EstimationofReliabilityforaTwoComponentSurvival Stress ...downloads.hindawi.com/archive/2011/721962.pdfmodel based on exponential distribution. Dan and Krausz [7] have obtained inference

Hindawi Publishing CorporationInternational Journal of Quality, Statistics, and ReliabilityVolume 2011, Article ID 721962, 8 pagesdoi:10.1155/2011/721962

Research Article

Estimation of Reliability for a Two Component SurvivalStress-Strength Model

S. B. Munoli1 and Rohit R. Mutkekar2

1 Department of Studies in Statistics, Karnatak University, Dharwad, Karnataka 580-003, India2 KLS-Institute of Management Education and Research, Belgaum, Karnataka 590-011, India

Correspondence should be addressed to S. B. Munoli, [email protected]

Received 14 December 2010; Revised 5 May 2011; Accepted 17 June 2011

Academic Editor: Mohammad Modarres

Copyright © 2011 S. B. Munoli and R. R. Mutkekar. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

The reliability function for a parallel system of two identical components is derived from a stress-strength model, where failure ofone component increases the stress on the surviving component of the system. The Maximum Likelihood Estimators of parametersand their asymptotic distribution are obtained. Further the Maximum Likelihood Estimator and Bayes Estimator of reliabilityfunction are obtained using the data from a life-testing experiment. Computation of estimators is illustrated through simulationstudy.

1. Introduction

Several authors have considered estimation of system relia-bility based on stress-strength models. Here are a few ref-erences of contributions towards these models. Church andHarris [1] considered estimation of reliability from stress-strength relationship. Downton [2] considered the case ofestimation of reliability for a stress-strength model undernormal distribution. Wani and Kabe [3] have consideredthe problem of estimation of system reliability where lifetime of each component has gamma distribution. Con-stantine et al. [4] considered estimation of stress-strengthrelationship under the assumption of stress-strength randomvariables following gamma distribution with known shapeparameters. Bhattacharya and Johnson [5] have consideredestimation of reliability in a multicomponent stress-strengthmodel. Kunchur and Munoli [6] have considered estimationof reliability for a multicomponent survival stress-strengthmodel based on exponential distribution. Dan and Krausz[7] have obtained inference for a multistep stress-strengthmodel of parallel system. Kunchur and Munoli [8] haveconsidered estimation of reliability in Freund’s model fora two component system. Hanagal [9] has consideredthe problem of estimation of system reliability in a twocomponent stress-strength model with cases on distribution

of stress as exponential and gamma. Kundu and Gupta[10] have obtained inference of stress-strength relationshipfor generalized exponential scale family of distribution.Bhattacharya [11] has proposed Bayesian approach to life-testing and reliability estimation. Draper and Guttman [12]have obtained the Bayes estimator of reliability in a multi-component stress-strength model.

In the present study, we are considering a system of twocomponents. The component survives as long as the stress onit is smaller than its strength. The system survives if at leastone component functions (parallel system). Here the stressand strength associated with the components of the systemare random variables. To carry out the inference, we assumecertain probability distribution for these random variables.Let the strength of the two components be X1 and X2,where X1, X2 are independently and identically distributedgamma random variables with shape parameter γ and scaleparameter μ. Let Y1, Y2 be the stress on the two components,respectively. Initially Y1, Y2 are independently and identicallydistributed as exponential random variables with parameterθ. Statistically dependent failures are typical for modernsystems, which involve complicated interactions amongcomponent parts, and it is reasonable to assume that thefailure of one component does change the stress on survivingcomponents. The strength on the other hand is dependent

2 International Journal of Quality, Statistics, and Reliability

on various inbuilt qualities of the component such as thetechnology by which it has been manufactured, the rawmaterials used, and so forth. Failure of one componentof the system need not change these inbuilt qualities ofthe surviving component of the system. Hence, changein the stress of the surviving component is assumed onfailure of other components of the system. That is, in atwo component parallel system, the distribution of stress ofthe surviving component changes upon the failure of theother component. Then the exponential distribution withparameter αθ(α > 0) follows. The system fails whenever boththe components of the system fail.

