Some new results on multivariate dispersive ordering of generalized order statistics

Let {X1,n∗,X2,n∗,…,Xn,n∗} be generalized order statistics based on a continuous distribution function FF with parameters kk and (m1,…,mn−1)(m1,…,mn−1). Chen and Hu (2007) [8] investigated the sufficient conditions on FF and on the parameters kk and mimi’s such that (X0,n∗,X1,n∗,…,Xn−1,n∗)≤disp(X1,n+1∗,X2,n+1∗,…,Xn,n+1∗)≤disp(X1,n∗,X2,n∗,…,Xn,n∗), where X0,n∗≡0, and ≤disp is the Shaked–Shanthikumar multivariate dispersive order. Since the order ≤disp does not possess the closure property under marginalization, one may naturally wonder whether the corresponding multivariate margins of the above random vectors are also ordered in the order ≤disp. This is answered affirmatively in this paper. Some comparison results for generalized order statistics from two samples are presented. Potential applications are also mentioned.