{"doi":"10.1214/aoms/1177730491","title":"On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other","abstract":"Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.","journal":"The Annals of Mathematical Statistics","year":1947,"id":3442,"datarank":35.14046904475806,"base_score":9.521714556684751,"endowment":9.521714556684751,"self_citation_contribution":1.428257183502713,"citation_network_contribution":33.71221186125535,"self_endowment_contribution":1.428257183502713,"citer_contribution":33.71221186125535,"corpus_percentile":99.6,"corpus_rank":713,"citation_count":13652,"citer_count":200,"citers_with_citation_signal":200,"citers_with_endowment":200,"datacite_reuse_total":0,"is_dataset":false,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"1947-03-01","authors":[{"id":36071,"name":"D. R. Whitney","orcid":null,"position":1,"is_corresponding":false},{"id":36072,"name":"Douglas R. Whitney","orcid":null,"position":2,"is_corresponding":false},{"id":36070,"name":"H. B. Mann","orcid":null,"position":0,"is_corresponding":true}],"reference_count":2,"raw_metadata":{"citation_network_status":"fetched"},"created_at":"2026-03-01T18:20:47.508186Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}