IJCRR - 4(21), November, 2012
Pages: 48-56
Date of Publication: 15-Nov-2012
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ESTIMATION OF RELIABILITY IN MULTICOMPONENT STRESS-STRENGTH BASED ON GENERALIZED INVERTED EXPONENTIAL DISTRIBUTION
Author: G. Srinivasa Rao
Category: General Sciences
Abstract:In this paper, we are mainly interested in estimating multicomponent stress- strength reliability. The system is regarded as alive only if at least s out of k (s such a system is obtained when strength, stress variates are given by generalized inverted exponential distribution with different shape parameters and common scale parameter. The research methodology adopted here is to estimate the parameters by using maximum likelihood estimation. The reliability is estimated using the maximum likelihood method of estimation when samples are drawn from strength and stress distributions. The reliability estimators are compared asymptotically. The results of small sample comparison of the reliability estimates are made through Monte-Carlo simulation. The simulation results indicates that the average bias and average MSE are decreases as sample size increases for both methods of estimation in reliability. The length of the confidence interval is also decreases as the sample size increases and coverage probability is close to the nominal value in all sets of parameters considered here. By using real data sets we well illustrate the procedure.
Keywords: Generalized inverted exponential distribution, reliability estimation, stress- strength, ML estimation, confidence intervals.
Full Text:
INTRODUCTION
Recently the two-parameter generalized inverted exponential distribution (GIED) has been proposed and studied extensively by the Abouammoh and Alshingiti (2009). The GIED has the following density function
where 1 2 , ,... X X Xk are identically indipendently distributed (iid) with common distribution function F x( ) and subjected to the common random stress Y. The probability in (4) is called reliability in a multicomponent stress-strength model [Bhattacharyya and Johnson (1974)]. The survival probabilities of a single component stressstrength version have been considered by several authors assuming various lifetime distributions for the stress-strength random variates. Enis and Geisser (1971), Downtown (1973), Awad and Gharraf (1986), McCool (1991), Nandi and Aich (1994), Surles and Padgett (1998), Raqab and Kundu (2005), Kundu and Gupta (2005& 2006), Raqab et al (2008), Kundu and Raqab (2009). The reliability in a multicomponent stress-strength was developed by Bhattacharyya and Johnson (1974), Pandey and Borhan Uddin (1985). Recently Rao and Kantam (2010) studied estimation of reliability in multicomponent stress-strength for the log-logistic distribution and Rao (2012) developed an estimation of reliability in multicomponent stress-strength based on generalized exponential distribution. Suppose a system, with k identical components, functions if at least s s k (1 ) ? ? components simultaneously operate. In its operating environment, the system is subjected to a stress Y which is a random variable with distribution function G(.) . The strengths of the components, that is the minimum stresses to cause failure, are independently and identically distributed random variables with distribution function F(.) . Then, the system reliability, which is the probability that the system does not fail, is the function Rsk, given in (4). The estimation of survival probability in a multicomponent stress-strength system when the stress and strength variates follow GIED is not paid much attention. Therefore, an attempt is made here to study the estimation of reliability in multicomponent stress-strength model with reference to GIED. The rest of the paper is organized as follows. In Section 2, we discussed the research methodology for expression of Rsk, and develop a procedure for estimating it. More specifically, we obtain the maximum likelihood estimates of the parameters. The MLE are employed to obtain the asymptotic distribution and confidence intervals for Rsk, . The results of small sample comparisons made through Monte Carlo simulations are in Section 3. Also, using real data, we illustrate the estimation process. Finally, the conclusion and comments are provided in Section 4.
The probability in (5) is called reliability in a multicomponent stress-strength model. If ? ? and are not known, it is necessary to estimate ? ? and to estimate Rsk, . In this paper we estimate ? ? and by ML method. The estimates are substituted in ? to get an estimate of Rsk, using equation (5). The theory of methods of estimation is explained below. It is well known that the method of maximum likelihood estimation (MLE) has invariance property. We have proposed ML estimator for the reliability of multicomponent stress-strength model by considering the estimators of the parameters of stress and strength distributions by ML method of estimation in GIED. Let X X Y 1 2 n 1 2 m ? ? ? <....
