Order Statistics of Samples from Multivariate Distributions

Abstract Let (X 1j , X 2j , ···, Xmj ), j = 1, 2, ···, n, be a sample of size n on an m-dimensional vector (X 1, X 2, ···, Xm ), m ≥ 2. Let the order statistics of the rth component be denoted by X r,1* ≤ X r,2* ≤ ··· ≤ X r,n *. In this article we investigate the distribution of the vector (X 1,n−i1*, X 2,n–i2*, ···, Xm,n–im *) for (i 1, i 2, ···, im ) not depending on n. The major emphasis is on asymptotic theory and a general formula is given for the asymptotic distribution of the vector above when each ij = 0. Necessary and sufficient condition is also given for the asymptotic independence of the components of the vector investigated. This extends results known for m = 2. In Section 4 examples are given for illustration.