A block diagram for the proposed model is given inFigure 1.

One can quote several examples for two componentparallel systems such as pair of kidneys, eyes, hands, andlegs, pair of elevators, and pair of engines in an aircraft. Toillustrate the function of the proposed model let us considerthe example of a pair of kidneys. Here the function of thekidneys is to purify the blood and thus help to maintainthe body in healthy condition. Here the pair of kidneysperform the same function as per their natural built-upmechanism (strength). Failure of one kidney increases thepurification work load on the surviving kidney (stress).Here the surviving kidney should carry out the purificationfunction the same as it was when both the kidneys werefunctioning. Taking this scenario under consideration, weconsider that failure of one component of the systemchanges only the stress and not the strength of the survivingcomponent.

The reliability function is derived in Section 2. Life-testing experiment is explained in Section 3. The MaximumLikelihood Estimators (MLEs) are obtained in the samesection. Section 4 deals with Bayes estimation of reliabilityfunction. Computations of estimators of reliability function(MLE and Bayes) along with the findings of the study arediscussed in Section 5. Some results that support the findingsof this research paper are proved in the appendix.

2. Reliability Function

In order to find the reliability function of the modeldiscussed in Section 1, let us consider that U is the minimum(Y1,Y2) and V is the maximum (Y1,Y2) and let W = V −U .

Here U follows exponential distribution with parameter(2θ) and W follows exponential distribution with parameter(αθ).

The reliability function of this system is given by

R = P[Max(X1,X2) > Max(Y1,Y2)]

= 1 +2 · μγΓγ

·⎡⎣ 2

(α− 2)

∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(αθ + 2μ

)(i+γ)

− α(α− 2)

·∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(2θ + 2μ

)(i+γ)

⎤⎦.

(1)

The details of these derivations are given in Lemma A.1.

3. Life-Testing Experiment

In order to obtain the estimators of R, suppose that “n”systems whose life distribution is characterized by thereliability function derived in Section 2 are put on life-testing experiment. Here,X1i,X2i(i = 1, 2 · · ·n) are observedand X1i, X2i are independently and identically distributedgamma random variables with shape parameter γ and scaleparameter μ. Also, the data of stress Uj = Min(Y1 j ,Y2 j)and Vj = Max(Y1 j ,Y2 j), ( j = 1, 2 · · ·m) are obtainedseparately from a simulation of conditions of the operatingenvironment. Uj , Wj ( j = 1, 2, 3 · · ·m) are exponentialrandom variables with parameters 2θ and αθ, respectively.

Now, the joint probability density function of therandom variables X1i, X2i(i = 1, 2 · · ·n), Uj , Wj ( j =1, 2 · · ·m) is given by

L =(μγ

Γγ

)2n

·⎧⎨⎩

n∏

i=1

(x1i)γ−1

⎫⎬⎭ ·

⎧⎨⎩

n∏

i=1

(x2i)γ−1

⎫⎬⎭

· e−μ(x′1+x′2) · (2θ)m · (αθ)m · e−2θu′ · e−αθw′ ,(2)

where

x′1 =n∑

i=1

x1i, x′2 =n∑

i=1

x2i, u′ =m∑

j=1

uj , w′ =m∑

j=1

wj.

