{"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":1.428257183502713,"base_score":9.521714556684751,"endowment":9.521714556684751,"self_citation_contribution":1.428257183502713,"citation_network_contribution":0.0,"self_endowment_contribution":1.428257183502713,"citer_contribution":0.0,"corpus_percentile":null,"corpus_rank":null,"citation_count":13652,"citer_count":0,"citers_with_citation_signal":0,"citers_with_endowment":0,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":0.0443,"is_oa":true,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":"1947-03-01","fair_score":17.7083,"fair_percentile":1.6710642040457344,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","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,"fair_f":20.0,"fair_a":30.0,"fair_i":12.5,"fair_r":8.3333,"fair_zscore":-2.4878,"fair_rationale":{"fair_score":17.71,"has_llm":true,"dimensions":{"F":{"name":"Findable","score":20.0,"criteria":[{"key":"f_has_doi","label":"Has a persistent DOI","kind":"deterministic","weight":1.0,"fraction":1.0,"signal":"DOI present","rationale":null},{"key":"f_repository_presence","label":"Indexed in repositories / literature DBs","kind":"deterministic","weight":1.0,"fraction":0.0,"signal":"datacite=0, pmcid=False, pmid=False","rationale":null},{"key":"f_persistent_ids","label":"Resolvable scholarly identifiers (OpenAlex)","kind":"deterministic","weight":0.5,"fraction":0.0,"signal":"no OpenAlex id","rationale":null},{"key":"f_metadata_richness","label":"Rich, machine-readable metadata","kind":"llm","weight":1.0,"fraction":0.0,"signal":null,"rationale":"No machine-readable metadata is provided; the paper only contains natural language text and mathematical notation."}]},"A":{"name":"Accessible","score":30.0,"criteria":[{"key":"a_open_access","label":"Open Access / files deposited","kind":"deterministic","weight":1.5,"fraction":1.0,"signal":"Open Access","rationale":null},{"key":"a_retrievable","label":"Free full text retrievable","kind":"deterministic","weight":1.0,"fraction":0.0,"signal":"0 OA location(s)","rationale":null},{"key":"a_access_protocol","label":"Clear data/code access protocol","kind":"llm","weight":1.0,"fraction":0.0,"signal":null,"rationale":"No protocol for accessing data or code is mentioned; the paper does not describe any such materials."}]},"I":{"name":"Interoperable","score":12.5,"criteria":[{"key":"i_linked_data","label":"Linked datasets / DataCite relations","kind":"deterministic","weight":1.0,"fraction":0.0,"signal":"linked_datasets=0, datacite=0","rationale":null},{"key":"i_standard_ids","label":"References data via standard accessions","kind":"deterministic","weight":1.0,"fraction":0.0,"signal":"accessions=0, trials=0","rationale":null},{"key":"i_standards","label":"Standard formats, vocabularies & identifiers","kind":"llm","weight":1.0,"fraction":0.25,"signal":null,"rationale":"The paper uses standard mathematical notation and statistical terms, but lacks formal identifiers or structured vocabularies for variables or results."}]},"R":{"name":"Reusable","score":8.33,"criteria":[{"key":"r_license","label":"Clear, open reuse license","kind":"deterministic","weight":1.5,"fraction":0.0,"signal":"no license","rationale":null},{"key":"r_downloads","label":"Demonstrated reuse (downloads)","kind":"deterministic","weight":0.5,"fraction":0.0,"signal":"downloads=0","rationale":null},{"key":"r_version","label":"Versioned / maintained","kind":"deterministic","weight":0.5,"fraction":0.0,"signal":"no version chain","rationale":null},{"key":"r_dataset","label":"Classified as a data resource","kind":"deterministic","weight":0.5,"fraction":0.0,"signal":"not a dataset","rationale":null},{"key":"r_reusability","label":"Data-availability statement, license & reproducibility","kind":"llm","weight":2.0,"fraction":0.167,"signal":null,"rationale":"No data-availability statement, license, or explicit reproducibility details are provided; only computed tables for small sample sizes are mentioned, but not shared as reusable artifacts."}]}},"suggestions":["Provide a machine-readable metadata file (e.g., JSON-LD) describing the paper's statistical methods and test.","Deposit the computed probability tables and recurrence relation code in a public repository with a clear access protocol and license.","Assign standard identifiers (e.g., DOIs, ORCIDs, RRIDs) to datasets, code, and key statistical terms to improve interoperability.","Add a formal data-availability statement specifying where supplementary materials are hosted and how to access them.","Include a reproducibility section with input parameters, software versions, and steps to regenerate the tables and moments."],"model":"deepseek/deepseek-v4-flash","agent_version":"fair_agent_v1","fulltext_source":"abstract_only"},"fair_model":"deepseek/deepseek-v4-flash","fair_agent_version":"fair_agent_v1","fair_fulltext_source":"abstract_only","fair_has_llm":true,"fair_computed_at":"2026-06-14T20:30:55.707565Z","clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}