Conditional ordering of order statistics

Abstract

a b s t r a c tFor any positive integers m and n, let X 1 , X 2 , . . . , X m∨n be independent random variables with possibly nonidentical distributions. Let X 1:n ≤ X 2:n ≤ · · · ≤ X n:n be order statistics of random variables X 1 , X 2 , . . . , X n , and let X 1:m ≤ X 2:m ≤ · · · ≤ X m:m be order statistics of random variables X 1 , X 2 , . . . , X m . It is shown that (X j:n , X j+1:n , . . . , X n:n ) given X i:m > y for j − i ≥ max{n − m, 0}, and (X 1:n , X 2:n , . . . , X j:n ) given X i:m ≤ y for j − i ≤ min{n − m, 0} are all increasing in y with respect to the usual multivariate stochastic order. We thus extend the main results in Dubhashi and Häggström (2008) [1] and Hu and Chen (2008) [2].

Stochastic properties of INID progressive Type-II censored order statistics

“…Boland et al [2] gave a counterexample to show that SI(X j:n |X 1:n ) does not in general hold for each j ≥ 1, while they proved that RTI (X j:n |X i:n ) and LTD (X i:n |X j:m ) whenever 1 ≤ i ≤ j ≤ n, (1.1) and that for any s < t and j given, P(X j:n > t|X i:n > s) is decreasing in i, (1.2) P(X j:n < s|X i:n < t) is increasing in i. (1.3) These results were extended by Dubhashi and Häggström [3], Hu and Xie [4], Hu and Chen [5], and Zhuang et al [6].…”

“…We use an idea exploited in the proof of Theorem 1 of [3] and in the proof of Theorem 2.1 of [6]. Two steps are considered below.…”

“…Remove assumption (2.3) in Step 1. The argument is the same as the one given in [3] or in [6]. First drop the disjointness assumption, and then remove the assumption of finite supports in (2.3) by using a standard limiting argument.…”

Revisiting Multivariate Likelihood Ratio Ordering Results for Order Statistics

“…In the case of independent and nonidentically distributed random variables, Zhao, Li, and Balakrishnan [34] and Kochar and Xu [21] provided several results in this direction. Again, it is possible to obtain additional results from general results in the setting of generalized order statistics; see, for example, Hu, Jin, and Khaledi [12], Xie and Hu [35], and Zhao and Balakrishnan [38] and Zhuang, Yao, and Hu [41]. Recently, Balakrishnan, Belzunce, Hami, and Khaledi [4] obtained several results for the likelihood ratio order of generalized order statistics.…”

“…Some other results about stochastic comparisons of conditional order statistic from independent and identically distributed random observations have been studied; see, for example, Langberg, Leon, and Proschan [22] and Belzunce, Franco, and Ruiz [5]. Again, it is possible to obtain additional results from general results in the setting of generalized order statistics; see, for example, Hu, Jin, and Khaledi [12], Xie and Hu [35], and Zhao and Balakrishnan [38] and Zhuang, Yao, and Hu [41]. In the case of independent and nonidentically distributed random variables, Zhao, Li, and Balakrishnan [34] and Kochar and Xu [21] provided several results in this direction.…”