Dualizing Le Cam's method, with applications to estimating the unseens

摘要: Le Cam's method (or the two-point method) is a commonly used tool for obtaining statistical lower bound and especially popular functional estimation problems. This work aims to explain give conditions tightness of in from perspective convex duality. Under variety settings it shown that maximization problem searches best bound, upon dualizing, becomes minimization optimizes bias-variance tradeoff among family estimators. For estimating linear functionals distribution our strengthens prior results Donoho-Liu \cite{DL91} (for quadratic loss) by dropping Holderian assumption on modulus continuity. exponential families extend those Juditsky-Nemirovski \cite{JN09} characterizing minimax risk loss under weaker assumptions family. We also provide an extension high-dimensional setting separable functionals. Notably, coupled with tools complex analysis, this particularly effective ``elbow effect'' -- phase transition parametric nonparametric rates. As main application we derive sharp rates Distinct elements (given fraction $p$ colored balls urn containing $d$ balls, optimal error number distinct colors $\tilde \Theta(d^{-\frac{1}{2}\min\{\frac{p}{1-p},1\}})$) Fisher's species $n$ iid observations unknown distribution, prediction unseen symbols next (unobserved) $r \cdot n$ \Theta(n^{-\min\{\frac{1}{r+1},\frac{1}{2}\}})$).