1974)) has been studied by Dey et al. [12]. Finally, the multicomponent stress-strength model based on progressively censored sample has been worked by Kohansal [13].The rest of the paper is organized as follows; we devote Section2 to study R|a,b in case of Kumaraswamy distribution. In a continuation of Section 2, the ML estimator of quantity R|a,b, its corresponding asymptotic distribution, and confidence interval are presented. Furthermore, two methods including Bayesian and bootstrap are applied to our recommended model. Finally, results are presented in Section 3.


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

In this section, quantity (2) is computed when distribution of components is Kumaraswamy.Theorem 1.
SupposeX∼Kuα,β1 and Y∼Kuα,β2 are independent random variables, then(4)R|a,b=β2β1+β2;a=b,β2β1+β21−bα1−aαβ1,a<b,1−β1β1+β21−aα1−bαβ2,a>b.Proof.
Use and substitute functionsfXx=αβ1xα−11−xαβ1−1, FXx=1−1−xαβ1, fYy=αβ2yα−11−yαβ2−1, and FYy=1−1−yαβ2 in equation (2). So, let a=b, then(5)R|a,b=1−FYa−∫a+∞FXyfYydy1−FXa1−FYb=1−FYa−∫a11−1−yαβ1αβ2yα−11−yαβ2−1dy1−aαβ1+β2=1−FYa−∫a1αβ2yα−11−yαβ2−1dy+∫a1αβ2yα−11−yαβ1+β2−1dy1−aαβ1+β2=1−FYa−∫a1fYydy+β2/β1+β2∫a1αβ1+β2yα−11−yαβ1+β2−1dy1−aαβ1+β2=β2/β1+β21−aαβ1+β21−aαβ1+β2=β2β1+β2.
And, ifa<b, then(6)R|a,b=1−FYb−∫b+∞FXyfYydy1−FXa1−FYb=1−FYb−∫b11−1−yαβ1αβ2yα−11−yαβ2−1dy1−aαβ11−bαβ2=β2β1+β21−bα1−aαβ1.
Eventually, ifa>b, we can calculate R|a,b like previous ones.(7)R|a,b=∫a+∞FYxfXxdx−FYb1−FXa1−FXa1−FYb=∫a11−1−xαβ2αβ1xα−11−xαβ1−1dx−FYb1−FXa1−aαβ11−bαβ2=1−aαβ11−bαβ2−β1/β1+β21−aαβ1+β21−aαβ11−bαβ2=1−β1β1+β21−aα1−bαβ2.
In continuation of this section, we find MLE ofR|a,b, and then asymptotic distribution of this estimator is found in order to construct its confidence interval.
LetX1,X2,…,Xn be a random sample of size n of Kuα,β1 and Y1,Y2,…,Ym be a random sample of size m of Kuα,β2 such that X and Y are independent. Then, the likelihood function is obtained.(8)Lα,β1,β2=αn+mβ1nβ2m∏i=1nxiα−1∏i=1n1−xiαβ1−1×∏j=1myjα−1∏j=1m1−yjαβ2−1.
Then, to facilitate the calculation, use the form of the likelihood function in [14].(9)lα,β1,β2=lnLα,β1,β2=n+mlnα+nlnβ1+mlnβ2+α−1lx+ly+β1−1lxx+β2−1lyy,where(10)lx=∑i=1nlnxi,ly=∑j=1mlnyj,lxx=∑i=1nln1−xiα,lyy=∑j=1mln1−yjα.
Therefore, the MLE parametersα,β1, and β2 are obtained by solving the following equations:(11)∂lα,β1,β2∂α=0,∂lα,β1,β2∂β1=0,∂lα,β1,β2∂β2=0.
Then,(12)β1^=−nlxx,β2^=−mlyy,and the MLE parameter α is obtained by solving the following nonlinear equation:(13)n+mα+lx+ly−β1−1^∑i=1nxiαlnxi1−xiα−β2−1^∑j=1myjαlnyj1−yjα=0.
The above equation is solved by numerically methods.
Therefore, the MLE ofR^|a,b becomes(14)R^|a,b=β2^β1^+β2^,a=b,β2^β1^+β2^1−bα^1−aα^β1^,a<b,1−β1^β1^+β2^1−aα^1−bα^β2^,a>b.


