Life Cycle Reliability and Safety Engineering RELIABILITY ESTIMATION OF S-OUT-OF-K SYSTEM FOR NON-IDENTICAL STRESS-STRENGTH COMPONENTS --Manuscript Draft-Manuscript Number:
LRSE-D-17-00032R1
Full Title:
RELIABILITY ESTIMATION OF S-OUT-OF-K SYSTEM FOR NON-IDENTICAL STRESS-STRENGTH COMPONENTS
Article Type:
Original
Funding Information: Abstract:
This paper takes into account the estimation of system reliability of a multi-component stress-strength model of a s-out-of-k system where both strength and stress are nonidentical components and follow Weibull and Burr-III distributions respectively. The reliability of such a system is obtained by the methods of maximum likelihood and Bayesian approach. We consider Bayes estimation under squared error loss function using conjugate priors for the parameters involved in the models. We propose Markov Chain Monte Carlo (MCMC) techniques to generate samples from the posterior distributions and in turn computing the Bayes estimator of the system reliability. Simulation study shows that Bayes estimator works better as compared to maximum likelihood estimator.
Corresponding Author:
Sanku Dey, Ph.D St. Anthony's College Shillong, ML INDIA
Corresponding Author Secondary Information: Corresponding Author's Institution:
St. Anthony's College
Corresponding Author's Secondary Institution: First Author:
Azeem Ali
First Author Secondary Information: Order of Authors:
Azeem Ali Shamma Khaliq, Ph.D Zeeshan Ali, Ph.D Sanku Dey, Ph.D
Order of Authors Secondary Information: Author Comments:
To The Editor-in-Chief Life Cycle Reliability and Safety Engineering
Dear Sir, I am submitting the revised version our paper entitled " RELIABILITY ESTIMATION OF S-OUT-OF-K SYSTEM FOR NON-IDENTICAL STRESS-STRENGTH COMPONENTS", co-authored with Azeem Ali , Shamma Khaliq, and Zeeshan Ali for consideration of possible publication in your esteemed journal: Life Cycle Reliability and Safety Engineering. . Thank you for your time and consideration. With regards, Sanku Dey
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Response to Reviewers: Response to the Reviewer # 1 for the manuscript entitled RELIABILITY ESTIMATION OF S- OUT - OF - K SYSTEM FOR NON-IDENTICAL STRESS-STRENGTH COMPONENTS" We express our sincere thanks to the Reviewer for his/her constructive suggestions and diligent review of the paper which in turn improved the paper. Below our response to the Reviewer's report. COMMENTS FOR THE AUTHOR: 1. Please provide a remark why Lindley's method has not been used to nd the Bayes estimate. Although the Bayes estimates can be easily obtained by Lindley's method, but HPD credible intervals cannot be constructed. In view of this, we proposed to use the MCMC technique to compute Bayes estimates as well as HPD credible intervals. In our future research work, we will try to incorporate Lindley's method. Since we have not done Lindleys method, we feel that, it would not be wise to put as a remark in the main text. 2. A real data analysis should be considered. If not kindly add some remark. Thanks for the comments. For non-identical components, it is very dicult to get data from the Industries, and conducting an experiment is not an easy task. Further, we have tried to get data sets from the published work, but we could not manage. In the revised version of the paper, additionally, we have incorporated two bootstrap confidence intervals (t-boot and p-boot) for the benifit of readers. We truly appreciate the comments. Thank you for your efforts to help improve our work.
Response to the Reviewer # 2 for the manuscript entitled \RELIABILITY ESTIMATION OF S- OUT - OF - K SYSTEM FOR NON-IDENTICAL STRESS-STRENGTH COMPONENTS" We express our sincere thanks to the Reviewer for his/her constructive suggestions and diligent review of the paper which in turn improved the paper. Below our response to the Reviewer's report. COMMENTS FOR THE AUTHOR: 1. It would be nice to see if the authors compare Bayes estimates for both noninformative and informative priors. As suggested, we have compared the Bayes estimates under both non-informative and informative priors in the revised version of the paper. Additionally, in the revised version of the paper, we have incorporated two bootstrap confidence intervals (t-boot and p-boot) for the benifit of the readers. We truly appreciate the comments. Thank you for your efforts to help improve our work.
