arXiv 2102.003562021-02math.STmath.PRstat.TH

Measuring association with Wasserstein distances

Johannes Wiesel

备注：: Forthcoming in "Bernoulli"

英文摘要：

Let $π\in Π(μ,ν)$ be a coupling between two probability measures $μ$ and $ν$ on a Polish space. In this article we propose and study a class of nonparametric measures of association between $μ$ and $ν$, which we call Wasserstein correlation coefficients. These coefficients are based on the Wasserstein distance between $ν$ and the disintegration $π_{x_1}$ of $π$ 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 $μ$ and $ν$. Throughout our analysis we make use of the so-called adapted/bicausal Wasserstein distance, in particular we rely on results established in [Backhoff, Bartl, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.

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