A characterization by linearity of the regression function based on order statistics

Abstract

We consider an infinite sequence X 1, X 2,… of independent random variables having a common continuous distribution function F( x). For 1⩽ i⩽ n, let { X i: n } denote the ith order statistic among X 1,…, X n . In this paper, we characterize the distributions for which the regression E(( X 1: n +⋯+ X n: n )/ n| X k: n = x) is a linear function of x.