(3)

The MLEs (the estimators that maximize the likelihoodfunction) of parameters γ, μ, θ, α are given in the followingexpressions, respectively,

γ̂ = n

2n[A]− [∑ni=1 log(x1i) +

∑ni=1 log(x2i)

] , (4)

μ̂ = 2n2

(x′1 + x′2)[2n{A} − {∑n

i=1 log(x1i) +∑n

i=1 log(x2i)}] ,

(5)

θ̂ = m

2u′, (6)

α̂ = 2u′

w′, (7)

where A denote log(x′1 + x′2)− log(2n)Using the invariance property of MLEs, the MLE of

reliability function R̂ is obtained by substituting the MLEsof parameters γ, μ, θ, and α in expression (1), that is,

R̂ = 1 +2 · μ̂γ̂Γγ̂

·⎡⎢⎣ 2

(α̂− 2)

∞∑

i=γ̂

(μ̂)ii!· Γ

(i + γ̂

)(α̂θ̂ + 2μ̂

)(i+γ̂)

− α̂(α̂− 2)

·∞∑

i=γ̂

(μ̂)ii!· Γ

(i + γ̂

)(

2θ̂ + 2μ̂)(i+γ̂)

⎤⎥⎦.

(8)

International Journal of Quality, Statistics, and Reliability 3

Stress Y1

Stress Y2

Component A(strength X1)

Component B(strength X2)

IfY1 > X1

Y2 > X2

A survives

False

A fails

TrueB fails

B survives

Component Awith strength X1

F

SStressY1 + Y2 Component B

with strength X2

S

S

F System failure S System survival

If

False

True

StressY1 + Y2

Figure 1

The details of these findings are given in the form ofLemma A.2.

From the asymptotic properties of MLEs under regu-larity conditions and multivariate central limit theorem, we

have [√n(γ̂ − γ),

√n(μ̂ − μ),

√m(θ̂ − θ),

√m(α̂ − α)] →

N4[0,A(γ,μ, θ,α)], where A(γ,μ, θ,α) are the correspond-ing elements of the inverse of Fisher Information Matrix“I(γ,μ, θ,α )” as

I(γ,μ, θ,α

) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

n(2γ + 1

)

2γ2

−2nμ

0 0

−2nμ

2nμ2

0 0

0 02mθ2

m

αθ

0 0m

αθ

m

α2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(9)

and its inverse is given by

I−1

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2n[(

2γ + 1)/γ2 − 4

] 2μn[(

2γ + 1)/γ2 − 4

] 0 0

2μn[(

2γ + 1)/γ2 − 4

] μ2(2γ + 1

)

n[(

2γ + 1)/γ2 − 4

] 0 0

0 0θ2

m

−αθm

0 0−αθm

2α2

m

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(10)

4. Bayes Estimation of Reliability Function

In order to obtain Bayes estimator of reliability function R̂B,assume that parameters γ and α are known (i. e., γ = γ0 and

α = α0) and consider the prior distribution of parameters μand θ [13] as

g(μ) = 1

Γp· e−μ · μp−1, p ≥ 0, 0 < μ <∞,

g(θ) = 1Γq· e−θ · θq−1, q ≥ 0, 0 < θ <∞.

(11)

The Bayes estimator of reliability function is obtained asthe posterior expectation of R and is given by

R̂B =∫∫∞

0R · f2

(μ, θ | x1i, x2i,uj ,wj

)dμdθ

=∫∫∞

0

⎡⎣1 +

2 · μγ0

Γγ0

·⎡⎣ 2

(α0 − 2)

∞∑

i=γ0

(μ)ii!· Γ

(i + γ0

)(α0θ + 2μ

)(i+γ0)

− α0

(α0 − 2)·∞∑

i=γ0

(μ)ii!· Γ

(i + γ0

)(2θ + 2μ

)(i+γ0)

⎤⎦⎤⎦

· f2(μ, θ | x1i, x2i,uj ,wj

)· dμdθ

= 1 + I1 − I2,(12)

where

I1 =∫∫∞

0

4 · μγ0

(α0 − 2) · Γγ0·∞∑

i=γ0

(μ)ii!· Γ

(i + γ0

)(α0θ + 2μ

)(i+γ0)

× f2(μ, θ | x1i, x2i,uj ,wj

)· dμdθ


= 4.μγ0

(α0 − 2) · Γγ0 · Γ(2nγ0 + p

) · Γ(2m + q)