### 2.1. Asymptotic Distribution ofR^|a,b

In this section, we compute the asymptotic distributions ofθ^=α^,β1^,β2^ and R^|a,b. Hence, the Fisher information matrix of θ=α,β1,β2 denoted by Jθ=EI,θ is given as follows, where I=Ii,ji,j=1,2,3 is the observed information matrix, i.e.,(15)Iθ=−∂2lα,β1,β2∂α2∂2lα,β1,β2∂α∂β1∂2lα,β1,β2∂α∂β2∂2lα,β1,β2∂β1∂α∂2lα,β1,β2∂β12∂2lα,β1,β2∂β1∂β2∂2lα,β1,β2∂β2∂α∂2lα,β1,β2∂β2∂β1∂2lα,β1,β2∂β22=I11I12I13I21I22I23I31I32I33,and the elements of Iθ are as follows:(16)I11=n+mα2+β1−1∑i=1nxiαln2xi1−xiα2+β2−1∑j=1myjαln2yj1−yjα2,I12=I21=∑i=1nxiαlnxi1−xiα,I13=I31=∑j=1myjαlnyj1−yjαI22=nβ12,I33=mβ22.The elements of the Fisher information matrix are obtained by taking the expectations of the observed matrix. The following integrals can be helpful when finding the elements of the Fisher information matrix:(17)∫01tx−11−ty−1lntdt=Bx,yψx−ψx+y,∫01tx−11−ty−1ln2tdt=Bx,yψx−ψx+y2+ψ′x−ψ′x+y,where Bx,y is the beta function, ψx is the digamma function, and(18)ψ′x=∂∂xψx,J11=n+mα2+nβ1β1−1α2B2,β1−2ψ2−ψβ12ψ′2−ψ′β12+mβ2β2−1α2B2,β2−2ψ2−ψβ22ψ′2−ψ′β22,J12=J21=nβ1αB2,β1−1ψ2−ψβ1+1,J13=J31=mβ2αB2,β2−1ψ2−ψβ2+1,J22=nβ12,J23=J32=0,J33=mβ22,where(19)ψ1=−γ,ψ2=1−γ,γ=−Γ′1,ψ′2=π6−1,ψ′βi=∑n=0∞1βi+n2,i=1,2.So,Jθ=J11J12J13J21J22J23J31J32J33. As n⟶∞ and m⟶∞, then by using of multivariate CLT of θ^, we have θ^⟶N3θ,Σ where θ^=α^,β1^,β2^ and Σ is inverse of the Fisher information matrix:(20)Σ=1detJθJ22J33−J23J32J13J32−J12J33J12J23−J13J22J23J31−J21J33J11J33−J13J31J13J21−J11J23J21J32−J22J31J12J31−J11J32J11J22−J12J21=1detJθJ22J33−J12J33−J13J22−J21J33J11J33−J13J31J13J21−J22J31J12J31J11J22−J12J21,such that(21)detJθ=J11J22J33+J12J23J31+J13J21J32−J13J22J31−J12J21J33−J11J23J32=J11J22J33−J13J22J31−J12J21J33=nmβ12β22α2n+m+nβ1β1−1B2,β1−2ψ2−ψβ12ψ′2−ψ′β12+mβ2β2−1B2,β2−2ψ2−ψβ22ψ′2−ψ′β2−nm2β12α2β2B2,β2−1ψ2−ψβ2+12−mn2β22α2β1B2,β1−1ψ2−ψβ1+12.Now, we can use the Delta method to express the following theorem.Theorem 2.
Asn⟶∞ and m⟶∞, then(22)R^|a,b−R|a,bσ1⟶N0,1;a=b,R^|a,b−R|a,bσ2⟶N0,1;a<b,R^|a,b−R|a,bσ3⟶N0,1;a>b,where σ12,σ22, and σ32 are obtained as the following equations:(23)σ12=β12β22nmβ1+β24A−C−DmA−C−2nmβ12β22B3B4ψ3ψ4+nA−D,σ22=1A−C−Dρ12α2+ρ22β12A−C+ρ32β22A−D−2ρ1ρ2αβ13−2ρ1ρ3αβ23+2ρ2ρ3β13β23B3B4ψ3ψ4,σ32=1A−C−Dγ12α2−2γ1γ2αβ13−2γ1γ3αβ23+2γ2γ3β13β23B3B4ψ3ψ4+γ22β12nA−C+γ32β22mA−D,where(24)B1=B2,β1−2,B2=B2,β2−2,B3=B2,β2−1,B4=B2,β1−1,ψ1=ψ2−ψβ1,ψ2=ψ2−ψβ2,ψ3=ψ2−ψβ2+1,ψ4=ψ2−ψβ1+1,ψ1′=ψ′2−ψ′β1,ψ2′=ψ′2−ψ′β2,A=n+m+nβ1β1−1B1ψ12ψ1′+mβ2β2−1B2ψ22ψ2′,C=mβ24B32ψ32,D=nβ14B42ψ42,ρ1=β1β2β1+β21−bα1−aαβ1−1aα1−bαlna−bα1−aαlnb1−aα2,ρ2=β2β1+β21−bα1−aαβ1ln1−bα1−aα−1β1+β2,ρ3=β1β1+β221−bα1−aαβ1,γ1=β1β2β1+β21−aα1−bαβ2−1aα1−bαlna−bα1−aαlnb1−bα2,γ2=−β2β1+β221−aα1−bαβ2,γ3=β1β1+β21−aα1−bαβ21β1+β2−ln1−aα1−bα.Proof.
Let(25)gx1,x2,x3=x3x2+x3;a=b,x3x2+x31−bx11−ax1x2;a<b,1−x2x2+x31−ax11−bx1x3;a>b,then(26)∂g∂x1=0;a=b,x3x2x2+x31−bx11−ax1x2−1ax11−bx1lna−bx11−ax1lnb1−ax12;a<b,x3x2x2+x31−ax11−bx1x3−1ax11−bx1lna−bx11−ax1lnb1−bx12;a>b,∂g∂x2=−x3x2+x32;a=b,x3x2+x31−bx11−ax1x2ln1−bx11−ax1−1x2+x3;a<b,−x3x2+x321−ax11−bx1x3;a>b,∂g∂x3=x2x2+x32;a=b,x2x2+x321−bx11−ax1x2;a<b,x2x2+x31−ax11−bx1x31x2+x3−ln1−ax11−bx1;a>b,gx˙=∂gx1,x2,x3∂x1,∂gx1,x2,x3∂x2,∂gx1,x2,x3∂x3.
Sinceθ^−θ⟶N0,Σ, by Cramer’s theorem,(27)gθ^−gθ⟶N0,gθ˙ΣgθT˙,where gθ˙=∂R|a,b/∂θ=∂R|a,b/∂α,∂R|a,b/∂β1,∂R|a,b/∂β2. Now, σ12,σ22, and σ32 are computed of gθ˙ΣgθT˙ for case of a=b, a<b, and a>b, respectively.
Using the previous theorem, the confidence interval forR|a,b can be obtained.Theorem 3.
(1 −α) 100% confidence interval for R|a,b is equal to(28)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 above equationsσ^i2wherei=1,2,3 are similar to σi2wherei=1,2,3 in Theorem 2 with substitutions α^, β^1, and β^2 instead of α, β1, and β2.


