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(· 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 nuP,p, truncation induces a bias of order O(nuP,pvarepsilon1-1/p). This bias creates a sharp information-theoretic detectability floor: when the signal α 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|SigmaP|(α-4nuP,pvarepsilon1-1/p)2√dright). 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(varepsilon). This reveals an intermediate regime in which estimation requires Theta(d) samples for uniform recovery, while testing recovers the classical Theta(√ 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.