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Click here to view linked References 1 2 3 4 5 6 7 8 9 10 11 RELIABILITY ESTIMATION OF S-OUT-OF-K SYSTEM FOR 12 NON-IDENTICAL STRESS-STRENGTH COMPONENTS 13 14 15 16 AZEEM ALI A , SHAMA KHALIQ B , ZEESHAN ALI C , SANKU DEY D 17 18 19 a 20 National College of Business Administration and Economics, 21 b,c Department of Statistics, Government College University Lahore, Pakistan, 22 d 23 Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India, 24 a
[email protected], b
[email protected], c
[email protected], 25 26 d
[email protected] 27 28 29 Abstract. This paper takes into account the estimation of system reliability 30 of a multi-component stress-strength model of a s − out − of − k system where 31 both strength and stress are non-identical components and follow Weibull and 32 33 Burr-III distributions respectively. The reliability of such a system is obtained 34 by the methods of maximum likelihood and Bayesian approach. We consider 35 36 Bayes estimation under squared error loss function using conjugate priors for 37 the parameters involved in the models. We propose Markov Chain Monte Carlo 38 39 (MCMC) techniques to generate samples from the posterior distributions and 40 in turn computing the Bayes estimator of the system reliability. Simulation 41 42 study shows that Bayes estimator works better as compared to maximum like43 lihood estimator. 44 45 KeywordsWeibull distribution; Laplace Transformation; Maximum Likeli46 hood Method; Bayesian Estimation 47 48 49 50 51 1. Introduction 52 53 The term, stress-strength, in the context of reliability, was introduced by Church 54 55 and Harris (1970). Since then, a lot of work has been done in this context by several 56 authors to discuss single-component stress-strength model in numerous lifetime 57 58 distributions. The stress-strength model, in its simplest form, defines the reliability 59 1 60 61 62 63 64 65
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of a component as the probability that the strength of the unit (X) is greater than the stress (Y ) imposed on it, that is, R = P (X > Y ). The estimation of stress-strength reliability is very common in the statistical literature. The reader is referred to Kotz et al. (2003) for other applications and motivations for the study of the stress-strength reliability. Bhattacharya and Johnson (1974) were first to discuss the reliability of a multicomponent stress-strength model. They studied the situation where a system, consisting of k components, functions when at least s(1 ≤ s ≤ k) of the components survive with a common random stress .This situation corresponds to the s−out−of −k : G system. Draper and Guttman (1978) considered stress -strength models with different distributions to estimate system reliability in which the component strengths were assumed to be i.i.d. random variables. Ebrahimi (1982) considered a series system consisting of p components where in the system was subjected to q different s-independent stresses. Pandey and Borhan Uddin (1992) studied a system having independent but non-identical distribution for strength subjected to the common stress; they assumed component strength is divided into two parts and both follow Weibull distributions but with different parametric values. Paul and Borhan Uddin (1997) considered that the strength has distributed non-identically subjected to common stress for estimating reliability of s − out − of − k system; they divided the component strength into two parts each having exponential distribution with different values of the parameter. In recent past, using classical method of estimation to estimate the reliability in multicomponent stress-strength for the log-logistic, inverse Rayleigh and Burr Type XII distributions were considered by Rao and Kantam (2010) and Rao et al. (2013, 2015) respectively; but recently, Kizilaslan and Nadar (2015), Dey et al. (2017) and Dey and Moala (2018) studied the estimation of reliability in multicomponent stress-strength based on both classical and Bayesian approaches for the Weibull, Kumaraswamy and Chen distributions respectively. A system having more than one component is called a multicomponent system. A multicomponent system may be a series system or a parallel system or a complex
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combination of series and parallel systems. A bridge structure is another example of a multicomponent system. The series and parallel systems are special cases of a general class of systems called s − out − of − k system. A system belonging to this class can be one of two types; (i) A system that fails with the failure of the s − th component, denoted by s − out − of − k : F system; or (ii) A system that functions as long as at least s components are working; and is denoted by s − out − of − k : G system, where 1 ≤ s ≤ k. Many examples can be given of multi-component systems. For example, it may be possible to drive a car with a V 8 engine if only four cylinders are firing. However, if less than four cylinders fire, then the automobile cannot be driven. Thus, the functioning of the engine may be represented by a 4 − out − of − 8 : G system. In the case of an automobile with four tires, for example, usually one additional spare tire is equipped on the vehicle. Thus, the vehicle can be driven as long as at least 4 − out − of − 5 tires are in good condition. Despite plenty of work done in reliability theory, to the best of our knowledge, so far it was not assumed that different parts of both stress and strength can be non-identical. In many practical situations, the components of a system may have different structures; therefore, it is not reasonable to assume similar functional form for each part of strength and stress. For example, if a rope is consolidated by two different ropes, the tensile strengths of both may not necessary be identically distributed. Similarly, it is not unrealistic to assume non-identical distributions for various parts of component stress which is imposed on component strength. Consequently, we are motivated to focus on a new perspective in reliability theory where the distribution of non-identical parts of both strength and stress may have different functional form for each part. Consider a system made up of k components. Suppose that out of those k components, k1 are of one category and their strengths can be assumed to have a common distribution function F1 . The remaining k2 = k − k1 components are of a different category and their common strength distribution is denoted by F2 . Similarly, the
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distribution functions of two different stress categories, say y1 and y2 , are G1 and G2 respectively. According to Johnson (1988), the system reliability , denoted by R, of s − out − of − k : G system is defined as:
(1.1) k2 Z∞ Z∞ k1 X X k1 k2 [1 − F1 ]j1 [F1 ]k1 −j1 [1 − F2 ]j2 [F2 ]k2 −j2 dG1 dG2 R= j j 1 2 j =s j =s 1
1
2
2
−∞ −∞
where the summation is over all possible pairs with 0 ≤ j1 ≤ k1 and 0 ≤ j2 ≤ k2 such that 1 ≤ s ≤ j1 +j2 ≤ k. It is important to note that the system reliability can be extended to more than two groups of components, but in this paper we assume only two categories. Let F1 and G1 follow Weibull distribution with parameters (σ, α1 ) and (σ, β1 ) respectively. Again, let F2 and G2 follow Burr-III distribution with parameters (α2 , θ) and (β2 , θ) respectively. Thus, we have F1 = 1 − exp(−α1 xσ1 ) ; x1 > 0, σ > 0, α1 > 0,
(1.2)
G1 = 1 − exp(−β1 y1σ ) ; y1 > 0, σ > 0, β1 > 0, −α2 F2 = (1 + x−θ ; x2 > 0, θ > 0, α2 > 0, 2 )
G2 = (1 + y2−θ )−β2 ; y2 > 0, θ > 0, β2 > 0, After substituting Eq. (??) into Eq. (??), we obtained simplified result for reliability of the system as: (1.3) R = ρβ2
j2 k1 k2 X X X k1 k2 j2 (−1)i B[j1 +ρ, k1 −j1 +1][α2 (k2 −j2 +i)+β2 ]−1 j j i 1 2 j =s j =s i=0 1
1
2
2
Eq. (??) can also be written as: (1.4) R = β 1 β2
j1 X j2 k1 k2 X X X j1 =s1 j2 =s2 l=0 i=0
(−1)
i+l
k1 k2 k1 − j1 j2 B[β1 + α1 (j1 + l), β2 + α2 (k2 − j2 + i)]−1 j1 j2 l i
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Where i, l, j1 and j2 are integers and ρ =
5 β1 α1 .
