A Cram\'er-Wold theorem for elliptical distributions

TL;DR

This paper extends the Cramér-Wold theorem specifically for elliptical distributions, showing that equality of distributions can be verified through a finite set of line projections, and introduces a statistical test based on this result.

Contribution

It proves a finite-sample version of the Cramér-Wold theorem for elliptical distributions, establishing the minimal number of projections needed and developing a new statistical test for distribution equality.

Findings

The minimal number of lines needed is (d^2+d)/2.

The proposed test performs well in simulations.

The theorem applies without moment finiteness assumptions.

Abstract

According to a well-known theorem of Cram\'er and Wold, if $P$ and $Q$ are two Borel probability measures on $\mathbb{R}^d$ whose projections $P_L,Q_L$ onto each line $L$ in $\mathbb{R}^d$ satisfy $P_L=Q_L$, then $P=Q$. Our main result is that, if $P$ and $Q$ are both elliptical distributions, then, to show that $P=Q$, it suffices merely to check that $P_L=Q_L$ for a certain set of $(d^2+d)/2$ lines $L$. Moreover $(d^2+d)/2$ is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. Our theorem contrasts with other variants of the Cram\'er-Wold theorem, in that no assumption is made about the finiteness of moments of $P$ and $Q$. We use our results to derive a statistical test for equality of elliptical distributions, and carry out a small simulation study of the test, comparing it with other tests from the literature. We also give an application to learning (binary classification), again illustrated with a small simulation