Conditional ordering of order statistics

For 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].