Noted that the above expressions
depend on α1 , α2 , β1 and β2 , but independent of θ and σ. The rest of the paper is organized as follows: In section 2, we obtain the reliability function by using maximum likelihood, and Bayesian estimation methods. In Section 3, a simulation study is carried out to compare the performance of these methods for the proposed model. Finally, a conclusion is presented in Section 4.
2. Estimation of R In this section, we estimate the unknown parameters of R by using maximum likelihood method and Bayesian estimation based on Weibull and Burr-III distributions.
2.1. Maximum Likelihood Estimation. Let a random sample, x11 , x12 , ..., x1n1 , of size n1 is drawn from W E(σ, α1 ) and another random sample, say x21 , x22 , ..., x2n2 , of size n2 is drawn from BurrIII(α2 , θ). Let us assume their corresponding order samples are x1(1) < x1(2) < ... < x1(n1 ) and x2(1) < x2(2) < ... < x2(n2 ) respectively. Consider y11 , y12 , ..., y1m1 be a random sample of size m1 drawn from W E(σ, β1 ) and y21 , y22 , ..., y2m2 be a random sample of size m2 drawn from BurrIII(β2 , θ). Then y1(1) < y1(2) < ... < y1(m1 ) and y2(1) < y2(2) < ... < y2(m2 ) are their corresponding order samples, respectively. Thus, the log-likelihood function of the observed sample for ψ = (α1 , α2 , β1 , β2 , σ, θ) can be written as:
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(2.1) l(ψ) =(n1 + m1 ) ln σ + (n2 + m2 ) ln θ + n1 ln α1 + n2 ln α2 + m1 ln β1 + m2 ln β2 + (σ − 1) m1 n2 m2 n1 m1 n1 X X X X X X σ ln x2i + ln y2j − α1 xσ1i − β1 y1j − ln y1j − (θ + 1) ln x1i + j=1
i=1
(α2 + 1)
n2 X
i=1
j=1
i=1
m2 X −θ ln 1 + x−θ − (β + 1) ln 1 + y2j 2 2i
i=1
j=1
The maximum likelihood estimates for the parameters involved in l(ψ) can be obtained by solving Eq. (??) numerically with the help of some statistical package such as R or Mathematica etc. Using invariant property of the maximum likelihood estimator, we can obtain the ˆ is given by: ML estimator of R, say R, (2.2) ˆ = ρˆβˆ2 R
j2 k1 k2 X X X k1 k2 j2 (−1)i B[j1 +ˆ ρ, k1 −j1 +1][αˆ2 (k2 −j2 +i)+βˆ2 ]−1 j j i 1 2 j =s j =s i=0 1
1
2
2
2.2. Bootstrap Confidence Interval. In this section, we propose two confidence intervals based on the bootstrap methods: (i) percentile bootstrap method (also called as Boot-p) based on the idea of Efron (1982), and (ii) bootstrap-t method (also referred as Boot-t) based on the idea of Hall (1988). We illustrate briefly how to estimate confidence intervals on R using both methods. (i) Boot-p method 1. Estimate ψ, say ψˆ by using Eq. (2.1) 2. Generate a bootstrap sample (x∗ , y ∗ ) using ψˆ Obtain the bootstrap estimate of ψ, say ψˆ∗ using the bootstrap sample. 3. Repeat step 10000 times. 4. Let U (x) = P (ψˆ∗ ≤ x) be the cumulative distribution function of ψˆ∗ . 5.Define ψˆBoot−p (x) = U −1 (x) for a given x. The approximate (1 − γ)100% boot-p
j=1
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confidence intervals for ψ are ˆ ( γ ), ψBoot−p (1 − γ ))) (ψBoot−p 2 2 (ii) Boot-t method 1. Using above bootstrap sample (x∗ , y ∗ ), compute bootstrap pivots as ψˆ∗ − ψˆ Ω∗ = q V ar(ψˆ∗ ) . 2. Repeat step 10000 times. 3. Let U (x) = P (Ω∗ ≤ x) be the cumulative distribution function of Ω∗ . For a q given x, Define ψBoot−t (x) = ψˆ + V ar(ψˆ∗ )U −1 (x). 4.The approximate (1 − γ)100% boot-t confidence intervals for ψ is ˆ ( γ ), ψBoot−t (1 − γ ))). (ψBoot−t 2 2 2.3. Bayesian Estimation. In this section, we consider a Bayesian approach based on MCMC methodology to obtain the estimator R, say R0 , under squared error loss function. However, any other loss function can also be incorporated in the present work. We use conjugate prior for the unknown parameters to get less convoluted mathematical expressions. Hence, let us assume independent gamma priors for the unknown parameters α1 , β1 and σ with hyper-parameters (ci , di ) , i = 1, 2, 3 respectively; and the same conjugate prior for unknown parameters α2 , β2 and θ with
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hyper-parameters (gi , hi ) , i = 1, 2, 3 respectively. The hyperparameters are specified and non-negative. Thus, we have prior densities as: (2.3)
π1 (α1 ; c1 , d1 ) =
dc11 c1 −1 α exp (−α1 d1 ) ; α1 > 0, d1 > 0, c1 > 0, Γc1 1
dc22 c2 −1 β exp (−β1 d2 ) ; β1 > 0, d2 > 0, c2 > 0, Γc2 1 dc3 π3 (σ; c3 , d3 ) = 3 σ c3 −1 exp (−σd3 ) ; σ > 0, d3 > 0, c3 > 0, Γc3
π2 (β1 ; c2 , d2 ) =
π4 (α2 ; g1 , h1 ) =
hg11 g1 −1 α exp (−α2 h1 ) ; α2 > 0, g1 > 0, h1 > 0, Γg1 2
π5 (β2 ; g2 , h2 ) =
hg22 g2 −1 β exp (−β2 h2 ) ; β2 > 0, g2 > 0, h2 > 0, Γg2 2
π6 (θ; g3 , h3 ) =
hg33 g3 −1 θ exp (−θh3 ) ; θ > 0, g3 > 0, h3 > 0, Γg3
