An identity on order statistics of a set of random variables

When an n × 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 ( X 1 , X 2 , … , X n ) and ( − X 1 , X 2 , … , X n ) are equal in distribution and prove a more general result that the expectations of L r ( 0 , X 1 , X 1 + X 2 , … , X 1 + X n ) and L r ( 0 , X 1 , … , X n ) are equal, where L r ( 0 , X 1 , … , X n ) is the r th order statistic of 0 , X 1 , … , X n for r = 1 , … , n + 1 .