·∞∑

i=γ0

Γ(i + γ0

)

i!·∫∫∞

0A1dμdθ

(13)

with

A1 =⎛⎝e−μ(x1

′+x2′+1) · μ2nγ0+p+i−1 · e−θ(2u′+α0w′+1)

(α0θ + 2μ

)(i+γ0)

⎞⎠

·(θ(2m+q−1) · (x1

′ + x2′ + 1)(2nγ0+p)

·(2u′ + α0w′ + 1)(2m+q)

),

I2 =∫∫∞

0

2 · α0 · μγ0

(α0 − 2) · Γγ0·∞∑

i=γ0

(μ)ii!· Γ

(i + γ0

)(2θ + 2μ

)(i+γ0)

× f2(μ, θ | x1i, x2i,uj ,wj

)· dμdθ

= 2 · α0 · μγ0

(α0 − 2) · Γγ0 · Γ(2nγ0 + p

) · Γ(2m + q)

·∞∑

i=γ0

Γ(i + γ0

)

i!·∫∫∞

0A2dμdθ

(14)

with

A2 =⎛⎝e−μ(x1

′+x2′+1) · μ2nγ0+p+i−1 · e−θ(2u′+α0w′+1)

(2θ + 2μ

)(i+γ0)

⎞⎠

·(θ(2m+q−1) · (x1

′ + x2′ + 1)(2nγ0+p)

·(2u′ + α0w′ + 1)(2m+q)

).

(15)

Substituting values of expressions (13), and (14) in expres-sion (12) we obtain the Bayes estimator of reliability fun-ction R̂B.

The details of these findings are given in the form ofLemma A.3.

5. Computation of Estimators

For the ith system, the random variables x1i, x2i (with respectto strength) and random variables ui,wi (with respect tostress) are generated independently as follows.

Step 1. Initialize j1 = 1, x1i = 0.0, x2i = 0.0, n = n0,γ = γ0 for the 1st and 2nd components of the system.Uniform random numbers U1[ j1], V1[ j1] are generated fromU(0, 1). Further for a given value of = μ0, exponentialrandom variables U2[ j1] = (−1/μ0) · ln(1 − U1[ j1]),V2[ j1] = (−1/μ0) · ln(1 − V1[ j1]) are obtained for the1st and the 2nd components of the ith system, respectively.Now, j1 is incremented by 1 and the process of generating

the exponential random variables is repeated for both thecomponents of the ith system. This repetition process iscontinued until j1 ≤ γ0 and the subsequent exponentialrandom variable values generated are noted for both thecomponents of the ith system. Here x1i = x1i + U2[ j1] andx2i = x2i + V2[ j1] for j1 = 1, 2 · · · γ0. Here x1i, x2i denotegamma random variables with shape parameter γ = γ0 andscale parameter μ = μ0. We also compute the values ln(x1i)and ln(x2i).

Step 2. The whole procedure in Step 1 is repeated for n = n0

number of systems, and the statistics x1′ = ∑n

i=1 x1i, x2′ =∑n

i=1 x2i,∑n

i=1 log(x1i),∑n

i=1 log(x2i) are computed.

Step 3. Initialize i = 1, ui = 0.0, wi = 0.0, m = m0. Uniformrandom numbers U3[i], U4[i] are generated from U(0, 1).Further for a given value of θ = θ0, α = α0 exponentialrandom variables ui = (−1/θ0)· ln(1−U3[i]), wi = (−1/α0 ·θ0) · ln(1−U4[i]) are obtained. The value of i is incrementedby 1, and the above process of generated exponential randomvariables is repeated. This repetition process is continueduntil i ≤ m and the subsequent exponential random variablevalues generated ui, wi for i = 1, 2 · · ·m are computed. Thestatistics u′ =∑m

i=1 ui, w′ =∑m

i=1 wi are computed.

Step 4. With the help of the statistics x1′, x2

′,∑n

i=1 log(x1i),∑ni=1 log(x2i), u′ and w′, the MLEs of parameters γ, μ, θ, α of

the model are obtained. Using these MLEs in the expressionof reliability function, the MLE of reliability function isobtained.

