Worst-case Nonparametric Bounds for the Student T-statistic

TL;DR

This paper precisely determines the worst-case nonparametric bounds for the Student T-statistic by solving an extremal problem involving mid-quantiles, providing explicit formulas and characterizations for optimal weights.

Contribution

It solves an open extremal problem exactly, characterizing the maximal bounds and optimal weight vectors for the T-statistic's nonparametric bounds.

Findings

Maximal bounds are achieved by k-sparse equal-weight vectors.

Optimal support size k is found via finite search over at most n candidates.

Provides explicit envelope M_n(α) and its universal limit as n grows.

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}) $\tilde Q(S(w);\alpha)$ 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 $\alpha<{1\over 2}$ 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 $M_n(\alpha)$ 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)