The corresponding likelihood function of Eq. (??) is given as: (2.4) n1 m1 n1 X X X L(ψ|x) =σ n1 +m1 θn2 +m2 α1n1 α2n2 β1m1 β2m2 exp((σ − 1)( ln x1i + ln y1j ) exp(−α1 xσ1i )) i=1
exp(−β1
m1 X
σ y1j ) exp(−(θ + 1)(
ln x2i +
n2 X
j=1
m2 X
i=1
ln(y2j ))
j=1
i=1
j=1
exp(−(α2 + 1)
n2 X
ln(1 + x−θ 2j )) exp(−(β2 + 1)
m2 X
−θ ln(1 + y2j ))
j=1
i=1
The product of Eqs. (??) and (??) provides the joint posterior density of ψ as:
(2.5) π(ψ|x1 , x2 , y1 , y2 ) =Aσ n1 +m1 +c3 −1 θn2 +m2 +g3 −1 α1n1 +c1 −1 α2n2 +g1 −1 β1m1 +c2 −1 β2m2 +g2 −1 n1 m1 n1 m1 X X X X σ exp((σ − 1)( ln x1i + ln y1j ) − σd3 ) exp(−α1 xσ1i + d1 ) exp(−β1 y1j + d2 ) i=1
exp(−(θ + 1)(
n2 X i=1
exp(−β2
m2 X j=1
j=1
ln x2i +
m2 X
i=1
ln y2j ) − θh3 ) exp(−α2
j=1
j=1
n2 X
ln(1 + x−θ 2j ) + h1 )
i=1 n2 X
−θ ln(1 + y2j ) + h2 ) exp(−(
i=1
ln(1 + x−θ 2i ) +
m2 X j=1
−θ ln(1 + y2j )))
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where A is the normalizing constant and its value can be obtained from Eq.(??). (2.6) A−1 =
Z
σ n1 +m1 +c3 −1 θn2 +m2 +g3 −1 α1n1 +c1 −1 α2n2 +g1 −1 β1m1 +c2 −1 β2m2 +g2 −1 exp(−α1
n1 X
xσ1i + d1 )
i=1
ψ
exp(−β1
m1 X
n1 m1 n2 X X X σ y1j + d2 ) exp((σ − 1)( ln x1i + ln y1j ) − σd3 ) exp(−α2 ln(1 + x−θ 2j ) + h1 )
j=1
exp(−β2
i=1
m2 X
j=1
−θ ln(1 + y2j ) + h2 )) exp(−(θ + 1)(
j=1
n2 X
i=1
ln x2i +
i=1
m2 X
ln y2j ) − θh3 ))
j=1
n2 m2 X X −θ exp(−( ln(1 + x−θ ) + ln(1 + y2j )))dψ 2i i=1
j=1
By using Eq. (??), we get the posterior marginal distributions of unknown parameters as: " (2.7)
n1 X
f (α1 |σ, x) = Gamma n1 + c1 ,
# xσ1i + d1
i=1
(2.8)
f (β1 |σ, x) = Gamma m1 + c2 ,
m1 X
σ y1j + d2
j=1
" (2.9)
f (α2 |θ, x) = Gamma n2 + g1 ,
n2 X
# ln 1 +
xθ2i
+ h1
i=1
(2.10)
f (β2 |θ, y) = Gamma m2 + g2 ,
m2 X
θ ln 1 + y2j + h2
j=1
(2.11)
n1 m1 X X ln x1i + ln y1j − σd3 f (σ|x, y) ∝ σ n1 +m1 +c3 −1 exp (σ − 1) i=1
j=1
(2.12)
n2 m2 n2 m2 X X X X −θ −θ f (θ|x, y) ∝ θn2 +m2 +g3 −1 exp −(θ + 1) ln x2i + ln y2j − θh3 − ln 1 + x2i + ln 1 + y2j i=1
j=1
i=1
j=1
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Thus, the Bayes estimates of R under squared error loss function becomes: (2.13) R0 = ρ0 β20
j2 k2 X k1 X X k1 k2 j2 (−1)i B[j1 +ρ0 , k1 −j1 +1][α20 (k2 −j2 +i)+β20 ]−1 j j i 1 2 j =s j =s i=0 1
1
2
2
3. Simulation Study In this section, we present some experimental results to examine the behavior of the proposed methods for different sample sizes. We consider the following sample sizes: n1 = (10,30, 50); n2 = (10,30, 50); m1 = (10,30, 50) ; m2 = (10,30, 50). In all cases, we take α1 = 0.1; α2 = 2; β1 = 3; β2 = 0.5; θ = 1.2; σ = 3.5. The values for s−out−of −k systems are taken as (s1 , s2 , k1 , k2 ) = (1, 1, 2, 2), (1, 2, 2, 2), (2, 1, 2, 2) and (2, 2, 2, 2). Additionally, for Bayesian analysis, we have considered the hyperparameter values for informative priors as : c1 = 0.05, d1 = 0.03; c2 = 0.01, d2 = 0.02; g1 = 0.04, h1 = 0.02; g2 = 0.07, h2 = 0.03 and for non-informative priors as zero i.e., c1 = d1 = c2 = d2 = g1 = h1 = g2 = h2 = 0. The comparison is made on the basis of Average Absolute Bias (AAB), Mean Squared Error (MSE), and Mean Absolute Percentage Error (MAPE). In achieving the required task, we perform the following steps: Step 1: The MLEs of (α1 , α2 , β1 , β2 ) can be obtained from Eq. (??). Step 2: The MLEs of R are computed by using the estimates of (α1 , α2 , β1 , β2 ), obtained in step 1 (see Table-3.1). Step 3: The Bayes estimates of R under squared error loss function using MCMC can be computed from equations Eqs.(??) to (??)(see Tables 3.3-3.4). Step 4: Repeat the above steps N (= 10000) times, the measures AAB, MSE, MAPE, 95% confidence intervals/credible intervals and median for each parameter is obtained. The simulated study shows that the proposed Bayes estimator for system reliability is more efficiently worked as compared to the maximum likelihood estimator. One can see from Tables 3.1 and 3.4 that the average bias for Bayes estimates under informative prior approaches to zero faster than MLEs when sample size become
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larger. These tables also suggest that the mean squared error for Bayes estimates for system reliability, in all cases tabulated in Table 3.4, is smaller than those tabulated in Table 3.1. Also we observe that Bayesian intervals become more tight than asymptotic confidence intervals. Hence, on the basis of limited simulated results, we can coclude that Bayes estimator of system reliability is superior to ML estimator of the system reliability. However, ML estimates perform better than Bayes estimates under non-informative priors. Further, we observe from Table 3.2 that Bootstrap confidence intervals performs little better than asymptotic confidence intervals.
