Probability inequalities via negative dependence for random variables conditioned on order statistics

Abstract

Distributions are studied which arise by considering independent and identically distributed random variables conditioned on events involving order statistics. It is shown that these distributions are negatively dependent in a very strong sense. Furthermore, bounds are found on the distribution functions. The conditioning events considered occur naturally in reliability theory as the time to system failure for k‐out‐of‐n systems. An application to systems formed with “second‐hand” components is given.

On stochastic comparison of random vectors

“…In this paper we provide conditions weaker than (1.3) that imply (1.4) and prove it using the standard construction (see Section 2). Using these results we obtain extensions of (i) a result of Block et at. (1984) concerning the stochastic monotonicity of independent and identically distributed (i.i.d.)…”

“…cally increasing in z and (ii) stochastically decreasing in k. Block et al (1984) proved (i) (see the Corollary to Lemma 4.2 there). They also used this result to show some negative dependence properties of T. In Section 5 we will show the negative dependence property of Z n,z,l,m.…”

“…Then clearly Block et at. (1984) consider the random vector Zj conditioned on event A as given in (3.4) and study its stochastic monotonicity and NOS properties.…”

Negative dependence in the balls and bins experiment with applications to order statistics

“…This study was initiated by Tukey [20] and Barlow and Proschan [2], developed by Karlin and Rinott [11], Block et al [3], Shanthikumar [19], Kim and David [14], and Boland et al [5], and has been continued until the more recent papers by Khaledi and Kochar [13], Hu and Zhu [9], Hu and Yang [8], Schmitz [17], and Avérous et al [1].…”

Further developments on sufficient conditions for negative dependence of random variables

“…Theorem 4.1 (Block et al, 1987;Shanthikumar, 1987). Let X be a random vector of n iid random variables with a continuous distribution.…”