Attainability of Two-Point Testing Rates for Finite-Sample Location Estimation

LeCam's two-point testing method yields perhaps the simplest lower bound for estimating the mean of a distribution: roughly, if it is impossible to well-distinguish a distribution centered at $\mu$ from the same distribution centered at $\mu+\Delta$, then it is impossible to estimate the mean by better than $\Delta/2$. It is setting-dependent whether or not a nearly matching upper bound is attainable. We study the conditions under which the two-point testing lower bound can be attained for univariate mean estimation; both in the setting of location estimation (where the distribution is known up to translation) and adaptive location estimation (unknown distribution). Roughly, we will say an estimate nearly attains the two-point testing lower bound if it incurs error that is at most polylogarithmically larger than the Hellinger modulus of continuity for $\tilde{\Omega}(n)$ samples.