Mean Testing under Truncation beyond Gaussian

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 $\varepsilon$-fraction of the probability mass. For distributions with $p$-th directional moments of magnitude at most $\nu_{P,p}$, truncation induces a bias of order $\mathcal{O}(\nu_{P,p}\varepsilon^{1-1/p})$. This bias creates a sharp information-theoretic detectability floor: when the signal $\alpha$ 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 $$ n = \mathcal{O}\left(\frac{\|\Sigma_P\|}{(\alpha-4\nu_{P,p}\varepsilon^{1-1/p})^2}\sqrt{d}\right). $$ We further identify a structural escape from this finite-moment bias barrier. Under a *directional median regularity* assumption, truncation bias improves to linear order $\mathcal{O}(\varepsilon)$. This reveals an intermediate regime in which estimation requires $\Theta(d)$ samples for uniform recovery, while testing recovers the classical $\Theta(\sqrt d)$ 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.