Reliability Estimation in Multicomponent Stress-strength Model based on Generalized Pareto Distribution

Parameshwar V Pandit^1, and Shubhashree Joshi¹

¹Department of Statistics, Bangalore University, Bangalore -560056

Cite this paper

Parameshwar V Pandit and Shubhashree Joshi. Reliability Estimation in Multicomponent Stress-strength Model based on Generalized Pareto Distribution. American Journal of Applied Mathematics and Statistics. 2018; 6(5):210-217. doi: 10.12691/AJAMS-6-5-5

Abstract

The paper deals with the estimation of multicomponent system reliability where the system has k components with their strengths X₁, X₂, … X_k being independently and identically distributed random variables and each component is experiencing a random stress Y. The s-out-of-k system is said to function if atleast s out of k (1 ≤ s ≤ k) strength variables exceed the random stress. The reliability of such a system is derived when both strength and stress variables follow generalized Pareto distribution. The system reliability is estimated using maximum likelihood and Bayesian approaches. The maximum likelihood estimators are derived under both simple random sampling and ranked set sampling schemes. Lindley's approximation technique is used to get approximate Bayes estimators. The reliability estimators obtained from both the methods are compared by using mean squares error criteria and real data analysis is carried out to illustrate the procedure.

Keywords

generalized Pareto distribution, stress-strength reliability, ranked set sampling, simple random sampling, maximum likelihood estimator, Bayes estimator

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	Birnbaum, Z. W. (1956). ’On a use of Mann-Whitney statistics’. Proceeding Third Berkley Symposium on Mathematical Statistics and Probability, 1, 13-17.
[2]	Birnbaum, Z. W. and McCarty, B.C. (1958). ‘A distribution-free upper confidence bounds for Pr(Y < X) based on independent samples of X and Y’. The Annals of Mathematical Statistics, 29(2), 558-562.
[3]	Raqab, M. Z., Madi, M. T. and Kundu, D (2008). Estimation of P(Y < X) for the 3-parameter generalized exponential distribution. Communications in Statistics-Theory and Methods, 37(18), 2854-2864.
[4]	Wong, A. (2012). Interval estimation of P(Y < X) for generalized Pareto distribution. Journal of Statistical Planning and Inference, 142, 601-607.
[5]	Angali, K. A, Latifi S. M, Hanagal, D. D. (2014). Bayesian estimation of bivariate exponential distributions based on LINEX and Quadratic loss functions: a survival approach with censored samples. Communications in Statistics-Theory and Methods, 43(1), 31-44.
[6]	Bhattacharyya, G. K. and Johnson, R. A. (1974). Estimation of reliability in multicomponent stress-strength model. Journal of American Statistical Association, 69, 966-970.
[7]	W. Kuo and M. J. Zuo. (2003). Optimal Reliability Modeling, Principles and Applications. New York, NY, USA: Wiley.
[8]	G. S. Rao and R.R.L. Kantam. (2010). Estimation of reliability in multicomponent stress-strength model: Log-logistic distribution. Electronic Journal of Applied Statistical Analysis, 3(2), 75-84.
[9]	G. S. Rao. (2012). Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution. Colombian Journal of statistics, 35(1), 67-76.
[10]	G. S. Rao. (2012). Estimation of reliability in multicomponent stress-strength model based on Rayleigh distribution. Probability Statistics forum, 5, 150-161.
[11]	G. S. Rao (2014). Estimation of reliability in multicomponent stress-strength model based on generalized Rayleigh distribution. Journal of Modern Applied Statistical Methods, 13(1), 367-379.
[12]	G. S. Rao, R. R. L. Kantam, K. Rosaiah, and J. P. Reddy (2013). Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution. Journal of Statistics and Applied Probability, 3, 261-267.
[13]	G. S. Rao, Aslam, M and Kundu, D (2015). Burr Type XII distribution parametric estimation and estimation of reliability in multicomponent stress-strength. Communications in Statistics-Theory and Methods, 44(23), 4953-4961.
[14]	Pandit, P. V and Kantu, Kala, J (2013). System reliability estimation in multicomponent exponential stress-strength models. International Journal of Reliability and Applications, 14(2), 97-105.
[15]	Nadar, M. and Kizilaslan, F. (2015). ‘Classical and Bayesian estimation of Reliability in Multicomponent Stress-Strength Model based on Weibull distribution’, Revista Clombiana de Estadistica, 38(2), 467-484.
[16]	S. Rezaei, R. Tahmasbi and M. Mahmoodi (2010). Estimation of Pr(X > Y) for generalized Pareto distribution. Journal of Statistical Planning and Inference, 140, 480-494.
[17]	G. A. McIntyre (1952). A method for unbiased selective sampling using ranked sets. Australian Journal of Agricultural Research, 3, 385-390.
[18]	Kvam, P. H, Samaniego, F, J. (1993). On maximum likelihood estimation based on ranked set sampling with application to reliability. In Basu: A., ed, Advanced in Reliability. Amsterdam: North- Holland, 215-229.
[19]	Kvam, P. H, Samaniego, F, J. (1994). Nonparametric maximum likelihood estimation based on ranked set sampling. American Journal of statistical Association, 89, 526-537.
[20]	Abu-Dayyeh, W. A., Muttlak, H. A. (1996). Using ranked set sampling for testing hypothesis on the scale parameter for exponential and uniform distribution. Pakistan Journal of statistics, 12, 131-138.
[21]	H. A. Muttlak (1997). Median ranked set sampling. Journal of Applied Statistical Science, 6, 245-255.
[22]	Yu, P. L. H., Lam. K (1997). Regression estimator in ranked set sampling. Biometrics, 53, 1070-1081.
[23]	S. Sengupta and S. Mukhuti (2008). Unbiased estimation of Pr(X > Y) using ranked set sample data. Statistics, 42, 223-230.
[24]	H. A. Muttlak, W. A. Abu-Dayyah, M. F. Saleh, and E. Al-Sawi (2010). Estimating P(Y < X) using ranked set sampling in case of the exponential distribution. Communications in Statistics: Theory and Methods, 39, 1855-1868.
[25]	M. A. Hussian (2014). Estimation of stress-strength model for generalized inverted exponential distribution using ranked set sampling. International Journal of Advance in Engineering and Technology, 6, 2354-2362.
[26]	A. S. Hassan, S. M. Assar, and M. Yahya (2015). Estimation of P(Y < X) for Burr distribution under several Modifications for ranked set sampling. Australian Journal of Basic and Applied Sciences, 9(1), 124-140.
[27]	D. V. Lindley (1980). Approximate Bayes method. Trabajos de Estadistica, 3, 281-288.
[28]	Badar, M. G. & Priest, A. M. (1982). Statistical aspects of fibre and bundle strength in hybrid composites. In T. Hayashi, K. Kawata, and S. Umekawa (eds.), Progress in Science and Engineering Composites, (pp. 1129-1136). Tokyo: ICCM-IV.