### 2.2. Bayesian Estimation ofR|a,b

In this section, we use the Bayesian method to approximate the posterior distribution for the values of interest in the multicomponent stress strength for the Kumarswamy distribution based on the MCMC method.Therefore, supposeπθ=παπβ1πβ2 is the joint prior density for θ=α,β1,β2, such that α∼Γa1,b1,β1∼Γa2,b2, and β2∼Γa3,b3.In these priors,Γai,bi is the gamma distribution with mean and variance ai/bi and ai/bi2, respectively, such that ai and bi are known parameters and ai,bi>0. We assume that the prior distributions α,β1, and β2 are independent.(29)πα∝αa1−1e−αb1,πβ1∝β1a2−1e−β1b2,πβ2∝β2a3−1e−β2b3.Then,πθ,x,y=παπβ1πβ2Lα,β1,β2.(30)∝αa1+m+n−1e−αb1β1a2+n−1e−β1b2β2a3+m−1e−β2b3β2m×∏i=1nxiα−1∏i=1n1−xiαβ1−1∏i=1nyjα−1∏i=1n1−yjαβ2−1.Hence,(31)πα|β1,β2,x,y∝Γa1+m+n,b1Lα,β1,β2,πβ1|α,β2,x,y∝Γa2+n,b2∏i=1n1−xiαβ1−1,πβ2|α,β1,x,y∝Γa3+m,b3∏j=1m1−yjαβ2−1.Bayesian inference for parametersα,β1, and β2 can be performed using the Metropolis–Hastings algorithm (see Chib and Greenberg [15]) considering the conditional distributions as the target densities.


### 2.3. Bootstrap Confidence Intervals

Similar toSection 2.2, we study confidence interval based on the percentile bootstrap method for the case of common shape parameter. The algorithm is as follows:Step 1: For samplesx1,…,xn and y1,…,ym, calculate θ^=α^,β1^,β2^.Step 2: Use (α^,β1^) and (α^, β2^) to generate bootstrap samples x1∗,…,xn∗ and y1∗,…,ym∗, respectively, and then using the generated samples, calculate the bootstrap estimate of R say R∗ as follows:(32)R∗^=β2^β1^+β2^;a=b,β2^β1^+β2^1−bα^1−aα^β1^;a<b,1−β1^β1^+β2^1−aα^1−bα^β2^;a>b.Step 3: Repeat the previous step with time N bootstrap to generateR^1∗,…,R^N∗.Now,R^boot∗=1/N∑i=1NR^i∗ and the approximate 100 1−α % confidence interval of RG is given by (R^α/2∗,R^1−α/2∗) where R^γ∗ shows quantile of order γ for R^1∗,…,R^N∗.