For the parameter values γ = γ0, μ = μ0, θ = θ0, andα = α0 the value of reliability function is also obtained.

The Bayes estimator of reliability function is obtainedusing the simulated values of x1

′, x2′,∑n

i=1 log(x1i),∑ni=1 log(x2i), u′ and w′ for given values of p and q.

Tables 1, 2, and 3 give the results of the above simulationexperiment for different values of n and m.

6. Conclusion

The reliability function for the proposed model is evaluatedin terms of stress-strength relationship rather than consider-ing the time factor, as it is realistic to observe reliability ofa system functioning under the influence of external factors(stress) as compared to longevity of working time associatedwith the system.

Though the expression for reliability function involvessum of infinite series, it is observed that for given values ofthe parameters the value of the reliability function stabilizesat the 15th value (i = 15) of the running variable involved inthe sums.

Stress and strength associated with the components ofthe system possess different physical properties; therebydata on stress and strength in the life-testing experiment isobserved separately based on their corresponding operativeenvironments.

The MLEs are sufficient, efficient and also maximizes thelikelihood of the joint distribution function. Further, using


Table 1: γ = 3, μ = 1.0, θ = 0.5, n = 5, m = 5, p = 10, and q = 10.

α R x′1 x′2∑n

i=1 log(x2i)∑n

i=1 log(x2i) u′ w′ R̂ R̂B Bias (MLE) Bias (Bayes)

0.5 0.476 13.01744 13.69945 3.94646 4.04823 4.88118 8.93451 0.619 0.562 2.05 × 10−2 7.40 × 10−3

1.0 0.676 15.00853 21.81265 4.64973 7.07659 3.29156 7.66388 0.798 0.744 1.49 × 10−2 4.62 × 10−3

2.0 0.774 16.13645 15.41186 5.42399 4.94180 5.01072 3.71295 0.866 0.792 8.46 × 10−3 3.24 × 10−4


α R x′1 x′2∑n

i=1 log(x2i)∑n


0.5 0.476 12.88996 17.43603 4.119318 5.50226 2.30250 20.4892 0.574 0.556 9.61 × 10−3 6.40 × 10−3

1.0 0.676 14.82036 16.22916 4.578078 5.22255 3.76145 10.4876 0.759 0.731 6.89 × 10−3 3.03 × 10−3

2.0 0.774 10.17496 16.28889 2.750896 5.10211 4.32817 4.35386 0.822 0.787 2.31 × 10−3 1.69 × 10−4


α R x′1 x′2∑n

i=1 log(x2i)∑n


0.5 0.476 23.09795 17.29346 6.941847 5.754879 4.19987 11.8371 0.512 0.494 1.29 × 10−3 3.24 × 10−4

1.0 0.676 12.50140 34.21716 2.203496 10.363429 1.61698 14.5710 0.747 0.724 5.04 × 10−3 2.30 × 10−3

2.0 0.774 20.37072 25.06768 7.069134 7.447618 4.01297 6.64931 0.807 0.780 1.09 × 10−3 3.60 × 10−5

the invariance property of MLE, it is easy to obtain the MLEof reliability function.

Bayes estimator is based on the prior informationobtained through certain pilot study that helps in synthe-sizing the information to be generated for the system underfunction. Hence, Bayes estimator of reliability function isobtained by considering certain prior information for theparameters. But the process of obtaining Bayes estimatorof reliability function is quite tedious as it involves lengthynumerical calculations.

From Tables 1, 2, and 3, it is clear that for greatervalues of “m” and “n” (large sample size) both MLE andBayes estimators perform better. In the majority of the casesboth the estimators overestimate the true value of reliabilityfunction “R.” Here we observe that Bayes estimator is a betterestimator in terms of bias for the given data set.

