A Note on Conditioning and Stochastic Domination for Order Statistics

Abstract

For an order statistic (X 1:n , . . . , X n:n ) of a collection of independent but not necessarily identically distributed random variables, and any i ∈ {1, . . . , n}, the conditional distribution of (X i+1:n , . . . , X n:n ) given X i:n > s is shown to be stochastically increasing in s. This answers a question by Hu and Xie (2006).

Efron’s monotonicity property for measures onR2

“…Efron's monotonicity property can also be viewed as a monotonicity property for the collection of conditional laws with respect to the stochastic order (Theorem 6.B.9. in Shaked and Shanthikumar [2007], see also Shanthikumar [1987a], Shanthikumar [1987b], Rinott and Samuel-Cahn [1991], Dubhashi and Häggström [2008], Zhuang, Yao and Hu [2010]).…”