Measuring association with Wasserstein distances

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 $\nu$. The analysis is 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. Throughout our analysis we make use of the so-called adapted/causal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglb\"ock, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our class of measures offers on alternative to the correlation coefficient proposed by [Dette, Siburg and Stoimenov (2013). A copula-based non-parametric measure of regression dependence. Scandinavian Journal of Statistics 40(1), 21-41] and [Chatterjee (2020). A new coefficient of correlation. Journal of the American Statistical Association, 1-21]. In contrast to these works, our approach also applies to probability laws on general Polish spaces.