Mean Testing under Truncation beyond Gaussian

Abstract

We characterize the fundamental limits of high-dimensional mean testing under arbitrary truncation, where samples are drawn from the conditional distribution $P(\cdot \mid S)$ for an unknown truncation set $S$ that may hide up to an $\epsilon$-fraction of the probability mass. For distributions with $p$-th directional moments of magnitude at most $\hat{I}{½}_{P,p}$, truncation induces a bias of order $O(\hat{I}{½}_{P,p}{\epsilon }^{1-1/p})$. This bias creates a sharp information-theoretic detectability floor: when the signal $\hat{I}\pm$ falls below this threshold, the null and alternative hypotheses are indistinguishable even with infinite data. Above this floor, we prove that a simple second-order test achieving near-optimal sample complexity $n=O\left(\frac{\Vert \hat{I}{\pounds }_P\Vert }{(\hat{I}\pm -4\hat{I}{½}_{P,p}{\epsilon }^{1-1/p}{)}^2}\sqrt{d}\right)$. We further identify a structural escape from this finite-moment bias barrier. Under a directional median regularity assumption, truncation bias improves to linear order $O(\epsilon )$. This reveals an intermediate regime in which estimation requires $\hat{I}˜(d)$ samples for uniform recovery, while testing recovers the classical $\hat{I}˜(\sqrt{d})$ rate once truncation bias is eliminated. Together, our results provide a unified framework for mean testing under truncation, connecting finite-moment, sub-Gaussian, and median-regular structural regimes.