4. Conclusion In this paper, we have provided a new perspective in reliability theory by assuming that both stress and strength consist of non-identical parts. We also assumed that these non-identical categories follow Weibull and Burr-III distributions. We have compared the reliability of MLEs and Bayes estimators with respect to the Average Absolute Bias (AAB), Mean Squared Error (MSE), and Mean Absolute Percentage Error (MAPE). We have also compared the confidence intervals attained using asymptotic distribution of the MLEs and the credible intervals obtained from the posterior distribution functions. The simulation study shows that Bayes estimates based on informative prior performs little better than Non Informative prior and the MLEs with regard to both biases and mean squared errors. The Bayes credible intervals for reliability are also of shorter length with competitive confidence intervals.
Acknowledgments The authors would like to thank the Editor-in-Chief, Associate Editor and two anonymous referees for careful reading and for comments which greatly improved the paper.
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References [1] Asgharzadeh A., Valiollahi R. and Raqab, M. Z. (2011), Stress strength reliability of Weibull distribution based on progressively censored samples. SORT, 35, 103-124. [2] Badar, M. G. and Priest, A. M. (1982). Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi, T., Kawata, K., Umekawa, S. (Eds.), Progress in Science and Engineering Composites, ICCM-IV, Tokyo, 1129-1136. [3] Bayes, T. (1763). An essay towards solving a problem in the doctrine of chances, Philosophical Transactions of the Royal Society of London 53, 370418. [4] Bhattacharyya, G.K., Johnson, R. A. (1974). Estimation of reliability in multicomponent stress-strength model, J. Am. Stat. Assoc., 69, 966-970. [5] Church, J. D. and Harris, B. (1970). The estimation of reliability from stressstrength relationships, Technometrics, 12, 49-54. [6] Dey, S., Mazucheli, J., Anis, M.Z. (2017). Estimation of reliability of multicomponent stress-strength for a Kumaraswamy distribution, Commun. Stat. Theo. Methods, 46, 1560-1572. [7] Dey, S. and Moala, Fernando A. (2018). Estimation of Reliability of Multicomponent Stress-Strength of a Bathtub Shape or Increasing Failure Rate Function. International Journal of Quality & Reliability Management (in press). [8] Draper, N.R. and Guttman, I. (1978). Bayesian analysis of reliability in multicomponent stress-strength models, Commun. Stat. Theory. Methods, 7, 441451. [9] Ebrahimi, N. (1982). Estimation of reliability for a series stress-strength system, IEEE Trans. Rel., R-31, 202-205. [10] Efron, B. (1982). The jackknife, the bootstrap and other re-sampling plans. Philadelphia, PA: SIAM, CBMSNSF Regional Conference Series in Applied Mathematics, 34.
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[11] Geisser, S. (1984). On prior distributions for binary trials, The American Statistician 38(4), 244-247. [12] Hall, P. (1988). Theoretical comparison of bootstrap confidence intervals. Annals of Statistics, 16, 927-953. [13] Johnson, R. A. (1988). Stressstrength models for reliability. In P. R. Krishnaiah, C. R. Rao (Eds.) In handbook of statistics (pp. 27-54). Vol. 7, NorthHolland: Elsevier. [14] Kizilaslan, F. and Nadar, M. (2015). Classical and Bayesian estimation of reliability in multi-component stress-strength model based on Weibull distribution, Rev. Colomb. Estad., 38(2), 67-484. [15] Kotz, S., Lumelskii, Y. and Pensky, M. (2003). The Stress-strength Model and its Generalizations: Theory and Applications, World Scientific. [16] Kundu, D. and Gupta, R. D. (2006). Estimation of P (Y < X) for Weibull distributions. IEEE Transcations on Reliability, 55(2), 270-280. [17] Kundu, D. and Raqab, M. Z. (2009). Estimation of R = P (Y < X) for three parameter Weibull distribution. Statistics and Probability Letters, 79, 18391846. [18] Rao, G.S. and Kantam, R.R.L. (2010). Estimation of reliability in multicomponent stress-strength model: log-logistic distribution, J. Appl. Stat. Sci., 3(2), 75-84. [19] Raqab, M. Z. and Kundu, D. (2005). Comparison of different estimators of P (Y < X) for a scaled Burr type X distribution. Communications in StatisticsSimulation and Computation, 34(2), 465-483. [20] Rao, G. S., Kantam, R. R. L., Rosaiah, K. and Reddy, J. P.(2013). Estimation of reliability in multicomponent stress strength model based on inverse Rayleigh distribution, J. Stat. Appl.Probab., 2(3), 261-267. [21] Rao, G. S., Aslam, M. and Kundu, D. (2015). Burr Type XII distribution parametric estimation and estimation of reliability of multicomponent stressstrength, Commun. Stat. Theo. Methods, 44(23),4953-4961.
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[22] Pandey, M. and Borhan Uddin, M.B. (1992). Reliability estimation of an s − out − of − k system with non-identical component strengths: the Weibull case, Reliab. Eng. Syst. Saf., 36(2), 109-116. [23] Paul, P. K. and Borhan Uddin, M. B.(1997). Estimation of reliability of stressstrength model with non-identical component strengths. Microelectronics Reliability, 37 (6), 923-927.