## 2.1. Asymptotic Distribution ofR^|a,b

In this section, we compute the asymptotic distributions ofθ^=α^,β1^,β2^ and R^|a,b. Hence, the Fisher information matrix of θ=α,β1,β2 denoted by Jθ=EI,θ is given as follows, where I=Ii,ji,j=1,2,3 is the observed information matrix, i.e.,(15)Iθ=−∂2lα,β1,β2∂α2∂2lα,β1,β2∂α∂β1∂2lα,β1,β2∂α∂β2∂2lα,β1,β2∂β1∂α∂2lα,β1,β2∂β12∂2lα,β1,β2∂β1∂β2∂2lα,β1,β2∂β2∂α∂2lα,β1,β2∂β2∂β1∂2lα,β1,β2∂β22=I11I12I13I21I22I23I31I32I33,and the elements of Iθ are as follows:(16)I11=n+mα2+β1−1∑i=1nxiαln2xi1−xiα2+β2−1∑j=1myjαln2yj1−yjα2,I12=I21=∑i=1nxiαlnxi1−xiα,I13=I31=∑j=1myjαlnyj1−yjαI22=nβ12,I33=mβ22.The elements of the Fisher information matrix are obtained by taking the expectations of the observed matrix. The following integrals can be helpful when finding the elements of the Fisher information matrix:(17)∫01tx−11−ty−1lntdt=Bx,yψx−ψx+y,∫01tx−11−ty−1ln2tdt=Bx,yψx−ψx+y2+ψ′x−ψ′x+y,where Bx,y is the beta function, ψx is the digamma function, and(18)ψ′x=∂∂xψx,J11=n+mα2+nβ1β1−1α2B2,β1−2ψ2−ψβ12ψ′2−ψ′β12+mβ2β2−1α2B2,β2−2ψ2−ψβ22ψ′2−ψ′β22,J12=J21=nβ1αB2,β1−1ψ2−ψβ1+1,J13=J31=mβ2αB2,β2−1ψ2−ψβ2+1,J22=nβ12,J23=J32=0,J33=mβ22,where(19)ψ1=−γ,ψ2=1−γ,γ=−Γ′1,ψ′2=π6−1,ψ′βi=∑n=0∞1βi+n2,i=1,2.So,Jθ=J11J12J13J21J22J23J31J32J33. As n⟶∞ and m⟶∞, then by using of multivariate CLT of θ^, we have θ^⟶N3θ,Σ where θ^=α^,β1^,β2^ and Σ is inverse of the Fisher information matrix:(20)Σ=1detJθJ22J33−J23J32J13J32−J12J33J12J23−J13J22J23J31−J21J33J11J33−J13J31J13J21−J11J23J21J32−J22J31J12J31−J11J32J11J22−J12J21=1detJθJ22J33−J12J33−J13J22−J21J33J11J33−J13J31J13J21−J22J31J12J31J11J22−J12J21,such that(21)detJθ=J11J22J33+J12J23J31+J13J21J32−J13J22J31−J12J21J33−J11J23J32=J11J22J33−J13J22J31−J12J21J33=nmβ12β22α2n+m+nβ1β1−1B2,β1−2ψ2−ψβ12ψ′2−ψ′β12+mβ2β2−1B2,β2−2ψ2−ψβ22ψ′2−ψ′β2−nm2β12α2β2B2,β2−1ψ2−ψβ2+12−mn2β22α2β1B2,β1−1ψ2−ψβ1+12.Now, we can use the Delta method to express the following theorem.Theorem 2.