RESULTS AND DATA ANALYSIS
Results from simulation study In this sub section we present some results based on Monte-Carlo simulations to compare the performance of the Rsk, using for different sample sizes. 3000 random sample of size 10(5)35 each from stress and strength populations are generated for ( , ) ? ? =(3.0,1.0), (2.5,1.0), (2.0,1.0), (1.5,1.0), (1.0,1.0), (1.0,1.5),(1.0,2.0), (1.0,2.5) and (1.0,3.0) as proposed by of Bhattacharyya and Johnson (1974). The MLE of scale parameter ? is estimated by iterative method and using ? the shape parameters ? ? and are estimated from (10) and (11). These ML estimators of ? ? and are then substituted in ? to get the multicomponent reliability for (s, k) = (1, 3), (2, 4). The average bias and average mean square error (MSE) of the reliability estimates over the 3000 replications are given in Tables 1 and 2. Average confidence length and coverage probability of the simulated 95% confidence intervals of Rsk, are given in Tables 3 and 4. The true value of reliability in multicomponent stressstrength with the given combinations of ( , ) ? ? for (s, k) = (1, 3) are 0.900, 0.882, 0.857, 0.818, 0.750, 0.667, 0.600, 0.545, 0.500 and for (s, k) = (2, 4) are 0.831, 0.802, 0.762, 0.701, 0.600, 0.485, 0.400, 0.336, 0.286. Thus the true value of reliability in multicomponent stress-strength decreases as ? increases for a fixed ? whereas reliability in multicomponent stress-strength increases as ? increases for a fixed ? in both the cases of (s, k). Therefore, the true value of reliability is decreases as ? increases and vice versa. The average bias and average MSE are decreases as sample size increases for both cases of estimation in reliability. Also the bias is negative in most of the combinations of the parameters in both situations of (s, k). It verifies the consistency property of the MLE of Rsk, . Whereas, among the parameters the absolute bias and MSE are increases as ? increases for a fixed ? in both the cases of (s, k) and the absolute bias and MSE are decreases as ? increases for a fixed ? in both the cases of (s, k). The length of the confidence interval is also decreases as the sample size increases. The coverage probability is close to the nominal value in all cases but less than 0.95. Overall, the performance of the confidence interval is quite good for all combinations of parameters. Whereas, among the parameters we observed the same phenomenon for average length and average coverage probability that we observed in case of average bias and MSE. Data Analysis In this sub section we analyze two real data sets and demonstrate how the proposed methods can be used in practice. The first data set reported by Lawless (1982) and second data set given by Fuller et al (1994) and these data sets are analyzed and fitted for various lifetime distributions. We fit the generalized inverted exponential distribution to the two data sets separately. The first data set (Lawless (1982); page 228) presented here arose in tests on endurance of deep groove ball bearings. The data presented are the number of million revolutions before failure for each of the 23 ball bearings in the life test and they are (Y): 17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04 and 173.40. The second data set was given by Fuller et al (1994) represents the data to predict the lifetime for a glass airplane window. The data are as follows (X): 18.83, 20.8, 21.657, 23.03, 23.23, 24.05, 24.321, 25.5, 25.52, 25.8, 26.69, 26.77, 26.78, 27.05, 27.67, 29.9, 31.11, 33.2, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, 44.045, 45.29, and 45.381. Abouammoh and Alshingiti (2009) studied the validity of the model for both data sets and they showed that by using different fitting procedures GIED fits quite well for both the data sets. We use the iterative procedure to obtain the MLE of ? using (12) and MLEs of ? ? and are obtained by substituting MLE of ? in (10) and (11). The final estimates for real data sets are ? ˆ = 75.047698, ˆ ? = 6.145771 and ˆ ? =141.565031. Base on estimates of ? ? and the MLE of Rsk, become 1,3 R ˆ =0.973428 and 2,4 R ˆ = 0.953899. The 95% confidence intervals for R1,3 become (0.959478, 0.987378) and for R2,4 become (0.929909, 0.977889).
CONCLUSIONS
In this paper, we have studied the multicomponent stress-strength reliability for generalized inverted exponential distribution when both of stress, strength variates follows the same population. Also, we have estimated asymptotic confidence interval for multicomponent stress-strength reliability. The simulation results indicates that the average bias and average MSE are decreases as sample size increases for both methods of estimation in reliability. Among the parameters the absolute bias and MSE are increases (decreases) as ? increases ( ? increases) in both the cases of (s, k). The length of the confidence interval is also decreases as the sample size increases and coverage probability is close to the nominal value in all sets of parameters considered here. Using real data, we illustrate the estimation process.
ACKNOWLEDGEMENT
Authors acknowledge the immense help received from the scholars whose articles are cited and included in references of this manuscript. The authors are also grateful to authors / editors /publishers of all those articles, journals and books from where the literature for this article has been reviewed and discussed.
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