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Table 3.1. Summary of the Simulation Study of the MLE of the reliability function of Non-identical Components for α1 = 0.1, β1 = 3, α2 = 2, β2 = 0.5, θ = 1.2, σ = 3.5 and 10000 replications (s1 , s2 , k1 , k2 ) R (n1 , n2 , m1 , m2 ) M ean AABias M SE M AP E 2.5% M edian 97.5% (1,1,2,2) 0.88709 (10,10,10,10) 0.875760.0113279 0.0001613 1.27697 0.86944 0.87576 0.88029 (10,10,30,30) 0.876230.0108580 0.0001510 1.22401 0.86966 0.87922 0.88265 (10,10,50,50) 0.885010.0079411 0.0001223 0.89519 0.88971 0.89222 0.89938 (30,30,10,10) 0.878870.0082243 0.0000843 0.92711 0.87483 0.87800 0.88234 (30,30,30,30) 0.881100.0059864 0.0000439 0.67484 0.87920 0.88071 0.88303 (30,30,50,50) 0.883970.0023268 0.0000143 0.26230 0.88706 0.88813 0.89088 (50,50,10,10) 0.873870.0132238 0.0002255 1.49069 0.86971 0.87486 0.87941 (50,50,30,30) 0.880920.0061703 0.0000481 0.69557 0.87779 0.88022 0.88381 (50,50,50,50) 0.884770.0026692 0.0000104 0.30089 0.88324 0.88437 0.88628 (1,2,2,2) 0.70967 (10,10,10,10) 0.684650.0250219 0.0007912 3.52585 0.67052 0.68502 0.69473 (10,10,30,30) 0.695020.0155882 0.0004060 2.19655 0.68563 0.69796 0.70514 (10,10,50,50) 0.704710.0113284 0.0002523 1.59630 0.69174 0.70592 0.71131 (30,30,10,10) 0.691930.0178055 0.0003977 2.50899 0.68341 0.68996 0.69968 (30,30,30,30) 0.696180.0134933 0.0002232 1.90135 0.69213 0.69553 0.70040 (30,30,50,50) 0.709520.0053846 0.0000759 0.75875 0.70965 0.71214 0.71846 (50,50,10,10) 0.680880.0287862 0.0010090 4.05628 0.67178 0.68287 0.69278 (50,50,30,30) 0.696060.0136112 0.0002328 1.91797 0.68937 0.69430 0.70231 (50,50,50,50) 0.704650.0057497 0.0000494 0.81019 0.70135 0.70401 0.70791 (2,1,2,2) 0.83333 (10,10,10,10) 0.815890.0174408 0.0004271 2.09290 0.81443 0.81734 0.81989 (10,10,30,30) 0.825230.0081880 0.0000789 0.98256 0.82188 0.82447 0.82828 (10,10,50,50) 0.829120.0044328 0.0000235 0.53193 0.82720 0.82870 0.83048 (30,30,10,10) 0.823030.0103041 0.0001393 1.23650 0.81789 0.82369 0.82831 (30,30,30,30) 0.826640.0067950 0.0000661 0.81541 0.82163 0.82715 0.83042 (30,30,50,50) 0.829330.0044339 0.0000288 0.53207 0.82570 0.83008 0.83181 (50,50,10,10) 0.825210.0081195 0.0000739 0.97434 0.82317 0.82467 0.82731 (50,50,30,30) 0.826950.0064404 0.0000591 0.77285 0.82263 0.82812 0.83005 (50,50,50,50) 0.829700.0043298 0.0000269 0.51957 0.82623 0.82876 0.83342 (2,2,2,2) 0.66667 (10,10,10,10) 0.637570.0290967 0.0010150 4.36448 0.62895 0.63264 0.64691 (10,10,30,30) 0.650460.0162146 0.0003377 2.43218 0.64432 0.65114 0.65760 (10,10,50,50) 0.658560.0100313 0.0001974 1.50468 0.64944 0.65872 0.66554 (30,30,10,10) 0.648600.0181101 0.0003948 2.71650 0.64135 0.64692 0.65542 (30,30,30,30) 0.653300.0133721 0.0002142 2.00580 0.64847 0.65033 0.65847 (30,30,50,50) 0.672570.0060257 0.0001038 0.90385 0.66690 0.66962 0.67534 (50,50,10,10) 0.637650.0290150 0.0010056 4.35223 0.62888 0.63968 0.64903 (50,50,30,30) 0.653380.0132882 0.0002301 1.99321 0.64536 0.65453 0.65987 (50,50,50,50) 0.660840.0072258 0.0000703 1.08387 0.65669 0.65895 0.65987
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
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Table 3.2. Summary of the Simulation Study of Boot-p and Boot-t of the reliability function of Non-identical Components for α1 = 0.1, β1 = 3, α2 = 2, β2 = 0.5, θ = 1.2, σ = 3.5 and 10000 replications (s1 , s2 , k1 , k2 ) (1,1,2,2)
(1,2,2,2)
(2,1,2,2)
(2,2,2,2)
R
(n1 , n2 , m1 , m2 )
0.88709 (10,10,10,10) (10,10,30,30) (10,10,50,50) (30,30,10,10) (30,30,30,30) (30,30,50,50) (50,50,10,10) (50,50,30,30) (50,50,50,50) 0.70967 (10,10,10,10) (10,10,30,30) (10,10,50,50) (30,30,10,10) (30,30,30,30) (30,30,50,50) (50,50,10,10) (50,50,30,30) (50,50,50,50) 0.83333 (10,10,10,10) (10,10,30,30) (10,10,50,50) (30,30,10,10) (30,30,30,30) (30,30,50,50) (50,50,10,10) (50,50,30,30) (50,50,50,50) 0.66667 (10,10,10,10) (10,10,30,30) (10,10,50,50) (30,30,10,10) (30,30,30,30) (30,30,50,50) (50,50,10,10) (50,50,30,30) (50,50,50,50)
