Brazilian Journal of Probability and Statistics

Hierarchical modelling of power law processes for the analysis of repairable systems with different truncation times: An empirical Bayes approach

Rodrigo Citton P. dos Reis, Enrico A. Colosimo, and Gustavo L. Gilardoni

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In the data analysis from multiple repairable systems, it is usual to observe both different truncation times and heterogeneity among the systems. Among other reasons, the latter is caused by different manufacturing lines and maintenance teams of the systems. In this paper, a hierarchical model is proposed for the statistical analysis of multiple repairable systems under different truncation times. A reparameterization of the power law process is proposed in order to obtain a quasi-conjugate bayesian analysis. An empirical Bayes approach is used to estimate model hyperparameters. The uncertainty in the estimate of these quantities are corrected by using a parametric bootstrap approach. The results are illustrated in a real data set of failure times of power transformers from an electric company in Brazil.

Article information

Braz. J. Probab. Stat., Volume 33, Number 2 (2019), 374-396.

Received: July 2015
Accepted: January 2018
First available in Project Euclid: 4 March 2019

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Zentralblatt MATH identifier

Bootstrap correction maximum a posterior density minimal repair multiple repairable systems rejection sampling reliability


dos Reis, Rodrigo Citton P.; Colosimo, Enrico A.; Gilardoni, Gustavo L. Hierarchical modelling of power law processes for the analysis of repairable systems with different truncation times: An empirical Bayes approach. Braz. J. Probab. Stat. 33 (2019), no. 2, 374--396. doi:10.1214/18-BJPS393.

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  • Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. New York: Springer.
  • Arab, A., Rigdon, S. E. and Basu, A. P. (2012). Bayesian inference for the piecewise exponential model for the reliability of multiple repairable systems. Journal of Quality Technology 44, 28–38.
  • Baker, R. D. (1996). Some new tests of the power law process. Technometrics 38, 256–265.
  • Barlow, R. and Hunter, L. (1960). Optimum preventive maintenance policies. Operations Research 8, 90–100.
  • Bhattacharjee, M., Arjas, E. and Pulkkinen, U. (2003). Modelling heterogeneity in nuclear power plant valve failure data. In Mathematical and Statistical Methods for Reliability (B. H. Lindqvist and K. A. Doksum, eds.) 341–353. World Scientific Publishing.
  • Carlin, B. P. and Gelfand, A. E. (1990). Approaches for the empirical Bayes confidence intervals. Journal of the American Statistical Association 85, 105–114.
  • Casella, G. (1985). An introduction to empirical Bayes data analysis. American Statistician 39, 83–87.
  • Cook, R. J. and Lawless, J. F. (2007). The Statistical Analysis of Recurrent Events. Statistics for Biology and Health. New York: Springer.
  • Devroye, L. (1986). Non-Uniform Random Variate Generation. New York: Springer.
  • Gaver, D. P. and O’Muircheartaigh, I. G. (1987). Robust empirical Bayes analyses of event rates. Technometrics 29, 1–15.
  • Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (2003). Bayesian Data Analysis. New York: Chapman and Hall.
  • George, E. I., Makov, U. E. and Smith, A. F. M. (1993). Conjugate likelihood distributions. Scandinavian Journal of Statistics 20, 147–156.
  • Gilardoni, G. L. and Colosimo, E. A. (2007). Optimal maintenance time for repairable systems. Journal of Quality Technology 39, 48–53.
  • Gilardoni, G. L. and Colosimo, E. A. (2011). On the superposition of overlapping Poisson processes and nonparametric estimation of their intensity function. Journal of Statistical Planning and Inference.
  • Gilardoni, G. L., Oliveira, M. D. and Colosimo, E. A. (2013). Nonparametric estimation and bootstrap confidence interval for the optimal maintenance time of a repairable system. Computational Statistics & Data Analysis 63, 113–124.
  • Giorgio, M., Guida, M. and Pulcini, G. (2014). Repairable system analysis in presence of covariates and random effects. Reliability Engineering & Systems Safety 131, 271–281.
  • Guida, M., Calabria, R. and Pulcini, G. (1989). Bayes inference for a non-homogeneous Poisson process with power intensity law. IEEE Transactions on Reliability 38, 603–609.
  • Guida, M. and Pulcini, G. (2005). Bayesian reliability assessment of repairable systems during multi-stage development programs. IIE Transactions 37, 1071–1081.
  • Hamada, M. S., Wilson, A. G., Reese, C. S. and Martz, H. F. (2008). Bayesian Reliability. Springer Series in Statistics. New York: Springer.
  • Huang, Y.-S. (2001). A decision model for deteriorating repairable systems. IIE Transactions 33, 479.
  • Kass, R. E. and Steffey, D. (1989). Approximate Bayesian inference in conditionally independent hierarchical models (parametric empirical Bayes models). Journal of the American Statistical Association 84, 717–726.
  • Laird, N. M. and Louis, T. A. (1987). Empirical Bayes confidence intervals based on bootstrap samples. Journal of the American Statistical Association 82, 739–750.
  • Lawless, J. F. (1987). Regression methods for Poisson process data. Journal of the American Statistical Association 82, 808–815.
  • Lindqvist, B. H. (2006). On the statistical modeling and analysis of repairable systems. Statistical Science 21, 532–551.
  • Lindqvist, B. H., Elvebakk, G. and Heggland, K. (2003). The trend-renewal process for statistical analysis of repairable systems. Technometrics 45, 31–44.
  • Mazzuchi, T. A. and Soyer, R. (1996). A Bayesian perspective on some replacement strategies. Reliability Engineering & Systems Safety 51, 295–303.
  • Morris, C. N. (1983). Parametric empirical Bayes inference: Theory and applications. Journal of the American Statistical Association 78, 47–55.
  • Oliveira, M. D., Colosimo, E. A. and Gilardoni, G. L. (2012). Bayesian inference for power law processes with applications in repairable systems. Journal of Statistical Planning and Inference 142, 1151–1160.
  • Pan, R. and Rigdon, S. E. (2009). Bayes inference for general repairable systems. Journal of Quality Technology 41, 82–94.
  • Pérez, C. J., Martín, J. and Rufo, M. J. (2006). Sensitivity estimations for Bayesian inference models solved by MCMC methods. Reliability Engineering & Systems Safety 91, 1310–1314.
  • Perkins, S., Cohen, M., Rahme, E. and Bernatsky, S. (2012). Melanoma and rheumatoid arthritis (brief report). Clinical Reumathology 31, 1001–1003.
  • Pesaran, M. H., Pettenuzzo, D. and Timmermann, A. (2006). Forecasting time series subject to multiple structural breaks. The Review of Economic Studies 73, 1057–1084.
  • R Core Team (2013). R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
  • Rigdon, S. E. and Basu, A. P. (2000). Statistical Methods for the Reliability of Repairable Systems. New York: John Wiley & Sons.
  • Ryan, K. J., Hamada, M. S. and Reese, C. S. (2011). A Bayesian hierarchical power law process model for multiple repairable systems with an application to supercomputer reliability. Journal of Quality Technology 43, 209–223.
  • Schilling, R. L. (2005). Measures, Integrals and Martingales. Cambridge: Cambridge University Press.