Measuring association with Wasserstein distances

TL;DR

This paper introduces Wasserstein correlation coefficients as nonparametric measures of association between probability measures on Polish spaces, with proven statistical properties and convergence rates.

Contribution

It proposes a new class of Wasserstein-based association measures and establishes their statistical consistency and convergence rates for compactly supported measures.

Findings

Wasserstein correlation coefficients effectively measure dependence.

Strong consistency of estimators is established.

Convergence rates are determined for compactly supported measures.

Abstract

Let $\pi\in \Pi(\mu,\nu)$ be a coupling between two probability measures $\mu$ and $\nu$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $\mu$ and $\nu$, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between $\nu$ and the disintegration $\pi_{x_1}$ of $\pi$ with respect to the first coordinate. We also establish basic statistical properties of this new class of measures: we develop a statistical theory for strongly consistent estimators and determine their convergence rate in the case of compactly supported measures $\mu$ and $\nu$. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglb\"ock, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.