Optimal Estimators for Heavy-Tailed Mean Estimation via Convex Analysis

We study optimal estimation of the location parameter of a distribution known only to lie in a symmetric moment class $\mathcal C_0$: the mean-zero distributions with bounded moment $\int\phi\, d\mathbb P\le B$ for a fixed even $\phi$. Our main result concerns the fixed-margin regime, where the error margin $\Delta$ is fixed as $n\to\infty$: we give an exact large-deviation characterization of the smallest worst-case probability $\beta_n(\Delta)$ of an error exceeding $\Delta$ that any measurable estimator can guarantee with $n$ observations. Its exponential rate is exactly a two-point Hellinger exponent over the class shifted to means $\pm\Delta$, $r(\Delta)=-\log\sup_{\mathbb P_{\pm\Delta}\in\mathcal C_{\pm\Delta}}\int\sqrt{d\mathbb P_{-\Delta}\, d\mathbb P_{\Delta}}$, achieved non-asymptotically, $\beta_n(\Delta)\le e^{-nr(\Delta)}$, by a monotone $M$-estimator synthesized from a two-parameter convex program. Lagrangian duality collapses the infinite-dimensional search over estimating functions to two multipliers, which determine a pair of envelopes characterizing the optimal estimating functions; the sandwich shape posited ad hoc in prior constructions emerges naturally. For bounded variance ($\phi(x)=x^2$, $B=\sigma^2$) the exponent is $r(\Delta)=\tfrac12\log(1+\Delta^2/\sigma^2)$. In the fixed-confidence regime, holding $\beta$ fixed and letting the optimal margin $\Delta_n(\beta)$ shrink with $n$, the same synthesis stays optimal to leading order for several concrete classes. As $\beta\downarrow0$ it attains the sharp constant $\sqrt2$ of Catoni for bounded variance and the constant $L(\alpha)$ of Lee and Bhatt et al. for bounded $\alpha$-moments, $\alpha\in(1,2)$, thereby shown tight; for slowly varying $\phi$ it is leading-order minimax at every fixed $\beta$. The least-favorable distributions are simple, supported on at most three atoms.