An identity on order statistics of a set of random variables.

Abstract

When an n × 1 random vector X = (X1, …, Xn ) T has a sign-invariant distribution, Strait [J. Multivariate Anal. 4 (1974) 494-496] 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.