An identity on order statistics of a set of random variables

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