Appendix

Lemma A.1. The reliability function given in expression (1) isderived as follows. The reliability function for the system understudy is given by

R = P[Max(X1, X2) > Max(Y1, Y2)]

= P[Max(X1, X2) > V]

= P[U + W < Max(X1, X2)]

= P[U < Max(X1, X2)− W]

= P[U < Z −W], where Z = Max(X1, X2)

=∫∞

0

∫ z

0

[1− e−2θ(z−w)

]· fW (w) · fZ(z)dwdz

=∫∞

0

∫ z

0

[1− e−2θ(z−w)

]· αθ · e−αθw · fZ(z)dwdz

=∫∞

0

∫ z

0αθ · e−αθw · fZ(z)dw dz −

∫∞0

∫ z

0e−2θ(z−w) · αθ

· e−αθw · fZ(z) dwdz

=∫∞

0

[1− e−αθz

]· fZ(z)dz − αθ

∫∞0

∫ z

0e−2θ(z−w)−αθw

· fZ(z)dwdz

= 1−∫∞

0e−αθz · fZ(z) dz − αθ

∫∞0

∫ z

0e−2θ(z−w)−αθw

· fZ(z)dwdz.

(A.1)

As Z = Max(X1, X2), its distribution is given by

fZ(z) = 2 · [F(z)] · f (z). (A.2)

Here F(z) represents the distribution function and f (z)represents probability density function of random variable Z,and as Z follows gamma distribution with shape parameter γand scale parameter μ, one has

fZ(z) = 2 ·[∫ z

0

μγ

Γγ· e−μx · xγ−1dx

]· μγ

Γγ· e−μz · zγ−1.

(A.3)

Using the relationship between incomplete gamma distri-bution and Poisson sum, one has

∫ z

0

μγ

Γγ· e−μx · xγ−1dx =

∞∑

i=γ

e−μz · (μz)ii!

. (A.4)


Substituting the above integral value in expression (A.3),one gets

f Z(z) = 2 ·⎡⎣∞∑

i=γ

e−μz · (μz)ii!

⎤⎦ · μγ

Γγ· e−μz · zγ−1

= 2 · μγ · e−2μz

Γγ·∞∑

i=γ

(μ)ii!· z(i+γ−1).

(A.5)

Now, substituting the value of f Z(z) from expression (A.5)in the expression for R (expression (A.1)), one will solve theintegrals associated with expression (A.1) as follows:

∫∞0e−αθz · fZ(z)dz

= 2 · μγΓγ

·∞∑

i=γ

(μ)ii!

·∫∞

0e−(αθ+2μ)z · z(i+γ−1)dz

= 2 · μγΓγ

·∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(αθ + 2μ

)(i+γ) ,

(A.6)

αθ∫∞

0

∫ z

0e−2θ(z−w)−αθw · fZ(z)dwdz

= α

α− 2

∫∞0e−2θz · fZ(z)dz

− α

α− 2

∫∞0e−αθz · fZ(z)dz,

(A.7)

where,

∫∞0e−2θz · fZ(z)dz = 2.μγ

Γγ·∞∑

i=γ

(μ)ii!

·∫∞

0e−(2θ+2μ)z · z(i+γ−1)dz

= 2 · μγΓγ

·∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(2θ + 2μ

)(i+γ) .

(A.8)

Using the results of expressions (A.8) and (A.6) in expression(A.7), one has

αθ∫∞

0

∫ z

0e−2θ(z−w)−αθw · fZ(z)dwdz

= α

α− 2·⎧⎨⎩

2 · μγΓγ

·∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(2θ + 2μ

)(i+γ)

⎫⎬⎭−

α

α− 2

·⎧⎨⎩

2 · μγΓγ

·∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(αθ + 2μ

)(i+γ)

⎫⎬⎭

= 2 · α · μγ(α− 2) · Γγ

×⎧⎨⎩∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(2θ + 2μ

)(i+γ) −∞∑

i=γ

(μ)ii!· Γ

(i + γ

)(αθ + 2μ

)(i+γ)

⎫⎬⎭.

