Statistics and Probability Letters 79 (2009) 977–983 Contents lists available at ScienceDirect Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro Prediction intervals for future records and order statistics coming from two parameter exponential distribution J. Ahmadi * , S.M.T.K. MirMostafaee Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, P. O. Box 91775-1159, Mashhad, Iran article info Article history: Received 6 July 2008 Received in revised form 27 November 2008 Accepted 2 December 2008 Available online 16 December 2008 MSC: primary 62G30 secondary 62E15 abstract We study the problem of predicting future records based on observed order statistics from two parameter exponential distribution. The prediction intervals for the future order statistics as well as for the total lifetime in a future sample of size m from two parameter exponential distribution are obtained on the basis of the first n records coming from the same distribution. © 2008 Elsevier B.V. All rights reserved. 1. Introduction Prediction of future events on the basis of the past and present knowledge is a fundamental problem of statistics, arising in many contexts and producing varied solutions. As in estimation, a predictor can be either a point or an interval predictor. Parametric and nonparametric predictions have been considered in the literature. In many practical data-analytic situations we are interested in using the observations from an initial sample to construct an interval that will have a present probability of containing some statistic based on a future sample of observations from the same underlying distribution. Such an interval is called a prediction interval for the statistic of interest. Suppose that X 1 ,..., X n are independent and identically distributed (iid) observations from an absolutely continuous cumulative distribution function (cdf) F (x) and probability density function (pdf) f (x). The order statistics of the sample are defined by the arrangement of X 1 ,..., X n from the smallest to the largest, denoted as X 1:n ≤ X 2:n ≤ ··· ≤ X n:n . These statistics have been used in a wide range of problems, including robust statistical estimation, detection of outliers, characterization of probability distributions and goodness-of-fit tests, entropy estimation, analysis of censored samples, reliability analysis, quality control and strength of materials; for more details, see Arnold et al. (1992) and David and Nagaraja (2003) and the references therein. Several authors have considered prediction intervals for future order statistics based on observed order statistics in parametric and nonparametric settings. See for example, Lawless (1977), Nagaraja (1984), Kaminsky and Rhodin (1985), Chou (1988), Nagaraja (1995), Raqab and Nagaraja (1995), Hsieh (1997), Kaminsky and Nelson (1998) and Abdel-Aty et al. (2007) and so on. Let X 1 , X 2 ,... be a sequence of iid random variables having an absolutely continuous cdf F (x) and pdf f (x). An observation X j is called an upper record value if its value exceeds that of all previous observations. Thus, X j is an upper record if X j > X i for every i < j. Denote the nth record (upper) by R n , the first upper record is set as R 1 = X 1 , which is referred to as the reference value or the trivial record. Then the marginal pdf of R n is given by f R n (r ) = [- log(1 - F (r ))] n-1 (n - 1)! f (r ), (1) * Corresponding author. E-mail addresses: ahmadi-j@um.ac.ir (J. Ahmadi), ta_mi182@stu-mail.um.ac.ir (S.M.T.K. MirMostafaee). 0167-7152/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2008.12.002