Bivariate Dependence Properties of Order Statistics

✍ Scribed by Philip J. Boland; Myles Hollander; Kumar Joag-Dev; Subhash Kochar

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 327 KB
• Volume: 56
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1996.0005

✦ Synopsis

If X 1 , ..., X n are random variables we denote by X (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 statistics X (i) and X ( j) . If in particular the random variables are independent with a common density or mass function, then X (i) and X ( j) are TP 2 dependent for any i and j. In this paper we consider the situation in which the random variables X 1 , ..., X n are independent but otherwise arbitrarily distributed. We show that for any it | 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 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 of i=1, ..., n for article no.