Order Statistics of Samples from Multivariate Distributions

Abstract

Let (X _1j , X _2j , ···, X_mj ), j = 1, 2, ···, n, be a sample of size n on an m-dimensional vector (X ₁, X ₂, ···, X_m ), 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*, ···, X_m,n–im *) for (i ₁, i ₂, ···, i_m ) 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 i_j = 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.