An identity on order statistics of a set of random variables

When an n × 1 random vector X = (X₁, …, X_n ) ^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 + X_n ) and max(0, X₁, …, X_n ) are equal. In this note we assume a weaker condition that (X₁, X₂, …, X_n ) and (-X₁, X₂, …, X_n ) are equal in distribution and prove a more general result that the expectations of L_r (0, X₁, X₁ + X₂, …, X₁ + X_n ) and L_r (0, X₁, …, X_n ) are equal, where L_r (0, X₁, …, X_n ) is the rth order statistic of 0, X₁, …, X_n for r = 1, …, n + 1.