Conditional ordering of order statistics

For any positive integers mm and nn, let X1,X2,…,Xm∨nX1,X2,…,Xm∨n be independent random variables with possibly nonidentical distributions. Let X1:n≤X2:n≤⋯≤Xn:nX1:n≤X2:n≤⋯≤Xn:n be order statistics of random variables X1,X2,…,XnX1,X2,…,Xn, and let X1:m≤X2:m≤⋯≤Xm:mX1:m≤X2:m≤⋯≤Xm:m be order statistics of random variables X1,X2,…,XmX1,X2,…,Xm. It is shown that (Xj:n,Xj+1:n,…,Xn:n)(Xj:n,Xj+1:n,…,Xn:n) given Xi:m>yXi:m>y for j−i≥max{n−m,0}j−i≥max{n−m,0}, and (X1:n,X2:n,…,Xj:n)(X1:n,X2:n,…,Xj:n) given Xi:m≤yXi:m≤y for j−i≤min{n−m,0}j−i≤min{n−m,0} are all increasing in yy 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].