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Measuring association with Wasserstein distances. (English) Zbl 07594079

Summary: 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, Beiglböck, Wiesel. Estimating processes in adapted Wasserstein distance. 2020]. Our approach applies to probability laws on general Polish spaces.

MSC:

62Gxx Nonparametric inference
62Hxx Multivariate analysis
90Cxx Mathematical programming

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References:

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