Symmetry of a distribution via symmetry of order statistics

In this paper, we establish the following characterization of symmetric absolutely continuous distributions and symmetric discrete distributions. Suppose X1,…,Xn is a random sample from a distribution with pdf/pmf fX(x), and X1:n,…,Xn:n are the corresponding order statistics. Then, Xr:n=d−Xn−r+1:n for some r=1,…,n if and only if fX(x)=fX(−x). Here, =d means that the two random variables have the same distribution. In the discrete case, we assume the support to be finite.