Asn⟶∞ and m⟶∞, then(22)R^|a,b−R|a,bσ1⟶N0,1;a=b,R^|a,b−R|a,bσ2⟶N0,1;a<b,R^|a,b−R|a,bσ3⟶N0,1;a>b,where σ12,σ22, and σ32 are obtained as the following equations:(23)σ12=β12β22nmβ1+β24A−C−DmA−C−2nmβ12β22B3B4ψ3ψ4+nA−D,σ22=1A−C−Dρ12α2+ρ22β12A−C+ρ32β22A−D−2ρ1ρ2αβ13−2ρ1ρ3αβ23+2ρ2ρ3β13β23B3B4ψ3ψ4,σ32=1A−C−Dγ12α2−2γ1γ2αβ13−2γ1γ3αβ23+2γ2γ3β13β23B3B4ψ3ψ4+γ22β12nA−C+γ32β22mA−D,where(24)B1=B2,β1−2,B2=B2,β2−2,B3=B2,β2−1,B4=B2,β1−1,ψ1=ψ2−ψβ1,ψ2=ψ2−ψβ2,ψ3=ψ2−ψβ2+1,ψ4=ψ2−ψβ1+1,ψ1′=ψ′2−ψ′β1,ψ2′=ψ′2−ψ′β2,A=n+m+nβ1β1−1B1ψ12ψ1′+mβ2β2−1B2ψ22ψ2′,C=mβ24B32ψ32,D=nβ14B42ψ42,ρ1=β1β2β1+β21−bα1−aαβ1−1aα1−bαlna−bα1−aαlnb1−aα2,ρ2=β2β1+β21−bα1−aαβ1ln1−bα1−aα−1β1+β2,ρ3=β1β1+β221−bα1−aαβ1,γ1=β1β2β1+β21−aα1−bαβ2−1aα1−bαlna−bα1−aαlnb1−bα2,γ2=−β2β1+β221−aα1−bαβ2,γ3=β1β1+β21−aα1−bαβ21β1+β2−ln1−aα1−bα.Proof.
Let(25)gx1,x2,x3=x3x2+x3;a=b,x3x2+x31−bx11−ax1x2;a<b,1−x2x2+x31−ax11−bx1x3;a>b,then(26)∂g∂x1=0;a=b,x3x2x2+x31−bx11−ax1x2−1ax11−bx1lna−bx11−ax1lnb1−ax12;a<b,x3x2x2+x31−ax11−bx1x3−1ax11−bx1lna−bx11−ax1lnb1−bx12;a>b,∂g∂x2=−x3x2+x32;a=b,x3x2+x31−bx11−ax1x2ln1−bx11−ax1−1x2+x3;a<b,−x3x2+x321−ax11−bx1x3;a>b,∂g∂x3=x2x2+x32;a=b,x2x2+x321−bx11−ax1x2;a<b,x2x2+x31−ax11−bx1x31x2+x3−ln1−ax11−bx1;a>b,gx˙=∂gx1,x2,x3∂x1,∂gx1,x2,x3∂x2,∂gx1,x2,x3∂x3.
Sinceθ^−θ⟶N0,Σ, by Cramer’s theorem,(27)gθ^−gθ⟶N0,gθ˙ΣgθT˙,where gθ˙=∂R|a,b/∂θ=∂R|a,b/∂α,∂R|a,b/∂β1,∂R|a,b/∂β2. Now, σ12,σ22, and σ32 are computed of gθ˙ΣgθT˙ for case of a=b, a<b, and a>b, respectively.
Using the previous theorem, the confidence interval forR|a,b can be obtained.Theorem 3.
(1 −α) 100% confidence interval for R|a,b is equal to(28)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 above equationsσ^i2wherei=1,2,3 are similar to σi2wherei=1,2,3 in Theorem 2 with substitutions α^, β^1, and β^2 instead of α, β1, and β2.


