Estimation a Stress-Strength Model for P ( Y r : n 1 < X k : n 2 ) Using the Lindley Distribution

The problem of estimation reliability in a multicomponent stress-strength model, when the system consists of k components have strength each component experiencing a random stress, is considered in this paper. The reliability of such a system is obtained when strength and stress variables are given by Lindley distribution. The system is regarded as alive only if at least r out of k (r < k) strength exceeds the stress. The multicomponent reliability of the system is given by Rr,k. The maximum likelihood estimator (MLE), uniformly minimum variance unbiased estimator (UMVUE) and Bayes estimator of Rr,k are obtained. A simulation study is performed to compare the different estimators of Rr,k. Real data is used as a practical application of the proposed model.


Introduction
The Lindley distribution originally developed by Lindley (1958Lindley ( , 1965) ) in the context of Bayesian statistics, is a counter example of fiducial statistics.The Lindley distribution has the following probability density function (p.d.f) The corresponding cumulative distribution function (c.d.f) is and the corresponding survival function is S(x; θ) = (1 + θ 1 + θ x)e −θx x > 0, θ > 0.
(3) Ghitany & Atieh (2008) studied the mathematical and statistical properties of the Lindley distribution.They have shown that this distribution is better a model than the well-known exponential distribution in some particular cases.Al-Mutairi, Ghitany & Kundu (2013) investigated the stress-strength model using the Lindley distribution and in this paper we will investigate the multicomponent stress-strength model of the Lindley distribution.Also Figure 1 shows that the Lindley distribution for different values of θ is positively skewed.Many authors have discussed the Lindley distribution as a model of lifetime data such as Krishna & Kumar (2011), Singh, Singh & Singh (2008) and Singh, Gupta &Sharma (2014), andAl-Mutairi et al. (2013) studied stress-strength model.Also the inverse Lindley distribution discussed as stress-strength model has been studied by Sharma, Singh, Singh & Agiwal (2014, 2015).

Iwase
In this paper, the system of reliability R r,k = P (Y r:n1 < X k:n2 ) in the Lindley distribution case is derived.Special cases of R r,k can be found in Section 2. The maximum likelihood estimator (MLE) of R r,k , the uniform minimum variance unbiased estimator (UMVUE) of R r,k , and the Bayes estimator of R r,k are obtained in Section 3. In Section 4, a simulation study is performed to compare the estimators of the reliability system.In Section 5, real data is used as a practical application of the proposed procedure.Finally, we conclude in Section 6.

System of Reliability
Let X and Y be two random variables as part of the Lindley distribution with parameters q and p, respectively.Suppose X 1 , . . ., X n2 and Y 1 , . . ., Y n1 are two independent samples from X and Y , respectively.The strength and the stress are assumed to be independent.Based on these assumptions, we find the system of reliability to be where, F Yr:n 1 (x) and f X:n2 (x) are the rth cumulative density function and kth probability density function of Y r:n1 and X k:n2 respectively.And, From ( 5) and ( 6) in (4), we obtain Based on some calculations and binomial theory, we obtain where Γ(.) is a gamma function.
We now present some special cases of R r,k with a different arrangement of the components.
1.For r = n 1 and k = 1, the minimum strength component is subjected to the maximum stress component.In this case, the probability R n1,1 is the reliability of a series system with an n 2 component 2. For r = n 1 and k = n 2 , the maximum strength component is subjected to the maximum stress component.Then, R n1,n2 is the reliability of a parallel system with an n 2 component 3. For r = 1 and k = 1, the minimum strength component is subjected to the minimum stress component.Then, 4. For r = n 1 and k = k, the kth strength order component is subjected to the maximum stress component. (11)

Different Estimators of R r,k
In this section the three estimation methods for R r,k that were applied were the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the Bayes estimator of R r,k using the Lindley approximation.

Maximum Likelihood Estimator
Let X 1 , . . ., X n2 be a random sample of the strengths of the n 2 systems that are distributed as Lindley random variables with parameter q and Y 1 , . . ., Y n1 .Let these be a random samples of stresses of n 1 systems that are distributed as Lindley random variables with the parameter p. Then the log likelihood function of the observed samples is x j Ghitany & Atieh (2008) showed that the maximum estimator of p and q, denoted by p and q, are qMLE Revista Colombiana de Estadística 40 (2017) 105-121 By using the invariance property of the maximum likelihood estimator the maximum estimator of R r,k can be obtain.This is denoted by RMLE r,k , replacing p and q in equation ( 7) by their maximum estimators.Hence RMLE r,k is given by Where pMLE and qMLE are defined in equation ( 12) and ( 13) respectively.Now, we find the asymptotic distribution of RMLE r,k because it is difficult to find the explicit distribution.To find the asymptotic distribution and the confidence interval of R r,k , we use the algorithm below.Algorithm: 1. Find the asymptotic variance of pMLE and qMLE as follows as is presented by Rao (1973) ) where Z α 2 is the upper α 2 -quantile of standard normal distribution.

