Multi-component stress-strength model for Weibull distribution in progressively censored samples
-
Akram Kohansal
,
Shirin Shoaee
Abstract
One of the important issues is risk assessment and calculation in complex and multi-component systems. In this paper, the estimation of multi-component stress-strength reliability for the Weibull distribution under the progressive Type-II censored samples is studied. We assume that both stress and strength are two independent Weibull distributions with different parameters. First, assuming the same shape parameter, the maximum likelihood estimation (MLE), different approximations of Bayes estimators (Lindley’s approximation and Markov chain Monte Carlo method) and different confidence intervals (asymptotic and highest posterior density) are obtained. In the case when the shape parameter is known, the MLE, uniformly minimum variance unbiased estimator (UMVUE), exact Bayes estimator and different confidence intervals (asymptotic and highest posterior density) are considered. Finally, in the general case, the statistical inferences on multi-component stress-strength reliability are derived. To compare the performances of different methods, Monte Carlo simulations are performed. Moreover, one data set for illustrative purposes is analyzed.
A Appendix
So the log-likelihood function from (2.2) is
The MLEs of 𝜃, 𝜆 and 𝛼, say
References
[1] N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications, Statistics for Industry and Technology, Birkhäuser, Boston, 2000. 10.1007/978-1-4612-1334-5Search in Google Scholar
[2] G. K. Bhattacharyya and R. A. Johnson, Estimation of reliability in a multicomponent stress-strength model, J. Amer. Statist. Assoc. 69 (1974), 966–970. 10.1080/01621459.1974.10480238Search in Google Scholar
[3] J. H. Cao and K. Cheng, An Introduction to the Reliability Mathematics, Higher Education, Beijing, 2006. Search in Google Scholar
[4] M.-H. Chen and Q.-M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Statist. 8 (1999), no. 1, 69–92. 10.1080/10618600.1999.10474802Search in Google Scholar
[5] S. Dey, J. Mazucheli and M. Z. Anis, Estimation of reliability of multicomponent stress-strength for a Kumaraswamy distribution, Comm. Statist. Theory Methods 46 (2017), no. 4, 1560–1572. 10.1080/03610926.2015.1022457Search in Google Scholar
[6] D. D. Hanagal, Estimation of reliability of a component subjected to bivariate exponential stress, Statist. Papers 40 (1999), no. 2, 211–220. 10.1007/BF02925519Search in Google Scholar
[7] F. Kizilaslan and M. Nadar, Classical and Bayesian estimation of reliability in multicomponent stress-strength model based on Weibull distribution, Rev. Colombiana Estadíst. 38 (2015), no. 2, 467–484. 10.15446/rce.v38n2.51674Search in Google Scholar
[8] F. Kızılaslan and M. Nadar, Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times, J. Stat. Comput. Simul. 86 (2016), no. 12, 2471–2493. 10.1080/00949655.2015.1119832Search in Google Scholar
[9] F. Kızılaslan and M. Nadar, Estimation of reliability in a multicomponent stress-strength model based on a bivariate Kumaraswamy distribution, Statist. Papers 59 (2018), no. 1, 307–340. 10.1007/s00362-016-0765-8Search in Google Scholar
[10] D. V. Lindley, Approximate Bayesian methods, Trabajos de Estadistica 3 (1980), 281–288. 10.1007/BF02888353Search in Google Scholar
[11] M. Modarres, M. P. Kaminskiy and V. Krivtsov, Reliability Engineering and Risk Analysis: A Practical Guide, CRC Press, Boca Raton, 2016. 10.1201/9781315382425Search in Google Scholar
[12] M. Nadar, A. Papadopoulos and F. Kızılaslan, Statistical analysis for Kumaraswamy’s distribution based on record data, Statist. Papers 54 (2013), no. 2, 355–369. 10.1007/s00362-012-0432-7Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
