Dependence properties of order statistics

Let X-1:n <= X-2:n <= ... X-n:n denote the order statistics of independent random variables X-1, X-2, ... X-n with possibly nonidentical distributions for each n. For fixed 1 <= j(1) < j(2) < ... < j(r) ,<= n and (X-1,...,X-r) is an element of R-r, it is shown that if j(1) - i >= max {0, n - m} and A(i,m,y) = {X-i:m > y}, then Delta(i)(y) = P[Xj(i:n) > x(1), X-j2:n > x(2), ... , X-jr:n > x(r)vertical bar A(i,m,y)] is increasing in y, and that if A(i,m,y) is either {X-i:m > y} or {X-i:m <= y}, then Delta(i)(y) is decreasing in i for fixed y < x1 < ... < x(r). It is also shown that if each X-k has a continuous distribution function, and if A(i,m,y) is either {X-i:m = y} or {Xi-1:m < y < X-i:m), then Delta(i)(y) is decreasing in i for fixed y < x(1) < ... < x(r), where Xm+1:m = + infinity. In particular, we obtain that RTI (X-j:n vertical bar X-i:m) for j - i >= max{n - m, 0} and LTD (X-j:n vertical bar X-i:m) for j - i <= min{n - m, 0}. We thus extend the main results in Boland et al. [1996. Bivariate dependence properties of order statistics. J. Multivariate Anal. 56, 75-89] and in Hu and Xie [2006. Negative dependence in the balls and bins experiment with applications to order statistics. J. Multivariate Anal. 97, 1342-1354]. (C) 2007 Elsevier B.V. All rights reserved.