Distribution and expectation bounds on order statistics from possibly dependent variates

Let X-1,X-2,...,X-n be n random variables with an arbitrary n-variate distribution. We say that the X's are maximally (resp. minimally) stable of order j (j is an element of {1, 2,..., n}), if the distribution F-(j) of max {X-k1,...,X-kj} (resp. G((j)) of min {X-ki,...,X-kj}) is the same, for any j-subset {k(i),...,k(j)} of {1,2,..., n}. Under the assumption of maximal (resp. minimal) stability of order j, sharp upper (resp. lower) bounds are given for the distribution F-k:n of the kth order statistic X-k:n, in terms of F-(j) (resp. G((j))), and the corresponding expectation bounds are derived. Moreover, some expectation bounds in the case of j-independent-F samples (i.e., when each j-tuple X-ki,...,X-kj is independent with a common marginal distribution F) are given. (C) 2001 Elsevier Science B.V. All rights reserved.