Ordering Conditional Distributions of Generalized Order Statistics

Abstract

The concept of generalized order statistics was introduced as a unified approach to a variety of models of ordered random variables. The purpose of this article is to establish the usual stochastic and the likelihood ratio orderings of conditional distributions of generalized order statistics from one sample or two samples, strengthening and generalizing the main results in Khaledi and Shaked [15], and Li and Zhao [17]. Some applications of the main results are also given.

Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

“…or exchangeable dependent components, especially k-outof-n systems. For instance, see [1], [8], [9], [10], [12], [13], [15], [21], [22], [24], [26], [27], and [28].…”

“…where Expression (8) indicates that the inactivity time (t − T | T ≤ t) of the system at time t is a mixture of the inactivity time (t − X i,i | X i,i ≤ t) of the order statistics at time t with coefficients β i (t) for i = 1, 2, . .…”

Conditional Ordering of k-out-of-n Systems with Independent But Nonidentical Components

“…Recently, Li and Zhao (2008) carried out a stochastic comparison on (1.1) and (1.2) of two (n − k + 1)-out-of-n systems and generalized the main results of Khaledi and Shaked (2006). It is worth mentioning here that Hu et al (2007) and Xie and Hu (2008) discussed conditional ordering of generalized order statistics, which includes (1.1) and (1.2) as special cases and, hence, extends some of the related results.…”

Multivariate likelihood ratio ordering of conditional order statistics

“…Kahledi and Shojaei [8] established the analogous comparison results for record values. Hu, et al [6] further obtained several comparison results for generalized order statistics. In particular, they obtained [X k:n − t|X l:n > t] ≤ lr [X k :n − t|X l :n > t]…”

“…for 1 ≤ l < k ≤ n. Recently, Hu, et al [5] built that [X k:n − t|X l:n > t] ≤ lr [X k+1:n − t|X l:n > t] ( 4 ) for 1 ≤ l < k < n and t ∈ R + , which is an open problem in [6]. From (2), (3), and (4), it can be concluded that, for 1 ≤ l < k ≤ n and t ∈ R + , [X k:n − t|X l:n > t] is increasing in k, and decreasing both in l and n according to the likelihood ratio order.…”