Estimation of reliability in a multicomponent stress- strength model based on generalized linear failure rate distribution

Abstract

In this paper, we consider the problem of estimation reliability in multicomponent stress-strength model, when the system consists of $k$-components have strength are given by independently and identically distributed random variables $X_{1},...,X_{k}$ each component experiencing a random stress governed by a random variable $Y$. The reliability such system is obtained when strength and stress variables are given by a generalized linear failure rate distribution. The system is regarded as alive only if at least $s$ out of $k$ $(s<k)$ strength exceed the stress. The multicomponent reliability of the system is given by $R_{s,k}=P[$ at least $s$ of $X_{1},...,X_{k}$ exceed $Y]$. The maximum likelihood estimator $(MLE)$ and Bayes estimator of $R_{s,k}$ are obtained. A simulation study is performed to compare the different estimators of $R_{s,k}$. Real data is used as a practical application of the proposed procedure.

Keywords

• Generalized linear failure rate,stress-strength model,maximum likelihood method,Bayes estimator,simulation

References

1. Al-Mutairi, D. K., Ghitany, M. E. and Kundu, D. \textit{ Inferences on stress-strength reliability from Lindley distributions}, Commun. Statist. Theory Methods 42(8) , 1443-1463, 2013.
2. Bhattacharyya, G. K. and Johnson, R. A. Estimation of reliability in a multicomponent stress-strength model, J. Amer. Statist. Assoc., 69, 966-970, 1974.
3. Birnbaum, Z.W. On a use of the Mann-Whitney statistic. In: Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability, 1, 13-17, University of California Press, Berkeley, CA., 1956
4. Chen, M.H., Shao, Q.M., Ibrahim and J.G. Monte Carlo methods in Bayesian computation, Springer-Verlag, New York, 2000 .
5. Eryilmaz, S. Multivariate stress-strength reliability model and its evaluation for coherent structures, J. Multivariate Anal., 99, 1878-1887, 2008.
6. Gentle, JE. Random number generation and Monte Carlo methods, Springer, New York, 1998.
7. Ghitany, M. E., Al-Mutairi, D. K. and Aboukhamseen, S. M. Estimation of the reliability of a stress-strength system from power Lindley distributions, Commun. Statist. Simul. Comput., 44 ,118-136, 2015.
8. Jeffrey, H. Theory of probability, 3rd ed., Oxford University Press, 1961.

9. Kotz, S., Lumelskii, Y., and Pensky, M. The stress-strength model and its generalization: theory and applications, World Scientific, Singapore, 2003.
10. Krishnamoorthy, K., Mukherjee, S. and Guo, H. Inference on reliability in two-parameter exponential stress-strength model, Metrika, 6 ,261-273, 2007.
11. Kundu, D. and Gupta, R. D. Estimation of $P(X
12. Kundu, D. and Gupta, R. D. Estimation of p(Y
13. Kundu, D. and Raqab, M. Z. Estimation of $R = P(Y < X)$ for three-parameter Weibull distribution, Statist. Probability Lett., 79, 1839-1846, 2009.
14. Lindley, D. V. Approximation Bayesian methods, Trabajos de Estadistica, 21, 223-237, 1980.
15. Mudholkar, G.S., Srivastava, D.K. and Freimer, M. The exponentiated Weibull family; a reanalysis of the bus motor failure data, Technometrics, 37, 436 - 445, 1995.
16. Pakdaman, Z. and Ahmadi, J. Stress- Strength reliability for$P[X_{r:n_{1},k:n_{2}}]$ in exponential case, Journal of the Turkish statistical association, 6(3), 92-102, 2013.
17. Pandey, M., Uddin, M. B. and Ferdous, J. Reliability estimation of an s-out-of-k system with non-identical component strengths: the Weibull case, Reliab. Eng. Sys. Saf., 36,109-116.
18. Rao, G. S. a and Kantam,R. R. L. Estimation of reliability in multicomponent stress-strength model: Log-logistic distribution, Electron. J. Appl. Statist. Anal, 3(2), 75-84, 2010.
19. Rao, G. S. Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution, Colombian J.Statist., 35(1), 67-76, 2012.
20. Rao, G. S. and Kantam,R. R. L., Rosaiah, K. and Reddy, J. P. Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution, J. Statist. Appl. Probability, 3, 261-267, 2013.
21. Raqab, M. Z. and Kundu, D. Comparison of different estimators of $p(Y
22. Robert, C.P. and Casella, G. Monte Carlo statistical methods, Second Edition, Springer, NewYork , 2004.
23. Saracoglu, B., Kinaci, I. and Kundu, D. On estimation of $R = p(Y < X)$ for exponential distribution under progressive type-II censoring, Journal of Statistical Computation and Simulation, 82(5), 729-744, 2012.
24. Sarhan, A. and Kundu, D. Generalized linear failure rate distribution, Commun. Statist. Theory Methods, 38, 642 - 660, 2009.
25. Shahsanaei, F and Daneshkhah, A. Estimation of stress strength model in generalized linear failure rate distribution, ArXiv Preprint $1312.0401$ v1, 2013.
26. Xia, Z. P., Yu, J. Y. , Cheng, L. D. , Liu, L. F. and Wang, W. M. Study on the breaking strength of jute fibers using modified Weibull distribution, Journal of Composites Part A: Applied Science and Manufacturing, 40, 54-59, 2009.