Abstract
The prediction of the unobserved units is typically based on the derivations of the predictive distributions of the individual observations. This technique is of little interest when one wishes to predict a function of missing or unobserved data such as the remaining testing time. In this article, based on a progressive type-II censored sample from the generalized Pareto (GP) distribution, we consider the problem of predicting times to failure of units in multiple stages. Importance sampling is used to estimate the model parameters, and Gibbs and Metropolis samplers are used to predict the testing times of the removed unfailed units. Data analyses involving the water-level exceedances by the River Nidd in North Yorkshire, England, have been performed and predictions of the total remaining level exceedances are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahsanullah, M. 1992. Inference and prediction problems of the generalized Pareto distribution based on record values. In Order statistics and nonparametrics: Theory and applications, ed. P.K. Sen and L. A. Salama, 47–57. Amsterdam, The Netherlands: Elsevier.
Asgharzadeh, A., M. Mohammadpour, and Z. M. Ganji. 2014. Estimation and reconstruction based on left censored data from Pareto model. Journal of the Iranian Statistical Society 13 (2):151–75.
Asgharzadeh, A., R. Valiollahi, and M. Z. Raqab. 2011. Stress-strength reliability of Weibull distribution based on progressively censored samples. Statistics & Operations Research Transactions 35 (2):103–24.
Balakrishnan, N. and R. Aggrawala. 2000. Progressive censoring, theory, methods and applications. Boston, MA: Birkhauser.
Balakrishnan, N., N. Kannan, C. T. Lin, and S. J. S. Wu. 2004. Inference for the extreme value distribution under progressive type-II censoring. Journal of Statistical Computation and Simulation 25:25–45.
Basak, I., P. Basak, and N. Balakrishnan. 2006. On some predictors of times to failures of censored items in progressively censored sample. Computational Statistics and Data Analysis 50:1313–37.
Bermudez, P. Z., and M. A. Turkman. 2003. Bayesian approach to parameter estimation of the generalized Pareto distribution. Test 12 (1):259–77.
Casella, G., and R. L. Berger. 2002. Statistical inference, 2nd ed. Pacific Grove, CA: Duxbury.
Castillo, E., and J. Daoudi. 2009. Estimation of the generalized Pareto distribution. Statistics and Probability Letters 79:684–88.
Castillo, E., A. S. Hadi, N. Balakrishnan, and J. M. Sarabia. 2004. Extreme value and related models with applications in engineering and science. Hoboken, NJ: Wiley.
David, H. A., and H. N. Nagaraja. 2003. Order statistics, 3rd ed. Hoboken, NJ: John Wiley & Sons.
Davison, A. C. 1984. Modelling excesses over high thresholds, with an application. In Statistical Extremes and Applications, ed. J. Tiago de Oliveira and D. Reidel, 461–482. Dordrecht.
Davison, A. C., and R. L. Smith. 1990. Models for exceedances over high thresholds. Journal of Royal Statistical Society B 52:393–442.
Gilks, W. R., S. Richardson, and D. J. Spiegelhalter, 1995. Markov chain Monte Carlo in practise. London, UK: Chapman & Hall.
Fligner, M. A., and D. A. Wolfe. 1979. Nonparametric prediction intervals for a future sample median. Journal of the American Statistical Association 74:453–56.
Freitas, N. D., P. Højen-Sørensen, M. Jordan, and S. Russell. 2001. Variational MCMC. Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence 70:120–27.
Hosking, J. R. M., and J. R. Wallis. 1987. Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29:339–49.
Kaminsky, K. S., and P. I. Nelson. 1998. Prediction of order statistics. In Handbook of statistics, order statistics: Applications, Vol. 17, ed. N. Balakrishnan and C. R. Rao, 43–50. Amsterdam, The Netherlands: North Holland.
Kim, C., J. Jung, and Y. Chung. 2011. Bayesian estimation for the exponentiated Weibull model under type II progressive censoring. Statistical Papers 52 (1):53–70.
Kotz, S., and S. Nadarajah. 2000. Extreme value distributions: Theory and applications. London, UK: Imperial College Press.
Lin, C. T., and W. Y. Wang. 2000. Estimation for the generalized Pareto distribution with censored data. Communications in Statistics — Simulation and Computation 29 (4):1183–213.
Mousa, M. A., and S. A. Al-Sagheer. 2006. Statistical inference for the Rayleigh model based on progressively type-II censored data. Statistics 40 (2):149–57.
Natural Environment Research Council. 1975. Flood studies report. London, UK: Water Environment Research Council.
Ng, H. K. T. 2005. Parameter estimation for a modified Weibull distribution for progressively type-II censored samples. IEEE Transactions on Reliability 54 (3):374–80.
Pickands, J. 1975. Statistical inference using extreme order statistics. Annals of Statistics 3:119–31.
Raqab, M. Z., and R. A. Al-Jarallah. 2015. Prediction intervals for future Weibull residual data. Journal of Statistical Computation and Simulation 86 (7):1320–33.
Raqab, M. Z., and M. T. Madi. 2002. Bayesian prediction of the total time on test using doubly censored Rayleigh data. Journal of Statistical Computation and Simulation 72:781–89.
Robert, C., and G. Casella. 2004. Monte Carlo statistical methods, 2nd ed. New York, NY: Springer-Verlag.
Sinharay, S. 2003. Assessing convergence of the Markov chain monte carlo algorithms: a review. Educational Testing Service, Research & Development Division Princeton, Princeton, NJ.
Smith, R. L. 1984. Threshold methods for sample extremes. In Statistical extremes and applications, ed. J.T. de Oliveira, 621–38. Dordrecht, The Netherlands: Reidel.
Tierney, L. 1994. Markov Chains for exploring posterior distributions. Annals of Statistics 22:1701–28.
Acknowledgments
The authors thank the editor and referees for their encouragement, comments, and helpful suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Raqab, M.Z., Bdair, O.M., Madi, M.T. et al. Prediction of the remaining testing time for the generalized Pareto progressive censoring samples with applications to extreme hydrology events. J Stat Theory Pract 12, 165–187 (2018). https://doi.org/10.1080/15598608.2017.1338168
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/15598608.2017.1338168