{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,2]],"date-time":"2026-01-02T07:45:25Z","timestamp":1767339925171,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"16","license":[{"start":{"date-parts":[[2022,8,16]],"date-time":"2022-08-16T00:00:00Z","timestamp":1660608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Research Foundation of Korea (NRF)","award":["2020R1A2C1A01100526","UIDB\/00297\/2020","UIDP\/00297\/2020"],"award-info":[{"award-number":["2020R1A2C1A01100526","UIDB\/00297\/2020","UIDP\/00297\/2020"]}]},{"name":"Portuguese national funds","award":["2020R1A2C1A01100526","UIDB\/00297\/2020","UIDP\/00297\/2020"],"award-info":[{"award-number":["2020R1A2C1A01100526","UIDB\/00297\/2020","UIDP\/00297\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>The Behrens\u2013Fisher problem occurs when testing the equality of means of two normal distributions without the assumption that the two variances are equal. This paper presents approaches based on the exact and near-exact distributions for the test statistic of the Behrens\u2013Fisher problem, depending on different combinations of even or odd sample sizes. We present the exact distribution when both sample sizes are odd and the near-exact distribution when one or both sample sizes are even. The near-exact distributions are based on a finite mixture of generalized integer gamma (GIG) distributions, used as an approximation to the exact distribution, which consists of an infinite series. The proposed tests, based on the exact and the near-exact distributions, are compared with Welch\u2019s t-test through Monte Carlo simulations, in particular for small and unbalanced sample sizes. The results show that the proposed approaches are competent solutions to the Behrens\u2013Fisher problem, exhibiting precise sizes and better powers than Welch\u2019s approach for those cases. Numerical studies show that the Welch\u2019s t-test tends to be a bit more conservative than the test statistics based on the exact or near-exact distribution, in particular when sample sizes are small and unbalanced, situations in which the proposed exact or near-exact distributions obtain higher powers than Welch\u2019s t-test.<\/jats:p>","DOI":"10.3390\/math10162953","type":"journal-article","created":{"date-parts":[[2022,8,17]],"date-time":"2022-08-17T03:15:27Z","timestamp":1660706127000},"page":"2953","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["An Exact and Near-Exact Distribution Approach to the Behrens\u2013Fisher Problem"],"prefix":"10.3390","volume":"10","author":[{"given":"Serim","family":"Hong","sequence":"first","affiliation":[{"name":"College of Liberal Studies, Seoul National University, Seoul 08826, Korea"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5628-0625","authenticated-orcid":false,"given":"Carlos A.","family":"Coelho","sequence":"additional","affiliation":[{"name":"NOVA Math (CMA-FCT\/UNL) and Mathematics Department, NOVA School of Science and Technology, NOVA University of Lisbon (FCT\/UNL), 2829-516 Caparica, Portugal"}]},{"given":"Junyong","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Statistics, Seoul National University, Seoul 08826, Korea"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,16]]},"reference":[{"key":"ref_1","unstructured":"Linnik, J.V. (1968). Statistical Problems with Nuisance Parameters (Scripta Technica, Trans.), American Mathematical Society. (Original Work Published 1966)."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1111\/j.1469-1809.1935.tb02120.x","article-title":"The fiducial argument in statistical inference","volume":"6","author":"Fisher","year":"1935","journal-title":"Ann. Eugen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1111\/j.1469-1809.1939.tb02205.x","article-title":"The comparison of samples with possibly unequal variances","volume":"9","author":"Fisher","year":"1939","journal-title":"Ann. Eugen."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1111\/j.1469-1809.1940.tb02236.x","article-title":"Note on the Behrens-Fisher formula","volume":"10","author":"Jeffreys","year":"1940","journal-title":"Ann. Eugen."},{"key":"ref_5","unstructured":"Jeffreys, H. (1961). Theory of Probability, Oxford University Press. [3rd ed.]."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1093\/biomet\/29.3-4.350","article-title":"The significance of the difference between two means when the population variances are unequal","volume":"29","author":"Welch","year":"1938","journal-title":"Biometrika"},{"key":"ref_7","first-page":"28","article-title":"The generalization of \u2018Student\u2019s\u2019 problem when several different population variances are involved","volume":"34","author":"Welch","year":"1947","journal-title":"Biometrika"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1006\/jmva.1997.1710","article-title":"The Generalized Integer Gamma distribution\u2014A basis for distributions in Multivariate Stastistics","volume":"64","author":"Coelho","year":"1998","journal-title":"J. Multivar. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/15598608.2007.10411821","article-title":"The wrapped Gamma distribution and wrapped sums and linear combinations of independent Gamma and Laplace distributions","volume":"1","author":"Coelho","year":"2007","journal-title":"J. Stat. Theory Pract."},{"key":"ref_10","unstructured":"Abramowitz, M., and Stegun, I.A. (1974). Handbook of Mathematical Functions, Dover. [9th ed.]."}],"container-title":["Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/16\/2953\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:10:30Z","timestamp":1760141430000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2227-7390\/10\/16\/2953"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,16]]},"references-count":10,"journal-issue":{"issue":"16","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["math10162953"],"URL":"https:\/\/doi.org\/10.3390\/math10162953","relation":{},"ISSN":["2227-7390"],"issn-type":[{"type":"electronic","value":"2227-7390"}],"subject":[],"published":{"date-parts":[[2022,8,16]]}}}