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On Bayesian interval prediction of future records. (English) Zbl 1040.62014

Summary: Based on the one-sample scheme, Bayesian prediction bounds for the \(s\)th future record value are obtained. All of the informative and future observations are assumed to be obtained from a general class of distributions which includes the Weibull, compound Weibull, Pareto, beta, Gompertz, and compound Gompertz among other distributions. The prior belief of the experimenter is measured by a proper general conjugate prior which was suggested by E. K. AL-Hussaini [J. Stat. Plann. Inference 79, 79–91 (1999; Zbl 0933.62018)].

MSC:

62F15 Bayesian inference
62N02 Estimation in survival analysis and censored data

Citations:

Zbl 0933.62018
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References:

[1] Ahsanullah, M. (1980). Linear prediction of record values for the two parameter exponential distribution.Annals of the Institute of Statistical Mathematics, 32:363–368. · Zbl 0456.62026 · doi:10.1007/BF02480340
[2] Ahsanullah, M. (1990). Estimation of the parameters of the Gumbel distribution based onm record values.Computational Statistics Quarterly, 6:231–239. · Zbl 0726.62037
[3] Ahsanullah, M. (1994). On record values from univariate distributions. In J. Galambos, J. Lechner, and E. Simiu, eds.,Proceedings of the Conference on Extreme Value Theory and Applications, vol. 3, pp. 1–16.
[4] Aitcheson, J. andDunsmore, I. (1975).Statistical Prediction Analysis. Cambridge University Press, Cambridge.
[5] Al-Hussaini, E. K. (1999a). Bayesian prediction under a mixture of two exponential components model based on type I censoring.Journal of Applied Statistical Science, 8:173–185. · Zbl 0916.62022
[6] Al-Hussaini, E. K. (1999b). Predicting observables from a general class of distributions.Journal of Statistical Planning and Inference, 79:79–91. · Zbl 0933.62018 · doi:10.1016/S0378-3758(98)00228-6
[7] Al-Hussaini, E. K. (2001). Prediction: advances and new research. InProceedings of Mathematics and the 21 st Century, pp. 223–245. World Scientific, Singapore. · Zbl 0996.62017
[8] Al-Hussaini, E. K. andJaheen, Z. (1992). Bayesian estimation of the parameters, reliability and failure rate functions of the Burr type XII failure model.Journal Statistical Computation and Simulation, 44:31–40. · Zbl 0775.62260 · doi:10.1080/00949659208811389
[9] Al-Hussaini, E. K., andJaheen, Z. F. (1995). Bayesian prediction bounds for the Burr type XII model.Communications in Statistics, Theory and Methods, 24(7):1829–1842. · Zbl 0937.62571 · doi:10.1080/03610929508831589
[10] Al-Hussaini, E. K., andJaheen, Z. F. (1996a). Bayesian prediction bounds for the Burr type XII distribution in the presence of outliers.Journal of Statistical Planning and Inference, 55:23–37. · Zbl 0859.62025 · doi:10.1016/0378-3758(95)00184-0
[11] Al-Hussaini, E. K., andJaheen, Z. F. (1996b). Parametric prediction bounds for the future median of the exponential distribution.Statistics, 32:267–275. · Zbl 0916.62023 · doi:10.1080/02331889908802667
[12] Al-Hussaini, E. K., Nigm, A. M., andJaheen, Z. (2001). Bayesian prediction based on finite mixtures of lomax components model and type I censoring.Statistics, 35(3):259–268. · Zbl 0979.62009 · doi:10.1080/02331880108802735
[13] Arnold, B., Balakrishnan, N., andNagaraja, H. N. (1998).Records, Wiley, New York.
