{"paper":{"title":"An Exact Distribution-Free Test for Means of Nonnegative Random Variables","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Nikos Vlassis, Philip S. Thomas","submitted_at":"2026-07-09T12:39:29Z","abstract_excerpt":"Let $X=(X_1,\\ldots,X_n)$ be independent nonnegative random variables, not necessarily identically distributed. Let $D=(D_0,D_1,\\ldots,D_n)\\sim\\operatorname{Dir}(1,\\ldots,1)$ be independent of $X$, and define $K(x)=\\mathbb{P}\\{\\sum_{i=1}^n x_iD_i\\le1\\}$. We prove that, for every $n\\ge1$, whenever $\\mathbb{E} X_i\\le1$ for every $i$, $\\mathbb{P}\\{K(X)\\le\\alpha\\}\\le\\alpha$ for all $0\\le\\alpha\\le1$. Thus $K(X)$ is a finite-sample, distribution-free $p$-value for testing the null hypothesis $\\mathbb{E}X_i \\le 1$ for all $i$. This proves a conjecture of Gaffke (2005)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08415","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.08415/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}