{"ID":2881470,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.13226","arxiv_id":"2508.13226","title":"Worst-case Nonparametric Bounds for the Student T-statistic","abstract":"We address the problem of finding worst-case nonparametric bounds for T-statistic by considering the extremal problem of maximising the mid-quantile (a special case of 'smoothed quantile' as discussed in \\cite{St77} and \\cite{W11}) $\\tilde Q(S(w);α)$ over nonnegative weight vectors $w\\in\\RR^n$ with $\\|w\\|_2=1$, where $S(w)=\\sum_{i=1}^n w_i \\varepsilon_i$ and $\\varepsilon_i$ are independent Rademacher variables. While classical results of Hoeffding [1] and Chernoff [2] may be used to provide sub-Gaussian upper bounds, and optimal-order inequalities were later obtained by the author [3,4], the associated extremal problem has remained unsolved. We resolve this problem exactly (for the Mid-Quantile and, trivially, the Continuous case): for each $α\u003c{1\\over 2}$ and each $n$, we determine the maximal value and characterise all maximising weights. The maximisers are $k$-sparse equal-weight vectors with weights $1/\\sqrt{k}$, and the optimal support size $k$ is found by a finite search over at most $n$ candidates. This yields an explicit envelope $M_n(α)$ and its universal limit as $n$ grows. Our results provide exact solutions to problems that have been studied through bounds and approximations for over sixty years, with applications to nonparametric inference, self-standardised statistics, and robust hypothesis testing under symmetry assumptions, including a conjecture by Edelman\\cite{edelman1990}, albeit for continuous distributions only (which he did not specify, which has been found to not always hold otherwise)","short_abstract":"We address the problem of finding worst-case nonparametric bounds for T-statistic by considering the extremal problem of maximising the mid-quantile (a special case of 'smoothed quantile' as discussed in \\cite{St77} and \\cite{W11}) $\\tilde Q(S(w);α)$ over nonnegative weight vectors $w\\in\\RR^n$ with $\\|w\\|_2=1$, where $...","url_abs":"https://arxiv.org/abs/2508.13226","url_pdf":"https://arxiv.org/pdf/2508.13226v2","authors":"[\"David Edelman\"]","published":"2025-08-17T15:47:15Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}