An identity on order statistics of a set of random variables

When an n x 1 random vector X = (X-1, ..., X-n)(T) has a sign-invariant distribution, Strait (1974) proved that the expectations of max(0, X-1, X-1 + X-2..... X-1 + X-n) and max(0, X-1, ...,X-n) are equal. In this note we assume a weaker condition that (X1, X2,, Xn) and (-X-1, X-2, ... X-n) are equal in distribution and prove a more general result that the expectations of L, (0, X-1, X-1 + X-2, ... , X-1 + Xi) and L, (0, X-1,..., X-n) are equal, where L,(0, X-1, ... X-5) is the rth order statistic of 0, X-1, ... Xn(,) for r = 1, ... , n+1. Published by Elsevier Inc.