2.5% 0.74646 0.81477 0.82052 0.77039 0.83115 0.84116 0.65615 0.79969 0.84363 0.44604 0.56554 0.57274 0.48615 0.59168 0.61381 0.32151 0.53534 0.61883 0.67867 0.78270 0.73470 0.70639 0.78889 0.76012 0.61803 0.73486 0.77165 0.42425 0.53399 0.53290 0.44776 0.56639 0.56244 0.30275 0.49497 0.57223
Boot − p Median 0.87865 0.88020 0.87151 0.87713 0.88545 0.88851 0.81472 0.86171 0.88926 0.69136 0.69230 0.67568 0.68960 0.70515 0.71461 0.56181 0.65552 0.71580 0.83668 0.86178 0.82541 0.81248 0.85476 0.82349 0.76807 0.80241 0.82669 0.65705 0.67706 0.63857 0.63798 0.67937 0.66291 0.52937 0.61003 0.66543
Boot − t 97.5% 2.5% 97.5% 0.96477 0.74640 0.96477 0.92365 0.75338 0.92065 0.91423 0.82046 0.91429 0.94733 0.77020 0.94754 0.93134 0.83106 0.93145 0.92143 0.84114 0.92149 0.88664 0.65610 0.88664 0.90775 0.79969 0.90778 0.98208 0.84353 0.92088 0.90150 0.44574 0.90164 0.79299 0.61114 0.83128 0.77575 0.75010 0.95321 0.85857 0.66350 0.83608 0.81274 0.59165 0.81283 0.79193 0.79091 0.96937 0.70883 0.32127 0.70884 0.75922 0.71273 0.93666 0.79032 0.79624 0.96779 0.93500 0.67839 0.93509 0.91015 0.72221 0.869831 0.88945 0.78841 0.94334 0.89260 0.76008 0.94637 0.90685 0.78882 0.90686 0.88376 0.81385 0.93753 0.83257 0.61771 0.83265 0.85563 0.78559 0.91011 0.87014 0.82539 0.92393 0.86844 0.42394 0.86850 0.77831 0.68881 0.81507 0.73753 0.75311 0.95806 0.80252 0.66776 0.92309 0.78768 0.56634 0.78769 0.74648 0.78275 0.96690 0.66466 0.30142 0.66463 0.70852 0.71527 0.92909 0.73887 0.79254 0.95929
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
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Table 3.3. Summary of the Simulation Study of the Bayes estimate of the reliability function of Non-identical Components using Non-Informative prior for α1 = 0.1, β1 = 3, α2 = 2, β2 = 0.5, θ = 1.2, σ = 3.5 and 10000 replications (s1 , s2 , k1 , k2 ) R (n1 , n2 , m1 , m2 ) M ean AABias M SE M AP E 2.5% M edian 97.5% (1,1,2,2) 0.88709 (10,10,10,10) 0.89751 0.027421 0.004828 1.38591 0.78014 0.808921 0.889213 (10,10,30,30) 0.88923 0.095382 0.009841 1.42308 0.85192 0.88801 0.90814 (10,10,50,50) 0.89501 0.029850 0.0097451 0.90475 0.87021 0.90152 0.92991 (30,30,10,10) 0.89387 0.019842 0.0025526 0.984391 0.87319 0.88990 0.92901 (30,30,30,30) 0.88561 0.037640 0.007794 0.698531 0.86521 0.91071 0.93719 (30,30,50,50) 0.99097 0.007265 0.001842 0.297319 0.878601 0.89974 0.91231 (50,50,10,10) 0.88987 0.201742 0.003269 1.549271 0.83291 0.87198 0.89921 (50,50,30,30) 0.88892 0.008232 0.009451 0.719360 0.85210 0.88022 0.90181 (50,50,50,50) 0.88477 0.003254 0.096438 0.318653 0.86203 0.888712 0.89072 (1,2,2,2) 0.70967 (10,10,10,10) 0.72465 0.027846 0.086241 1.94693 0.62502 0.71987 0.73421 (10,10,30,30) 0.75502 0.023285 0.082501 2.203501 0.65484 0.70991 0.72504 (10,10,50,50) 0.71471 0.139321 0.07943 1.963710 0.68901 0.72840 0.75139 (30,30,10,10) 0.73073 0.029819 0.087421 2.620329 0.65201 0.71672 0.74081 (30,30,30,30) 0.80218 0.030813 0.073290 2.041701 0.67285 0.70194 0.72051 (30,30,50,50) 0.71952 0.073291 0.072821 0.807621 0.64391 0.71931 0.76837 (50,50,10,10) 0.76088 0.038640 0.074392 2.23120 0.61986 0.70518 0.73171 (50,50,30,30) 0.73606 0.062591 0.068234 1.99901 0.66212 0.71619 0.74132 (50,50,50,50) 0.77765 0.092516 0.084210 0.824501 0.68231 0.70991 0.73999 (2,1,2,2) 0.83333 (10,10,10,10) 0.85089 0.097159 0.026892 2.173035 0.78312 0.82021 0.84719 (10,10,30,30) 0.83993 0.092619 0.092512 1.032841 0.80172 0.84137 0.86283 (10,10,50,50) 0.83712 0.021818 0.018361 0.082731 0.79151 0.82876 0.85981 (30,30,10,10) 0.83503 0.020301 0.020916 1.72041 0.79752 0.83751 0.85141 (30,30,30,30) 0.87464 0.092017 0.072130 0.87529 0.80330 0.83126 0.86142 (30,30,50,50) 0.83433 0.01689 0.020964 0.984301 0.81920 0.83170 0.86193 (50,50,10,10) 0.87321 0.018301 0.009801 0.99918 0.80163 0.85001 0.84281 (50,50,30,30) 0.88095 0.02897663 0.091520 0.882201 