[14] Arnold, B., andPress, S. J. (1983). Bayesian inference for Pareto populations.Journal of Econometrics, 21:287–306. · Zbl 0503.62027 · doi:10.1016/0304-4076(83)90047-7
[15] Arnold, B. andPress, S. J. (1989). Bayesian estimation and prediction for Pareto data,Journal of the American Statistical Association, 84:1079–1084. · Zbl 0702.62026 · doi:10.2307/2290086
[16] Arnold, B. C. (1983).Pareto Distributions, vol. I. International Cooperative Publishing House, Fairland, Maryland. · Zbl 1169.62307
[17] Balakrishnan, N., Ahsanullah, M. andChan, P. S. (1995). On the logistic record values and associated inference.Journal of Applied Statistical Science, 2:233–248. · Zbl 0873.62013
[18] Balakrishnan, N. andChan, P. S. (1994). Record values from Rayleigh and Weibull distributions and associated inference. In J. Galambos, J. Lechner, and E. Simiu, eds.,Proceedings of the Conference on Extreme Value Theory and Applications, vol. 3, pp. 41–51.
[19] Bernardo, J. M. andSmith, A. F. M. (1994).Bayesian Theory. Wiley, New York.
[20] Berred, A. M. (1998). Prediction of record values.Communications in Statistics, Theory and Methods, 27:2221–2240. · Zbl 0907.62037 · doi:10.1080/03610929808832224
[21] Corcuera, J. M. andGiummolè, F. (1999). A generalized Bayes rule for prediction.Scandinavian Journal of Statistics, 26:265–279. · Zbl 0934.62027 · doi:10.1111/1467-9469.00149
[22] Doganakso, N. andBalakrishnan, N. (1997). A useful property of best linear unbiased prediction with applications to life testing.The American Statistician, 51:22–28. · Zbl 04537177 · doi:10.2307/2684687
[23] Dunsmore, I. R. (1974). The Bayesian predictive distribution in life testing models.Technometrics, 16:455–460. · Zbl 0283.62093 · doi:10.2307/1267677
[24] Dunsmore, I. R. (1976). Asymptotic prediction analysis.Biometrika, 63:627–630. · Zbl 0345.62009 · doi:10.1093/biomet/63.3.627
[25] Dunsmore, I. R. (1983). The future occurrence of records.Annals of the Institute of Statistical Mathematics, 35:267–277. · Zbl 0522.62027 · doi:10.1007/BF02480982
[26] Dunsmore, I. R. andAmin, Z. H. (1998). Some prediction problems concerning samples from the Pareto distribution.Communications in Statistics, Theory and Methods, 27:1221–1238. · Zbl 0901.62041 · doi:10.1080/03610929808832155
[27] Flinger, M. A. andWolfe, D. A. (1976). Some applications of sample analogues to probability integral transform and a coverage property.The American Statistician, 30:78–85. · Zbl 0369.62015 · doi:10.2307/2683799
[28] Flinger, M. A. andWolfe, D. A. (1979a). Methods for obtaining distribution-free prediction interval for the median of a future sample.Journal of Quality and Technology, 11:192–198.
[29] Flinger, M. A. andWolfe, D. A. (1979b). Nonparametric prediction intervals for a future sample median.Journal of the American Statistical Association, 74:453–456. · Zbl 0416.62017 · doi:10.2307/2286354
[30] Geisser, S. (1975). The predictive sample reuse method with applications.Journal of the American Statistical Association, 70:320–328. · Zbl 0321.62077 · doi:10.2307/2285815
[31] Geisser, S. (1984). Predicting Pareto and exponential observables.Canadian Journal of Statistics, 12:143–152. · Zbl 0553.62002 · doi:10.2307/3315178
[32] Geisser, S. (1985). Interval prediction for Pareto and exponential observables.Journal of Econometrics, 29:173–185. · Zbl 0576.62044 · doi:10.1016/0304-4076(85)90038-7
[33] Geisser, S. (1986). Predictive analysis. In S. Kotz, N. L. Johnson, and C. B. Read, eds.,Encyclopedia of Statistical Sciences, vol. 7, pp. 158–170. Wiley, New York.
