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

We consider an infinite sequence X1,X2,… of independent random variables having a common continuous distribution function F(x). For 1⩽i⩽n, let {Xi:n} denote the ith order statistic among X1,…,Xn. In this paper, we characterize the distributions for which the regression E((X1:n+⋯+Xn:n)/n|Xk:n=x) is a linear function of x.