An identity on order statistics of a set of random variables

Jian-Lun Xu

Journal of Multivariate Analysis, 2014, vol. 129, issue C, 243-244

Abstract: 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.

Keywords: Expectation; Order statistics; Sign-invariant distribution (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1016/j.jmva.2014.04.016

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