Uniformly minimum variance unbiased estimator (UMVUE) of R r,k
To find the UMVUE of R r,k which is denoted by RU r,k , we need to prove the following theorem.
Theorem 1.If X 1 , . . ., X n is a random sample from the Lindley distribution with parameter θ, then the probability density function of Now, let X 1 , . . ., X n2 be a random sample of the strengths of n 2 systems that are distributed as Lindley random variables with parameter q, let Y 1 , . . ., Y n1 be a random sample of stresses of n 1 systems that are distributed as Lindley random variables with parameter p.Also, let U = n2 i=1 X i and V = n1 j=1 Y j be complete and sufficient statistics for p and q, respectively.Hence RU r,k can obtained as where, φ(X where, 1 < i < n 2 and 1 < j < n 1 .Using binomial theorem we get where, When calculating the integral equation ( 15), when u ≤ v and u > v, we get RU r,k .

Bayes estimator ofR r,k
To find the Bayesian estimators of unknown parameters p, q, and the stressstrength reliability model R r,k , which is denoted by RB r,k , we consider a noninformative and an informative gamma prior for unknown parameters p and q (see Jeffrey 1961).Let X 1 , . . ., X n2 be a random sample of the strengths of n 2 systems that are distributed as Lindley random variables with parameter q, and let Y 1 , . . ., Y n1 be a random sample of the stresses of n 1 systems that are distributed as Lindley random variables with parameter p.We assume p and q have gamma prior distributions of the following forms and, where, a 1 , a 2 , b 1 , and b 2 are known.
The joint posterior distribution of p and q is defined by where, k = 1/ ∞ 0 ∞ 0 L(p, q | data)π(p)π(q) dpdq, and The Bayes estimator of any parametric function R r,k under square error loss function (SELF) can be written as We have no closed form for RB r,k , hence numerical computations are needed.(1980) proposed an approximation technique to find the Bayes estimators of stress-strength parameters p, q and R r,k under the squared error loss function, which are given by θ * l,SELF = θl,MLE + ρl σll + 0.5( Llll σll ) = θ l,M LE , l = 1, 2 and,

Lindley
where, θ1,MLE = pMLE , θ2,MLE = qMLE , ) is the asymptotic expected Fisher information matrix, θ is the joint prior wing a methodology given by Jeffrey (1961), which ca be presented by the following formula or π(θ 1 , θ 2 ) = π g (θ 1 , θ 2 ) is the joint prior when θ 1 and θ 2 have prior gamma distribution as in equation ( 16) and ( 17), respectively, which is then given as where: Where, . Note that all terms and derivatives written with a hat are calculated by replacing θ 1 , θ 2 , and R with their maximum likelihood estimators.

Simulation Study
In this section we perform a simulation study to: 1. Study the behavior of RMLE r,k by using different sample sizes.The average bias and average mean square error are computed.Also the average confidence length of the simulated 95% confidence intervals are computed.
2. We Lindley's approximation to compute RB r,k .Also average bias and average mean square error are computed.To study the behavior of RMLE r,k
2. For the given sample sizes and given parameters in (1) generate random samples from the Lindley distribution 3. Computep M LE ,q M LE and RMLE r,k .

Compute Bias and mean square error (MSE).
Note 1.To avoid the difficulty of computations, we take r = 1, k = 3 and perform the study for R 1,3 .
From Table 1, we can observe that the bias decreases as p decreases and q becomes fixed; it also decreases as q increases and p becomes fixed.Also, MSE decreases as the sample sizes increases.From Table 2, we can observe that the average confidence length decreases as p decreases and q becomes fixed; it also decreases as q increases and p becomes fixed.

Data Analysis
To decide whether the proposed model in the previous section can be used in practice, we consider two real data sets reported by Lawless (1982) and Proschan (1963).The first data set is obtained from Lawless (1982) and it represents the number of revolutions before the failure of 23 ball bearings in life tests, which are as follows: Data Set I: 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.44, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12,105.84, 127.92, 128.04, 173.40.The second data set is obtained from Proschan (1963) and represents the times between the successive failures of 15 air conditioning (AC) units in a Boeing 720 airplane, which are as follows: Data Set II:12,21,26,27,29,29,48,57,59,70,74,153,326,386,502.To estimate the stress strength model using the above data sets, we use the following steps: 1. Check the validity of the Lindley distribution for given data sets by using the Kolmogrov-Smirnov (K-S) test.
2. Find the maximum likelihood estimators for p and q.
3. Compute the maximum likelihood estimator of R r,k and asymptotic confidence interval.

Conclusions
In this paper, we have considered the problem of estimation reliability in a multicomponent stress-strength model R r,k = P [Y r:n1 < X k:n2 ] for which the stress and strength variables are given by a Lindley distribution.The three estimation methods of R r,k applied were the maximum likelihood, the uniformly minimum variance unbiased, and the Bayes estimators.By simulation we made a comparison between the maximum likelihood and Bayes estimators.In both estimators the mean square error decreases as sample sizes increases.Also, the maximum likelihood estimator has a mean square error that is less than the Bayes estimator, as can be seen in Table 1 and Table 3. Real data was used as a practical application of the proposed model.Finally we recommend that the Lindley distribution is used as the multicomonent stress-strength model.

Figure 2 :
Figure 2: Fitted Lindley distribution for data set I.

Figure 3 :
Figure 3: Fitted Lindley distribution for data set II.

Table 1 :
The average bias and average mean square error for different sample sizes for R1,3.
we can observe that the MSE of RMLE

Table 3 :
The average bias and average mean square error for different sample sizes for R1,3(Bayes estimator).

Table 4 :
The model fitting summary for both the data sets.

Table 5 :
Data analysis results.