(A.9)

Substituting the results of expressions (A.6) and (A.9) inthe expression (A.1), one gets the reliability function given inexpression (1).

Lemma A.2. The MLEs of parameters γ,μ, θ, and α given inexpressions (4), (5), (6), and (7) and further used to determinethe MLE of reliability function in expression (8) are derived asfollows.

The log-likelihood function of expression (2) in is

log L = 2n log

(μγ

Γγ

)+ log

⎧⎨⎩

n∏

i=1

(x1i)γ−1

⎫⎬⎭ + log

⎧⎨⎩

n∏

i=1

(x2i)γ−1

⎫⎬⎭

− μ(x1′ + x2

′) + m log(2θ)

+ m log(αθ)− 2θu′ − αθw′

= 2nγ logμ− 2n log(Γγ)

+(γ − 1

) n∑

i=1

log(x1i)

+(γ − 1

) n∑

i=1

log(x2i)− μ(x1′ + x2

′)

+ m log(2θ) + m log(αθ)− 2θu′ − αθw′.

(A.10)

Now,

∂ logL∂μ

= 0

=⇒ 2nγμ− (x′1 + x′2

) = 0

=⇒ 2nγμ= (x′1 + x′2

)

=⇒ log(2n) + log

(γ

μ

)= log

(x′1 + x′2

).

(A.11)

Similarly,

∂ logL∂γ

= 0

=⇒ 2n logμ− 2n∂

∂γ

(log(Γγ))

+n∑

i=1

log(x1i) +n∑

i=1

log(x2i) = 0.

(A.12)


Now

∂

∂γ

(log(Γγ)) log γ − 1

2γ

=⇒ 2n logμ− 2n

{log γ − 1

2γ

}

+n∑

i=1

log(x1i) +n∑

i=1

log(x2i) = 0

=⇒ 2n logμ− 2n log γ +n

γ+

n∑

i=1

log(x1i) +n∑

i=1

log(x2i) = 0

=⇒ 2n[log γ − logμ

] = n

γ+

n∑

i=1

log(x1i) +n∑

i=1

log(x2i)

=⇒ 2n log

(γ

μ

)= n

γ+

n∑

i=1

log(x1i) +n∑

i=1

log(x2i).

(A.13)

Solving (A.11) and (A.13) simultaneously one obtains theMLEs of parameters of γ and μ given in expression (4) and (5),respectively.

Now again,

∂ logL∂θ

= 0

=⇒ m

θ+m

θ− 2u′ − αw′ = 0

=⇒ 2mθ− 2u′ − αw′ = 0.

(A.14)

Similarly,

∂ logL∂α

= 0

=⇒ m

α− θw′ = 0.

(A.15)

Solving (A.14) and (A.15) simultaneously one obtains theMLEs of parameters of θ and α given in expression (6) and (7),respectively.

Lemma A.3. The posterior distribution of μ and θ, that is,f2(μ, θ | x1i, x2i,uj ,wj) used in expression (12), is derived asfollows.

The joint probability density function of random variables,X1i, X2i (i = 1, 2 · · ·n), Uj , Wj ( j = 1, 2 · · ·m), μ and θ is

given byX1i, X2i (i = 1, 2 · · ·n), Uj , Wj ( j = 1, 2 · · ·m), μand θ is given by

f(x1i, x2i,uj ,wj ,μ, θ

)

=⎧⎨⎩

(μγ0

Γγ0

)2n

·⎧⎨⎩

n∏

i=1

(x1i)γ0−1

⎫⎬⎭ ·

⎧⎨⎩

n∏

i=1

(x2i)γ0−1

⎫⎬⎭ · e

−μ(x′1+x′2)

⎫⎬⎭

.

{(2θ)m · (α0θ)m · e−2θu′ · e−α0θw′ · 1

Γp

·e−μ · μp−1 · 1Γq· e−θ · θq−1

}.

