Stochastic Orderings of Order Statistics of Independent Random Variables with Different Scale Parameters Baha-Eldin Khaledi Department of Statistics Shahid Beheshti University, Tehran, Iran Razi University, Kermanshah, Iran e-mail: bkhaledi@hotmail.com Subhash Kochar Department of Mathematics and Statistics Portland State University Portland, Oregon 97201, USA e-mail: kochar@pdx.edu Abstract This is a survey paper on recent results on stochastic comparisons of order statistics of n independent random variables differing in their scale parameters. Most of the results obtained so far are for the Weibull and the Gamma distributions. Key-Words : Proportional hazards family, hazard rate ordering, Schur functions, majoriza- tion and p-larger ordering. 1 Introduction Let X 1 ,...,X n be n random variables and let X (i) denote their ith order statistic, i = 1,...,n. Order statistics arise naturally at number of places in applications. A k-out-of-n system of n components functions if at least k of the n components function. The time of a k-out-of-n system of n components with lifetimes X 1 ,...,X n corresponds to the (n − k +1)th order statistic. Thus, the study of lifetimes of k-out-of-n systems is equivalent to studying the stochastic properties of order statistics. In particular, a 1-out-of-n system corresponds to a parallel system and a n-out-of-n system corresponds to a series system. Lot of work has been done in the literature on different aspects of order statistics when the observations are independently and identically distributed (i.i.d.). In many practical situations, like in reliability theory, however, the observations are not necessarily i.i.d. Because of the complicated nature of the problem, not much work has been done for the non-i.i.d. case. Some interesting partial ordering results on order statistics when the parent observations are independent with proportional hazard rates have been obtained by Pledger and Proschan (1971), Proschan and Sethuraman (1976), Boland, El-Neweihi and Proschan (1994), Dykstra,