## 2.2. Bayesian Estimation ofR|a,b

In this section, we use the Bayesian method to approximate the posterior distribution for the values of interest in the multicomponent stress strength for the Kumarswamy distribution based on the MCMC method.Therefore, supposeπθ=παπβ1πβ2 is the joint prior density for θ=α,β1,β2, such that α∼Γa1,b1,β1∼Γa2,b2, and β2∼Γa3,b3.In these priors,Γai,bi is the gamma distribution with mean and variance ai/bi and ai/bi2, respectively, such that ai and bi are known parameters and ai,bi>0. We assume that the prior distributions α,β1, and β2 are independent.(29)πα∝αa1−1e−αb1,πβ1∝β1a2−1e−β1b2,πβ2∝β2a3−1e−β2b3.Then,πθ,x,y=παπβ1πβ2Lα,β1,β2.(30)∝αa1+m+n−1e−αb1β1a2+n−1e−β1b2β2a3+m−1e−β2b3β2m×∏i=1nxiα−1∏i=1n1−xiαβ1−1∏i=1nyjα−1∏i=1n1−yjαβ2−1.Hence,(31)πα|β1,β2,x,y∝Γa1+m+n,b1Lα,β1,β2,πβ1|α,β2,x,y∝Γa2+n,b2∏i=1n1−xiαβ1−1,πβ2|α,β1,x,y∝Γa3+m,b3∏j=1m1−yjαβ2−1.Bayesian inference for parametersα,β1, and β2 can be performed using the Metropolis–Hastings algorithm (see Chib and Greenberg [15]) considering the conditional distributions as the target densities.


## 2.3. Bootstrap Confidence Intervals

Similar toSection 2.2, we study confidence interval based on the percentile bootstrap method for the case of common shape parameter. The algorithm is as follows:Step 1: For samplesx1,…,xn and y1,…,ym, calculate θ^=α^,β1^,β2^.Step 2: Use (α^,β1^) and (α^, β2^) to generate bootstrap samples x1∗,…,xn∗ and y1∗,…,ym∗, respectively, and then using the generated samples, calculate the bootstrap estimate of R say R∗ as follows:(32)R∗^=β2^β1^+β2^;a=b,β2^β1^+β2^1−bα^1−aα^β1^;a<b,1−β1^β1^+β2^1−aα^1−bα^β2^;a>b.Step 3: Repeat the previous step with time N bootstrap to generateR^1∗,…,R^N∗.Now,R^boot∗=1/N∑i=1NR^i∗ and the approximate 100 1−α % confidence interval of RG is given by (R^α/2∗,R^1−α/2∗) where R^γ∗ shows quantile of order γ for R^1∗,…,R^N∗.


