BIVARIATE DEPENDENCE PROPERTIES OF ORDER-STATISTICS

If X(1),...,X(n) are random variables we denote by X((1)) less than or equal to X((2))...less than or equal to X((n)) their respective order statistics. In the case where the random variables are independent an d identically distributed, one may demonstrate very strong notions of dependence between any two order statistics X((i)) and X((j)). If in p articular the random variables are independent with a common density o r mass function, then X((i)) and X((j)) are TP2 dependent for any i an d j. In this paper we consider the situation in which the random varia bles X(1),...,X(n) are independent but otherwise arbitrarily distribut ed. We show that for any i < j and t fixed, P[X((f)) > t \ X((i)) > s] is an increasing function of s. This is a stronger form of dependence between X((i)) and X((j)) than that of association, but we also show that among the hierarchy of notions of bivariate dependence this is th e strongest possible under these circumstances. It is also shown that in this situation, P[X((f)) > t \ X((i)) > s] is a decreasing function of i = 1,...,n for any fixed s < t. We give various applications of t hese results in reliability theory, counting processes, and estimation of conditional probabilities. We also consider the situation where X( 1),...,X(n) represent a random sample of size II drawn without replace ment from a linearly ordered finite population. In this case it is sho wn that X((i)) and X((j)) are TP2 dependent for any i and j, and tile implications are discussed. (C) 1996 Academic Press, Inc.