November 2022 Measuring association with Wasserstein distances
Johannes C.W. Wiesel
Author Affiliations +
Bernoulli 28(4): 2816-2832 (November 2022). DOI: 10.3150/21-BEJ1438

Abstract

Let πΠ(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 πx1 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.

Acknowledgements

The author would like to thank the Bodhi Sen and Giovanni Puccetti for helpful discussions. Furthermore the author would like to thank the anonymous referees, an Associate Editor and the Editor for their constructive comments that improved the quality of this paper.

Citation

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Johannes C.W. Wiesel. "Measuring association with Wasserstein distances." Bernoulli 28 (4) 2816 - 2832, November 2022. https://doi.org/10.3150/21-BEJ1438

Information

Received: 1 February 2021; Published: November 2022
First available in Project Euclid: 17 August 2022

zbMATH: 07594079
MathSciNet: MR4474563
Digital Object Identifier: 10.3150/21-BEJ1438

Keywords: (bicausal/adapted) Wasserstein distance , Correlation , empirical measure , independence , Measure of association , Optimal transport

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Vol.28 • No. 4 • November 2022
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