## 3. Simulation Study

In this section, we want to simulate the results for different sample sizes to better illustrate the methods presented in this paper (classical estimation, Bayesian estimation, and bootstrap). Codes of this section have been provided in Appendix. The two known parametersa=0.1 and b=0.2 are used in our study. For other parameters in the model, we consider two cases α=5,β1=3.5, and β2=3.25 which lead to R|a,b=0.480959271 and α=1.25,β1=7, and β2=4.2 that their corresponding R|a,b is 0.20582159557. In this simulation, sample sizes n,m=5,5,10,10,15,15,20,20,35,35,50,50,70,70,100,100, 5,15,25,15,20,35,30,60,100,55 are used. All results are the mean of 5000 iteration. Estimation has been accomplished by three methods such as MLE, Bayesian, and bootstrap. For comparing these methods, we compute bias and MSE. In addition, coverage probabilities (CPs) and length of confidence interval have been computed. Our findings are represented in Tables 1–6. As these tables demonstrate, Bayesian estimation is the best method among three used methods with respect to criteria MSE, CP, and length of confidence interval. However, ML estimation has less bias than two other studied methods.Table 1
Classical estimation:α=5,β1=3.5,β2=3.25, a=0.1,b=0.2, and R|a,b=0.480959271 for equal sample sizes n and m.
(m, n)R^|a,bBias (R^|a,b)MSE (R^|a,b)C.ICPLength(5, 5)0.48420.00330.0320(0.2225, 0.7459)0.80000.5235(10, 10)0.51350.03070.0097(0.3040, 0.7228)0.95000.4188(15, 15)0.48330.00050.0082(0.3115, 0.6549)0.94870.3433(20, 20)0.4754−0.00680.0050(0.3248, 0.6259)0.96000.3012(35, 35)0.4778−0.00320.0028(0.3625,0.5930)0.96300.2305(50, 50)0.48090.00000.0030(0.3846, 0.5772)0.93000.1927(70, 70)0.485890.004930.00178(0.4038, 0.5679)0.930000.16402(100, 100)0.4786−0.00230.0010(0.4098, 0.5474)0.97000.1376Table 2
Bayesian and bootstrap estimations:α=5,β1=3.5,β2=3.25, a=0.1,b=0.2, and R|a,b=0.480959271 for equal sample sizes n and m.
(n, m)R^bayes|a,bBias (R^bayes|a,b)MSE (R^bayes|a,b)C.IbayesCPbayesLengthbayesC.IbootCPbootLengthboot(5, 5)0.49290.01200.003(0.4188, 0.5497)10.1310(0.1228, 0.8455)10.7227(10, 10)0.49670.01570.0003(0.4363, 0.5498)10.1135(0.2709, 0.7559)10.4850(15, 15)0.49750.01660.0003(0.4412, 0.5498)10.1086(0.2905, 0.6759)10.3853(20, 20)0.49850.01750.0003(0.4446, 0.5498)10.1052(0.3092, 0.6415)10.3323(35, 35)0.49890.01800.0003(0.4469, 0.5498)10.1030(0.3538, 0.6017)10.2479(50, 50)0.49900.01810.0003(0.4475, 0.5498)10.1023(0.3772, 0.5846)10.2074(70, 70)0.494670.013710.00088(0.44507, 0.54427)10.09920(0.40004, 0.5717)10.17169(100, 100)0.01900.50000.0004(0.4499, 0.5499)10.1000(0.4138, 0.5434)10.1297Table 3
Classical estimation:α=5,β1=3.5,β2=3.25, a=0.1,b=0.2, and R|a,b=0.480959271 for different sample sizes n and m.
