Stochastic comparisons of order statistics and their concomitants

Abstract

Let X-1:n <= X-2:n ... <= X-n:n be the order statistics from some sample, and let Y-[1 (:n]) Y-[2:n],..., Y-[n:n], be the corresponding concomitants. One purpose of this paper is to obtain results that stochastically compare, in various senses, the random vector (X-r:n, Y-[r:n]) to the random vector (Xr+1:n Y[r+1:n]), r = 1, 2,..., n - 1. Such comparisons are called one-sample comparisons. Next, let S-1:n <= S-2:n ... <= S-n:n be the order statistics constructed from another sample, and let T-[1:n], T-[2:n],...,T-[n:n] be the corresponding concomitants. Another purpose of this paper is to obtain results that stochastically compare, in various senses, the random vector (X-r:n, Y-[r:n]). with the random vector (S-r:n, T-[r:n]), r = 1, 2,..., n. Such comparisons are called two-sample comparisons. It is shown that some of the results in this paper strengthen previous results in the literature. Some applications in reliability theory are described. (C) 2013 Elsevier Inc. All rights reserved.