Stochastic comparisons of order statistics and their concomitants

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 ( X r + 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.