Published
Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution
Estimación clásica y bayesiana de la confiabilidad de un modelo estrés-fuerza basado en la distribución Weibull
DOI:
https://doi.org/10.15446/rce.v38n2.51674Keywords:
Stress-Strength Model, System Reliability, Weibull Distribution (en)DistribuciónWeibull, Modelo estrés-fuerza, Sistema de confiabilidad. (es)
In this study, we consider a multicomponent system which has k independent and identical strength components X1,...,Xk and each component is exposed to a common random stress Y when the underlying distributions are Weibull. The system is regarded as operating only if at least s out of k (1 ≤ s ≤ k) strength variables exceeds the random stress. We estimate the reliability of the system by using frequentist and Bayesian approaches. The Bayes estimate of the reliability has been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms. The asymptotic confidence interval and the highest probability density credible interval are constructed for the reliability. The comparison of the reliability estimators is made in terms of the estimated risks by the Monte Carlo simulations.
En este estudio, consideramos un sistema multicomponente con k componentes de fuerza independientes y cada componente expuesto a un estrés común aleatorio Y cuando seconsidera una distribución Bernoulli. El sistema se considera operativo si por lo menos s de los k (1 s k) exceden el estrés aleatorio. Se estima la confiabilidad del sistema usando métodos bayesianos y frecuentistas. La estimación de Bayes de la confiabilidad ha sido desarrollada usando una aproximación de Lindley y métodos MCMC debido a la falta de formas explícitas. El intervalo de confianza asintótico y el intervalo de la densidad deprobabilidad más alta se construyen para la confiabilidad. La comparación de los estimadores de confiabilidad se hace en término de los riesgos estimados por medio de simulaciones Monte Carlo.
References
Bhattacharyya, G. K. & Johnson, R. A. (1974), ‘Estimation of reliability in multicomponent stress-strength model’, Journal of the American Statistical Association 69, 966–970.
Birnbaum, Z. W. (1956), ‘On a use of Mann-Whitney statistics’, Proceeding Third Berkeley Symposium on Mathematical Statistics and Probability 1, 13–17.
Birnbaum, Z. W. & 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.
Chen, M. H. & Shao, Q. M. (1999), ‘Monte Carlo estimation of Bayesian credible and HPD intervals’, Journal of Computational and Graphical Statistics 8(1), 69–92.
Eryilmaz, S. (2008), ‘Multivariate stress-strength reliability model and its evaluation for coherent structures’, Journal of Multivariate Analysis 99, 1878–1887.
Eryilmaz, S. (2010), ‘On system reliability in stress-strength setup’, Statistics and Probability Letters 80, 834–839.
Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. (2003), Bayesian Data Analysis, 2 edn, Chapman & Hall, London.
Gradshteyn, I. S. & Ryzhik, I. M. (1994), Table of Integrals, Series and Products, fifth edn, Academic Press, Boston.
Hanagal, D. D. (1999), ‘Estimation of system reliability’, Statistical Papers 40, 99–106.
Hanagal, D. D. (2003), ‘Estimation of system reliability in multicomponent series stress-strength models’, Journal of Indian Statistical Association 41, 1–7.
Jae, J. K. & Eun, M. K. (1981), ‘Estimation of reliability in a multicomponent stress-strength model in Weibull case’, Journal of the Korean Society for Quality Management 9(1), 3–11.
Kotz, S., Lumelskii, Y. & Pensky, M. (2003), The Stress-Strength Model and its Generalizations: Theory and Applications, World Scientific, Singapore.
Kundu, D. & Gupta, R. D. (2005), ‘Estimation of P(Y < X) for generalized exponential distribution’, Metrika 61, 291–308.
Kundu, D. & Gupta, R. D. (2006), ‘Estimation of P(Y < X) for Weibull distribution’, IEEE Transactions on Reliability Analysis 52(2), 270–280.
Kundu, D. & Raqab, M. Z. (2009), ‘Estimation of R = P(Y < X) for threeparameter Weibull distribution’, Statistics and Probability Letters 79, 1839–1846.
Kuo, W. & Zuo, M. J. (2003), Optimal Reliability Modeling, Principles and Applications, John Wiley & Sons, New York.
Lindley, D. V. (1980), ‘Approximate Bayes method’, Trabajos de Estadistica 3, 281–288.
Nadar, M., Kizilaslan, F. & Papadopoulos, A. (2014), ‘Classical and Bayesian estimation of P(Y < X) for Kumaraswamy’s distribution’, Journal of Statistical Computation and Simulation 84(7), 1505–1529.
Rao, C. R. (1965), Linear Statistical Inference and Its Applications, John Wiley & Sons, New York.
Rao, G. S. (2012a), ‘Estimation of reliability in multicomponent stress-strength model based on generalized exponential distribution’, Revista Colombiana de Estadística 35(1), 67–76.
Rao, G. S. (2012b), ‘Estimation of reliability in multicomponent stress-strength model based on generalized inverted exponential distribution’, International Journal of Current Research and Review 4(21), 48–56.
Rao, G. S. (2012c), ‘Estimation of reliability in multicomponent stress-strength model based on Rayleigh distribution’, ProbStat Forum 5, 150–161.
Rao, G. S., Aslam, M. & Kundu, D. (2014), ‘Burr Type XII distribution parametric estimation and estimation of reliability in multicomponent stress-strength model’, Communication in Statistics-Theory and Methods 1.
Rao, G. S. & Kantam, R. R. L. (2010), ‘Estimation of reliability in multicomponent stress-strength model: log-logistic distribution’, Electronic Journal of Applied Statistical Analysis 3(2), 75–84.
Rao, G. S., Kantam, R. R. L., Rosaiah, K. & Reddy, J. P. (2013), ‘Estimation of reliability in multicomponent stress-strength model based on inverse Rayleigh distribution’, Journal of Statistics Applications & Probability 3, 261–267.
Tierney, L. (1994), ‘Markov chains for exploring posterior distributions’, The Annals of Statistics 22(4), 1701–1728.
Dimensions
PlumX
Article abstract page views
Downloads
How to Cite
License
Copyright (c) 2015 Revista Colombiana de Estadística
This work is licensed under a Creative Commons Attribution 4.0 International License.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
