Moment Expansions of the Energy Distance

Abstract

The energy distance is used to test distributional equality, and as a loss function in machine learning. While D^2(X, Y)=0 only when X Y, the sensitivity to different moments is of practical importance. This work considers D^2(X, Y) in the case where the distributions are close. In this regime, D^2(X, Y) is more sensitive to differences in the means X-Y, than differences in the covariances . This is due to the structure of the energy distance and is independent of dimension. The sensitivity to on versus off diagonal components of is examined when X and Y are close to isotropic. Here a dimension dependent averaging occurs and, in many cases, off diagonal correlations contribute significantly less. Numerical results verify these relationships hold even when distributional assumptions are not strictly met.

Tasks

Reproductions