Some new results on multivariate dispersive ordering of generalized order statistics

Let { X 1 , n ∗ , X 2 , n ∗ , … , X n , n ∗ } be generalized order statistics based on a continuous distribution function F with parameters k and ( m 1 , … , m n − 1 ) . Chen and Hu (2007) [8] investigated the sufficient conditions on F and on the parameters k and m i ’s such that ( X 0 , n ∗ , X 1 , n ∗ , … , X n − 1 , n ∗ ) ≤ disp ( X 1 , n + 1 ∗ , X 2 , n + 1 ∗ , … , X n , n + 1 ∗ ) ≤ disp ( X 1 , n ∗ , X 2 , n ∗ , … , X n , n ∗ ) , where X 0 , 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.