METHODS FOR SYMMETRIZING RANDOM VARIABLES

Let X be a random variable with nonsymmetric density p(x).We give the symmetric density q(x) closest to it in the sense ofKulback–Liebler and Hellinger distances. (All symmetries are around zero.) For the first distance, we show that q(x) is proportional to the geometric mean of p(x) and p(−x). For example, a symmetrized shifted exponential is a centered uniform, and a symmetrized shifted gamma is a centered beta random variable. For the second distance, q(x) is proportional to the square of the arithmetic mean of p(x)1/2 and p(−x)1/2. Sample versions are also given for each.We also give the optimal random function f such that f (X) is symmetrically distributed and minimizes |f (X) − X|. Finally, we show how to optimize the Hellinger distance for vector X subject to supersymmetry and for scalar X subject to being monotone about zero in each half-line.