0.80721 0.81419 0.84906 (50,50,50,50) 0.84070 0.0051815 0.072021 0.552912 0.77186 0.80184 0.84539 (2,2,2,2) 0.66667 (10,10,10,10) 0.67593 0.020901 0.003201 4.501723 0.60954 0.70191 0.73715 (10,10,30,30) 0.66856 0.038271 0.007213 2.78323 0.61485 0.67541 0.70539 (10,10,50,50) 0.69701 0.042715 0.017842 1.927104 0.60374 0.65915 0.698324 (30,30,10,10) 0.69023 0.029161 0.045173 2.900101 0.61300 0.67163 0.69936 (30,30,30,30) 0.680192 0.027007 0.052001 2.027183 0.62094 0.65931 0.67891 (30,30,50,50) 0.71001 0.0089251 0.018927 0.959267 0.65372 0.68021 0.72643 (50,50,10,10) 0.690210 0.038207 0.032231 2.58293 0.61954 0.67197 0.69692 (50,50,30,30) 0.701021 0.058152 0.089276 2.02617 0.62640 0.67213 0.71433 (50,50,50,50) 0.689012 0.018180 0.02917 1.823101 0.62785 0.68313 0.73725
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
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Table 3.4. Summary of the Simulation Study of the Bayes Estimates of the reliability function of Non-identical Components using informative prior for α1 = 0.1, β1 = 3, α2 = 2, β2 = 0.5, θ = 1.2, σ = 3.5 and 10000 replications (s1 , s2 , k1 , k2 ) R (n1 , n2 , m1 , m2 ) M ean AABias M SE M AP E 2.5% M edian 97.5% (1,1,2,2) 0.88709 (10,10,10,10) 0.883780.0033141 0.0000142 0.37359 0.88206 0.88364 0.88544 (10,10,30,30) 0.884380.0029209 0.0000116 0.32927 0.88281 0.88404 0.88565 (10,10,50,50) 0.887060.0010438 0.0000024 0.11766 0.88605 0.88714 0.88817 (30,30,10,10) 0.882670.0044276 0.0000269 0.49912 0.88023 0.88273 0.88499 (30,30,30,30) 0.883920.0031709 0.0000133 0.35745 0.88326 0.88382 0.88534 (30,30,50,50) 0.887790.0007454 0.0000009 0.08403 0.88737 0.88774 0.88808 (50,50,10,10) 0.885570.0016770 0.0000054 0.18904 0.88407 0.88603 0.88702 (50,50,30,30) 0.886530.0008066 0.0000010 0.09093 0.88587 0.88656 0.88721 (50,50,50,50) 0.886980.0006169 0.0000005 0.06954 0.88641 0.88697 0.88759 (1,2,2,2) 0.70967 (10,10,10,10) 0.702800.0068654 0.0001108 0.96740 0.69924 0.70268 0.70631 (10,10,30,30) 0.704140.0061588 0.0001001 0.86783 0.70083 0.70342 0.70692 (10,10,50,50) 0.709630.0033188 0.0000640 0.46766 0.70605 0.70976 0.71312 (30,30,10,10) 0.704510.0053684 0.0000932 0.75647 0.70074 0.70532 0.70776 (30,30,30,30) 0.707590.0022779 0.0000572 0.32098 0.70639 0.70764 0.70882 (30,30,50,50) 0.708890.0015371 0.0000544 0.21659 0.70809 0.70929 0.71035 (50,50,10,10) 0.703820.0058512 0.0000930 0.82450 0.70131 0.70436 0.70612 (50,50,30,30) 0.705470.0043246 0.0000802 0.60938 0.70237 0.70503 0.70918 (50,50,50,50) 0.708360.0020242 0.0000383 0.28523 0.70756 0.70890 0.71011 (2,1,2,2) 0.83333 (10,10,10,10) 0.824670.0087349 0.0000931 1.04819 0.82141 0.82398 0.82693 (10,10,30,30) 0.825230.0080156 0.0000743 0.96188 0.82253 0.82500 0.82739 (10,10,50,50) 0.833130.0019010 0.0000053 0.22812 0.83243 0.83395 0.83625 (30,30,10,10) 0.824450.0088837 0.0001097 1.06605 0.81975 0.82395 0.83008 (30,30,30,30) 0.830620.0027205 0.0000105 0.32646 0.82939 0.82089 0.83198 (30,30,50,50) 0.832430.0019170 0.0000055 0.23005 0.83110 0.83227 0.83429 (50,50,10,10) 0.826050.0073620 0.0000654 0.88344 0.82362 0.82521 0.82777 (50,50,30,30) 0.828440.0061172 0.0000577 0.73407 0.82294 0.82859 0.83435 (50,50,50,50) 0.830160.0041010 0.0000302 0.49213 0.82792 0.82991 0.83418 (2,2,2,2) 0.66667 (10,10,10,10) 0.655800.0108736 0.0001899 1.63103 0.65112 0.65475 0.66005 (10,10,30,30) 0.657120.0096234 0.0001602 1.44351 0.65303 0.65651 0.66021 (10,10,50,50) 0.666410.0024448 0.0000525 0.36671 0.66693 0.66858 0.67012 (30,30,10,10) 0.657110.0096219 0.0001723 1.44327 0.65218 0.65754 0.66317 (30,30,30,30) 0.663680.0030135 0.0000583 0.45203 0.66179 0.66412 0.66581 (30,30,50,50) 0.666280.0020348 0.0000507 0.30522 0.66710 0.66806 0.66965 (50,50,10,10) 0.658940.0077428 0.0001177 1.16142 0.65651 0.65949 0.66105 (50,50,30,30) 0.661160.0066302 0.0001095 0.99453 0.65479 0.66026 0.66748 (50,50,50,50) 0.662220.0049133 0.0000630 0.73699 0.66025 0.66320 0.66639
