TL;DR
This paper unifies various nonparametric two-sample tests through the lens of Wasserstein distance, connecting classical and modern methods across univariate and multivariate settings.
Contribution
It establishes new links between diverse two-sample tests, including univariate and multivariate methods, using Wasserstein distance as a central framework.
Findings
Connections between Kolmogorov-Smirnov, PP/QQ plots, and ROC/ODC curves.
Introduction of a smoothed Wasserstein distance for testing.
Identification of previously unnoticed links in the literature.
Abstract
Nonparametric two sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being intelligently designed and analyzed, both for the unidimensional and the multivariate setting. Our contribution is to tie together many of these tests, drawing connections between seemingly very different statistics. In this work, our central object is the Wasserstein distance, as we form a chain of connections from univariate methods like the Kolmogorov-Smirnov test, PP/QQ plots and ROC/ODC curves, to multivariate tests involving energy statistics and kernel based maximum mean discrepancy. Some connections proceed through the construction of a \textit{smoothed} Wasserstein distance, and…
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