{"ID":6267185,"CreatedAt":"2026-07-10T01:11:38.759438437Z","UpdatedAt":"2026-07-13T01:02:08.706470581Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.08415","arxiv_id":"2607.08415","title":"An Exact Distribution-Free Test for Means of Nonnegative Random Variables","abstract":"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 $i$, $\\mathbb{P}\\{K(X)\\leα\\}\\leα$ for all $0\\leα\\le1$. 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).","short_abstract":"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 $\\mathbb{E} X_i\\le1$ fo...","url_abs":"https://arxiv.org/abs/2607.08415","url_pdf":"https://arxiv.org/pdf/2607.08415v1","authors":"[\"Nikos Vlassis\",\"Philip S. Thomas\"]","published":"2026-07-09T12:39:29Z","proceeding":"math.ST","tasks":"[\"math.ST\",\"math.PR\"]","methods":"[]","has_code":false}