[34] Geisser, S. (1990). On hierarchical Bayes procedures for predicting simple exponential survival.Biometrics, 46:225–230. · Zbl 0715.62066 · doi:10.2307/2531646
[35] Geisser, S. (1993).Predictive Inference: An Introduction. Chapman and Hall, London. · Zbl 0824.62001
[36] Guilbaud, O. (1983). Nonparametric prediction intervals for sample medians in the general case.Journal of the American Statistical Association, 78:937–941. · Zbl 0535.62044 · doi:10.2307/2288207
[37] Howlader, H. A. andHossain, A. (1995). On Bayesian estimation and prediction from Rayleigh based on type II censored data.Communications in Statistics, Theory and Methods, 24:2249–2259. · Zbl 0937.62574
[38] Johnson, N. L., Kotz, S. andBalakrishnan, N. (1994).Continuous Univariate Distributions, vol. 1, Wiley, New York, 2nd ed. · Zbl 0811.62001
[39] Johnson, N. L., Kotz, S. andBalakrishnan, N. (1995).Continuous Univariate Distributions, vol. 2, Wiley, New York, 2nd ed. · Zbl 0821.62001
[40] Johnson, R. A., Evans, J. W., andGreen, D. W. (1999). Nonparametric Bayesian predictive distributions for future order statistics.Statistics and Probability Letters, 41:247–254. · Zbl 0933.62041 · doi:10.1016/S0167-7152(98)00161-8
[41] Kaminsky, K. S. andNelson, P. I. (1998). Prediction on order statistics. In N. Balakrishnan and C. R. Rao, eds.,Handbook of Statistics, vol. 17, pp. 431–450, Elsevier Science, Amsterdam. · Zbl 0922.62039
[42] Lee, J. C. andLiao, Y. L. (1999). A note on Bayesian estimation and prediction for the beta-binomial model.Journal of Statistical Computation and Simulation, 63:73–91. · Zbl 0941.62031 · doi:10.1080/00949659908811950
[43] Lingappaiah, G. S. (1978). Bayesian approach to the prediction problem in the exponential population.IEEE Transactions on Reliability, R-27:222–225. · Zbl 0381.62086 · doi:10.1109/TR.1978.5220332
[44] Lingappaiah, G. S. (1979). Bayesian approach to prediction and the spacings in the exponential distribution.Annals of the Institute of Statistical Mathematics, 31:391–401. · Zbl 0445.62044 · doi:10.1007/BF02480296
[45] Lingappaiah, G. S. (1980). Intermittent life testing and Bayesian approach to prediction with spacing in the exponential model.Statistica, 40:477–490. · Zbl 0472.62101
[46] Lingappaiah, G. S. (1986). Bayes prediction in the exponential life-testing when sample size is a random variable.IEEE Transactions on Reliability, 35:106–110. · Zbl 0602.62089 · doi:10.1109/TR.1986.4335366
[47] Lingappaiah, G. S. (1989). Bayes prediction of maxima and minima in exponential life tests in the presence of outliers.Journal of Indusrial Mathematical Society, 39:169–182. · Zbl 0711.62028
[48] Lwin, T. (1972). Estimating the tail of the Paretian law.Skadinavian Aktuarietidskr, 55:170–178. · Zbl 0275.62024
[49] Maritz, J. S. andLwin, T. (1989).Empirical Bayes Methods, Chapman and Hall, London, 2nd ed. · Zbl 0731.62040
[50] Nagaraja, H. N. (1984). Asymptotic linear prediction of extreme order statistics.Annals of the Institute of Statistical Mathematics, 36:289–299. · Zbl 0553.62049 · doi:10.1007/BF02481971
[51] Nagaraja, H. N. (1995). Prediction problems. In N. Balakrishnan and A. P. Basu, eds.,The Exponential Distribution: Theory and Applications, pp. 139–163. Gordon and Breach, New York.
[52] Patel, J. K. (1989). Prediction intervals-a review.Communications in Statistics, Theory and Methods, 18:2393–2465. · Zbl 0696.62160 · doi:10.1080/03610928908830043
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