title: An identity on order statistics of a set of random variables.
abstract: When ann× 1 random vectorX= (X1, …,Xn)Thas 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 ofLr(0,X1,X1+X2, …,X1+Xn) andLr(0,X1, …,Xn) are equal, whereLr(0,X1, …,Xn) is therth order statistic of 0,X1, …,Xnforr= 1, …,n+ 1.
authors.affiliation: Biometry Research Group, National Cancer Institute, 9609 Medical Center Drive, Room 5E124, Bethesda, MD 20892-9789, USA.
authors.forename: Jian-Lun
authors.initials: JL
authors.lastname: Xu
keywords: Expectation; Order statistics; Sign-invariant distribution