{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:56:40Z","timestamp":1760237800325,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T00:00:00Z","timestamp":1655769600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The well-known Mann\u2013Whitney\u2013Wilcoxon (MWW) statistic is based on empirical distribution estimates. However, the data are often drawn from smooth populations. Therefore, the smoothness characteristic is not preserved. In addition, several authors have pointed out that empirical distribution is often an inadmissible estimate. Thus, in this work, we develop smooth versions of the MWW statistic based on smooth distribution function estimates. This approach preserves the data characteristics and allows the efficiency of the procedure to improve. In addition, our procedure is shown to be robust against a large class of dependent observations. Hence, by choosing a rectangular array of known distribution functions, our procedure allows the test to be a lot more reflective of the real data.<\/jats:p>","DOI":"10.3390\/axioms11070300","type":"journal-article","created":{"date-parts":[[2022,6,21]],"date-time":"2022-06-21T21:55:19Z","timestamp":1655848519000},"page":"300","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Smooth Versions of the Mann\u2013Whitney\u2013Wilcoxon Statistics"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5872-607X","authenticated-orcid":false,"given":"Netti","family":"Herawati","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Lampung, Bandar Lampung 35141, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ibrahim A.","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Statistics, Oklahoma State University, Stillwater, OK 74078, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1214\/aoms\/1177730491","article-title":"On a test whether one of two random variables is stochastically larger than the other","volume":"18","author":"Mann","year":"1947","journal-title":"Ann. 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