{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T16:43:45Z","timestamp":1792601025193,"version":"4.5.0"},"reference-count":47,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2017,1,26]],"date-time":"2017-01-26T00:00:00Z","timestamp":1485388800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2013Smirnov 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\u2019s classical and continued importance, we aim to provide useful connections for theorists and practitioners familiar with one subset of methods but not others.<\/jats:p>","DOI":"10.3390\/e19020047","type":"journal-article","created":{"date-parts":[[2017,1,26]],"date-time":"2017-01-26T09:55:23Z","timestamp":1485424523000},"page":"47","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":375,"title":["On Wasserstein Two-Sample Testing and Related Families of Nonparametric Tests"],"prefix":"10.3390","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0497-311X","authenticated-orcid":false,"given":"Aaditya","family":"Ramdas","sequence":"first","affiliation":[{"name":"Departments of Statistics and Computer Science, University of California, Berkeley, CA 94703, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicol\u00e1s","family":"Trillos","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Brown University, Providence, RI 02912, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Cuturi","sequence":"additional","affiliation":[{"name":"Laboratory of Statistics, CREST, ENSAE, Universit\u00e9 Paris-Saclay, Malakoff 92240, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2017,1,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1016\/j.jspi.2006.06.003","article-title":"A nonparametric test for similarity of marginals\u2014with applications to the assessment of population bioequivalence","volume":"137","author":"Freitag","year":"2007","journal-title":"J. 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