(n, m)R^|a,bBias (R^|a,b)MSE (R^|a,b)C.ICPLength(5, 15)0.4040−0.11340.0413(0.2015, 0.6065)0.50000.4050(20, 15)0.4787−0.00220.0065(0.3243, 0.6330)0.93150.3087(20, 35)0.50070.02170.0047(0.3661, 0.6352)0.89740.2691(30, 60)0.4755−0.00550.0034(0.3680, 0.5829)0.89000.2149(100, 55)0.48200.00100.0015(0.4004, 0.5634)0.96000.1630Table 4
Bayesian and bootstrap Estimations:α=5,β1=3.5,β2=3.25, a=0.1,b=0.2, and R|a,b=0.480959271 for different sample sizes n and m.
(n, m)R^bayes|a,bBias (R^bayes|a,b)MSE (R^bayes|a,b)C.IbayesCPbayesLengthbayesC.IbootCPbootLengthboot(5, 15)0.49690.01590.0003(0.4361, 0.5498)1.00000.1137(0.1341, 0.6738)1.00000.5398(20, 15)0.49810.01710.0003(0.4437, 0.54981)1.0000.1061(0.3076, 0.6497)1.00000.3421(20, 35)0.49860.01770.0003(0.4453, 0.5498)1.00000.1045(0.3539, 0.6474)1.00000.2935(30, 60)0.49900.01810.0003(0.4473, 0.5498)1.00000.1025(0.3598, 0.5911)1.00000.2313(100, 55)0.50000.01900.0004(0.4499, 0.5499)1.00000.1000(0.3955, 0.5683)1.00000.1728Table 5
Classical estimation:α=1.25,β1=7,β2=4.2, a=0.1,b=0.2, and R|a,b=0.20582159557.
(n, m)R^|a,bBias (R^|a,b)MSE (R^|a,b)C.ICPLength(5, 5)0.1962−0.01650.0138(0.0169, 0.4094)0.78950.4265(25, 25)0.21480.00900.0036(0.1055, 0.3239)0.93000.2184(30, 30)0.2009−0.00490.0028(0.1043, 0.2975)0.88000.1932(40, 40)0.2024−0.00340.0017(0.1180, 0.2868)0.95000.1689Table 6
Bayesian and bootstrap estimations:α=1.25,β1=7,β2=4.2, a=0.1,b=0.2, and R|a,b=0.20582159557.
(n, m)R^bayes|a,bBias (R^bayes|a,b)MSE (R^bayes|a,b)C.IbayesCPbayesLengthbayesC.IbootCPbootLengthboot(5, 5)0.34620.14040.0233(0.1438, 0.5486)0.91000.3048(-0.0571, 0.4496)1.00000.5069(25, 25)0.30550.09970.0119(0.1337, 0.4772)1.00000.2435(0.0974, 0.3320)1.00000.2346(30, 30)0.30350.09770.0115(0.1297, 0.4773)1.00000.2477(0.0933, 0.3085)1.00000.2152(40, 40)0.30010.09420.0110(0.1231, 0.4769)1.00000.2538(0.1093, 0.2955)1.00000.1863


## 4. Discussion and Conclusion

The conditional stress-strength model as an interesting extension of the stress-strength model in reliability was studied for Kumaraswamy distribution. Three estimation methods are applied for statistical inference in this model.The stress-strength model has been investigated for many distributions such as generalized logistic distribution, generalized failure rate distribution, Rayleigh, and half-normal distribution (see, for instance, Rasekhi et al. [16] and Alamri et al. [17]). They may be applied for the conditional stress-strength model, too. We are going to study this model for generalized logistic distribution in the next step.The generalized stress-strength parameter (RG) introduced by Saber et al. [18] is given by(33)RG=PY<X<Z.This quantity is related to a system with three components whileR is defined for systems which have two components.For future studies, a possible extension of (33) is recommended in the following:(34)RG|a,b,c=PY<X<Z|X>a,Y>b,Z>c.



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*Source: 1087871-2021-07-31.xml*