Conditional ordering of order statistics

For any positive integers m and n, let X1,X2,...,Xm[logical or]n be independent random variables with possibly nonidentical distributions. Let X1:n<=X2:n<=...<=Xn:n be order statistics of random variables X1,X2,...,Xn, and let X1:m<=X2:m<=...<=Xm:m be order statistics of random variables X1,X2,...,Xm. It is shown that (Xj:n,Xj+1:n,...,Xn:n) given Xi:m>y for j-i>=max{n-m,0}, and (X1:n,X2:n,...,Xj:n) given Xi: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].