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Entropy 2017, 19(2), 47; doi:10.3390/e19020047

On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests

2
Department of Mathematics, Brown University, Providence, RI 02912, USA
3
Laboratory of Statistics, CREST, ENSAE, Université Paris-Saclay, Malakoff 92240, France
1
Departments of Statistics and Computer Science, University of California, Berkeley, CA 94703, USA
*
Author to whom correspondence should be addressed.
Received: 28 May 2016 / Accepted: 26 December 2016 / Published: 26 January 2017
(This article belongs to the Special Issue Statistical Significance and the Logic of Hypothesis Testing)
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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 designed and analyzed, both for the unidimensional and the multivariate setting. In this short survey, we focus on test statistics that involve the Wasserstein distance. Using an entropic smoothing of the Wasserstein distance, we connect these to very different tests including multivariate methods involving energy statistics and kernel based maximum mean discrepancy and univariate methods like the Kolmogorov–Smirnov test, probability or quantile (PP/QQ) plots and receiver operating characteristic or ordinal dominance (ROC/ODC) curves. Some observations are implicit in the literature, while others seem to have not been noticed thus far. Given nonparametric two-sample testing’s classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others. View Full-Text
Keywords: two-sample testing; wasserstein distance; entropic smoothing; energy distance; maximum mean discrepancy; QQ and PP plots; ROC and ODC curves two-sample testing; wasserstein distance; entropic smoothing; energy distance; maximum mean discrepancy; QQ and PP plots; ROC and ODC curves
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Ramdas, A.; Trillos, N.G.; Cuturi, M. On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests. Entropy 2017, 19, 47.

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