STRESS-STRENGTH RELIABILITY for P(Xr:n1}<Yk:n2) in the EXPONENTIAL CASE

Abstract

This paper deals with the estimation problem of the multicomponent stress-strength reliability parameter when  stress, strength variates are given by two independent one-parameter exponential distributions with different parameters. It is  assumed that Y₁,...,Y_n2 are the random strengths of n₂ components  subjected to random stresses X₁,...,X_n1. Our study is concentrated on  the probability P(X_r:n1<Y_k:n2) and   the problem of frequentist and Bayesian estimation of   P(X_r:n1<Y_k:n2)  based on  X and Y-samples are  discussed.  Some special cases are considered and the small sample comparison of the reliability estimates is made through Monte Carlo simulation.

Keywords

• Stress-strength reliability,Squared error loss function,Uniformly minimum variance unbiased estimator,Maximum likelihood estimator,Parallel and series systems

References

1. Abramowitz, M. and Stegun, I. A. (1992), Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Reprint of the 1972 edition, Dover Publications, New York.
2. Ahmad, K. E., Fakhry, M. E. and Jaheen Z. F. (1997), Empirical Bayes estimation of P(X < Y ) and characterization of Burr type-X model, Journal of Statistical Planing and Inference, 64, 297–308.
3. Basu, D. (1955), On statistics independent of a complete sufficient statistic, Sankhya, 15, 377–380.
4. Bhattacharyya, G. K. and Johnson, R. A.(1974), Estimation of reliability in a multicomponent stressstrength model, Journal of the American Statistical Association, 69, 966–970.
5. Chao, A. (1982), On comparing estimators of P(X < Y ) in the exponential case, IEEE Transactions on Reliability, 31, 389–392.
6. David, H. A. and Nagaraja, H. N. (2003), Order Statistics, John Wiley & Sons, New York.
7. DasGupta, A. (2008), Asymptotic Theory of Statistics and Probability, Springer, New York.
8. Enis, P. and Geisser, S. (1971), Estimation of probability that Y < X, Journal of the American Statistical Association, 66, 162–168.

9. Eryilmaz, S. (2008a), Consecutive k-out-of-n: G system in stress strength set up, Communication in Statistics-Simulation and Computation, 37, 579–589.
10. Eryilmaz, S. (2008b), Multivariate stress-strength reliability model and its evaluation for coherent structures, Journal of Multivariate Analysis, 99, 1878–1887.
11. Eryilmaz, S. (2010), On system reliability in stress-strength setup, Statistics and Probability Letters, 80, 834–839.
12. Kelley, G. D., Kelley, J. A. and Schuncandy, W. R. (1976), Efficient estimation of P(X < Y ) in the exponential case, Technometrics, 18, 395–404.
13. Kotz, S., Lumelskii, Y., Pensky, M. (2003) The Stress-Strength Model and its Generalizations: Theory and Applications, World Scientific, Singapore.
14. Kundu, D. and Gupta, R. D. (2005), Estimation of P(X < Y ) for the generalized exponential distribution, Metrika, 61, 291–308.
15. Lindley, D. V. (1980), Approximation Bayesian methods, Trabajos de Estadistica, 21, 223–237
16. Pandey, M., Uddin, M. B. and Ferdous, J. (1992), Reliability estimation of an s-out-of-k system with non-identical component strengths: the Weibull case, Reliability Engineering and System Safety, 36, 109–116.
17. Rao, C. R. (1973), Linear Statistical Inference and Its Applications, John Wiley and Sons, New York.
18. Reiser, B. and Guttman, I. (1987), A comparison of three estimators for P(Y < X) in the normal case, Computational Statistics and Data Analysis, 5, 59–66.
19. Sara¸coˇglu, B. and Kaya, M. F. (2007), Maximum likelihood estimation and confidence intervals of system reliability for Gompertz distribution in stress strength models, Sel¸cuk Journal of Applied Mathematics, 8, 25–36.
20. Shah, S. P. and Sathe, Y. S. (1981), On estimating P(X > Y ) for the exponential distribution, Communication in Statistics-Theory and Methods, 10, 39–47.
21. Tong, H. (1974), A note on the estimation of P(X < Y ) in the exponential case, Technometrics, 16, 617–625.