Reliability Estimators for the Components of Series and Parallel Systems: The Weibull Model

Felipe L. Bhering, Carlos A. de B. Pereira, Adriano Polpo

Department of Statistics, Federal University of S?o Carlos, S?o Carlos, Brazil.
Department of Statistics, University of S?o Paulo, S?o Paulo, Brazil.

DOI: 10.4236/am.2014.511157                

Abstract

This paper presents a hierarchical Bayesian approach to the estimation of components’ reliability (survival) using a Weibull model for each of them. The proposed method can be used to estimation with general survival censored data, because the estimation of a component’s reliability in a series (parallel) system is equivalent to the estimation of its survival function with right- (left-) censored data. Besides the Weibull parametric model for reliability data, independent gamma distributions are considered at the first hierarchical level for the Weibull parameters and independent uniform distributions over the real line as priors for the parameters of the gammas. In order to evaluate the model, an example and a simulation study are discussed.

Keywords

Bayesian Analysis, Hierarchical Model, Left Censored Data, Right Censored Data

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Kaplan, E.L. and Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481. http://dx.doi.org/10.1080/01621459.1958.10501452
[2]	Salinas-Torres, V., Pereira, C.A.B. and Tiwari, R. (2002) Bayesian Nonparametric Estimation in a Series System or a Competing Risks Model. Journal of Nonparametric Statistics, 14, 449-458. http://dx.doi.org/10.1080/10485250213114
[3]	Polpo, A. and Sinha, D. (2011) Correction in Bayesian Nonparametric Estimation in a Series System or a Competing-Risks Model. Statistics & Probability Letters, 810, 1756-1759. http://dx.doi.org/10.1016/j.spl.2011.07.023
[4]	Rodrigues, A.S., Dias, T.C.M., Lauretto, M. and Polpo, A. (2012) Reliability Analysis in Series Systems: An Empirical Comparison between Bayesian and Classical Estimators. 31st International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Waterloo, AIP Conference Proceedings, 1443, 214-221.
[5]	Polpo, A. and Pereira, C.A.B. (2009) Reliability Nonparametric Bayesian Estimation in Parallel Systems. IEEE Transactions on Reliability, 580, 364-373. http://dx.doi.org/10.1109/TR.2009.2019493
[6]	Bhattacharya, D. and Samaniego, F.J. (2010) Estimating Component Characteristics from System Failure-Time Data. Naval Research Logistics (NRL), 570, 380-389.
[7]	Polpo, A., Coque, M.A. and Pereira, C.A.B. (2009) Statistical Analysis for Weibull Distributions in Presence of Right and Left Censoring. The Proceedings of 8th International Conference on Reliability, Maintainability and Safety (ICRMS), 1, 219-223.
[8]	Bhering, F., Polpo, A. and Pereira, C.A.B. (2012) A Hierarchical Weibull Bayesian Model for Series and Parallel Systems. AIP Conference Proceedings, 1490, 59-66. http://dx.doi.org/10.1063/1.4759589
[9]	McLachlan, G.J. and Krishnan, T. (2008) The EM Algorithm and Extensions. John Wiley & Sons, Hoboken. http://dx.doi.org/10.1002/9780470191613