(A.16)

Integrating f (x1i, x2i,uj ,wj ,μ, θ) with respect to μ and θ overtheir respective range one gets, the joint probability distributionof X1i, X2i(i = 1, 2 · · ·n), Uj , Wj ( j = 1, 2 · · ·m) as

f1(x1i, x2i,uj ,wj

)

=⎡⎣ 1(

Γγ0)2n ·

⎧⎨⎩

n∏

i=1

(x1i)γ0−1

⎫⎬⎭ ·

⎧⎨⎩

n∏

i=1

(x2i)γ0−1

⎫⎬⎭

⎤⎦

.

[2m · α0

m

Γp · Γq · Γ(2nγ0 + p

)

(x′1 + x′1 + 1)(2nγ0+p)

· Γ(2m + q

)

(2u′ + α0w′ + 1)(2m+q)

].

(A.17)

Dividing f (x1i, x2i,uj ,wj ,μ, θ) in expression (A.16) byf1(x1i, x2i,uj ,wj) in expression (A.17) one gets the posteriordistribution of μ and θ f2(μ, θ | x1i, x2i,uj ,wj) as

f2(μ, θ | x1i, x2i,uj ,wj

)

={e−μ(x1

′+x2′+1) · μ2nγ0+p−1 · e−θ(2u′+α0w′+1)

Γ(2nγ0 + p

) · Γ(2m + q)

}

.{θ(2m+q−1) · (x′1 + x′2 + 1

)(2nγ0+p)

·(2u′ + α0w′ + 1)(2m+q)

}.

(A.18)

Acknowledgment

The authors thank the reviewers for many helpful commentsand suggestions on an earlier version which substantially hasimproved this paper.

References

[1] J. D. Church and B. Harris, “The estimation of reliability fromstress-strength relationship,” Technometrics, vol. 12, no. 1, pp.49–54, 1970.

[2] F. Downton, “The estimation of P(Y < X) in the normal case,”Technometrics, vol. 15, pp. 551–558, 1973.


[3] J. K. Wani and D. G. Kabe, “Point estimation of reliabilityof a system comprised of k elements from the same gammamodel,” Technometrics, vol. 13, pp. 859–864, 1971.

[4] K. Constantine, S. Tse, and M. Karson, “Estimation of P(Y <X) in the gamma distribution case,” Communications inStatistics, vol. 15, no. 2, pp. 365–388, 1986.

[5] G. K. Bhattacharya and R. A. Johnson, “Estimation of reliabil-ity in a multi component stress-strength model,” Journal of theAmerican Statistical Association, vol. 69, no. 348, pp. 966–970,1974.

[6] S. H. Kunchur and S. B. Munoli, “Estimation of reliability fora multi component survival stress-strength model based onexponential distribution,” Communications in Statistics, vol.22, no. 3, pp. 769–779, 1993.

[7] S. Dan and A. S. Krausz, “A multi-step stress-strength model ofa parallel system,” IEEE Transactions on Reliability, vol. R-35,no. 1, pp. 119–123, 1986.

[8] S. H. Kunchur and S. B. Munoli, “Estimation of reliability inFreunds model for two component system,” Communicationsin Statistics, vol. 23, no. 11, pp. 3273–3283, 1994.

[9] D. D. Hanagal, “Estimation of system reliability in a two com-ponent stress-strength models,” Economic Quality Control, vol.11, pp. 145–154, 1996.

[10] D. Kundu and R. D. Gupta, “Estimation of P(Y < X) forgeneralized exponential distribution,” Metrika, vol. 61, no. 3,pp. 291–308, 2005.

[11] S. K. Bhattacharya, “Bayesian approach to life testing andreliability estimation,” Journal of the American StatisticalAssociation, vol. 62, no. 317, pp. 48–62, 1967.

[12] N. R. Draper and I. Guttman, “Bayesian analysis of reliabilityin multi component stress-strength models,” Communicationsin Statistics, vol. A7, pp. 441–451, 1978.

[13] G. E. P. Box and G. C. Tiao, Bayesian Inference in StatisticalAnalysis, Addison Wesley, 1973.

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