Bivariate Dependence Properties of Order Statistics

IfX1, …,Xnare random variables we denote byX(1)⩽X(2)⩽…⩽X(n)their respective order statistics. In the case where the random variables are independent and identically distributed, one may demonstrate very strong notions of dependence between any two order statisticsX(i)andX(j). If in particular the random variables are independent with a common density or mass function, thenX(i)andX(j)areTP2dependent for anyiandj. In this paper we consider the situation in which the random variablesX1, …,Xnare independent but otherwise arbitrarily distributed. We show that for anyi<jandtfixed,P[X(j)>t∣X(i)>s] is an increasing function ofs. This is a stronger form of dependence betweenX(i)andX(j)than that of association, but we also show that among the hierarchy of notions of bivariate dependence this is the strongest possible under these circumstances. It is also shown that in this situation,P[X(j)>t∣X(i)>s] is a decreasing function ofi=1, …,nfor any fixeds<t. We give various applications of these results in reliability theory, counting processes, and estimation of conditional probabilities. We also consider the situation whereX1, …,Xnrepresent a random sample of sizendrawn without replacement from a linearly ordered finite population. In this case it is shown thatX(i)andX(j)areTP2dependent for anyiandj, and the implications are discussed.