# Nonparametric Prediction Intervals for Future Order Statistics in a Proportional Hazard Model

**ABSTRACT** Consider k independent random samples with different sample sizes such that the ith sample comes from the cumulative distribution function (cdf) F i = 1 − (1 − F)α i , where α i is a known positive constant and F is an absolutely continuous cdf. Also, suppose that we have observed the maximum and minimum of the first k samples. This article shows how one can construct the nonparametric prediction intervals for the order statistics of the future samples on the basis of these information. Three schemes are studied and in each case exact expressions for the prediction coefficients of prediction intervals are derived. Numerical computations are given for illustrating the results. Also, a comparison study is done while the complete samples are available.

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**ABSTRACT:**Let X i,j (1 ≤ i ≤ k, 1 ≤ j ≤ n i ) be independent random variables and for a fixed i, X i,j 's, (1 ≤ j ≤ n i ) be identically distributed random variables with survival function , where α i is a known positive constant. Also, suppose M i and M′ i , respectively, denote the maximum and minimum of the ith sample. This article investigates the nonparametric confidence intervals for an arbitrary quantile of the distribution F and tolerance limits based on these statistics. Various cases have been studied and in each case, the nonparametric confidence intervals are obtained and exact expressions for the confidence coefficients of these confidence intervals are derived. A data set representing the time of successive failures of the air conditioning system on Boeing 720 jet aircraft is used to illustrate the results. Finally, the accuracy of the proposed procedure has been investigated, when α i 's are unknown via a simulation study.Communication in Statistics- Theory and Methods 06/2008; 37(10-10):1525-1542. DOI:10.1080/03610920801893897 · 0.28 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The paper deals with the Bayesian prediction intervals for generalized order statistics (GOS) based on a certain class of exponential-type distributions. This class of distributions includes several important distributions such as Weibull, Burr-XII and Pareto distributions. A general class of prior density functions is used and the predictive reliability function is obtained in the one sample case. The investigation of multiply type-II censored GOS samples generalizes results of ordinary order statistics (OS), sequential order statistics (SOS) and censored GOS. The special case of Pareto distributed observations is considered and completed with numerical results.Statistics: A Journal of Theoretical and Applied Statistics 12/2007; 41(6):495-504. DOI:10.1080/02331880701223357 · 1.59 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We discuss the problem of predicting, on the basis of a sample from a two parameter exponential distribution, the s'th smallest observation Ys in a future sample of n observations for the same distribution, and the mean Y of the future sample. It is shown how to obtain prediction intervals for Ys and Y, based on a Type II censored sample from the distribution.Technometrics 11/1977; 19(4):469-472. DOI:10.1080/00401706.1977.10489587 · 1.79 Impact Factor