标题： 对函数估计的Le Cam方法进行对偶化，以及在估计未观测事物中的应用 显示英文标题

标题： Dualizing Le Cam's method for functional estimation, with applications to estimating the unseens

摘要： Le Cam方法（或两点法）是用于获得统计下界的一种常用工具，尤其在函数估计问题中非常流行。 本工作旨在从凸对偶性的角度解释并给出Le Cam下界在函数估计中的紧性条件。 在各种设置下，展示了搜索最佳两点下界的最大化问题，在对偶化后变为一个最小化问题，该问题在一组估计量中优化偏差-方差权衡。 对于估计分布的线性泛函，我们的工作通过去掉Donoho-Liu\cite{DL91}（针对二次损失）的连续模的Hölder假设，加强了先前的结果。 对于指数族，我们的结果通过在指数族上较弱的假设下表征二次损失的最小最大风险，扩展了Juditsky-Nemirovski\cite{JN09}的结果。 我们还提供了对可分离泛函估计的高维设置的扩展。 值得注意的是，结合复分析的工具，这种方法在表征“拐点效应”——从参数速率到非参数速率的相变方面特别有效。 作为主要应用，我们推导了不同元素问题的精确极小极大率（给定一个分数$p$的颜色球从一个包含$d$个球的罐子中取出，估计不同颜色数量的最佳误差是$\tilde \Theta(d^{-\frac{1}{2}\min\{\frac{p}{1-p},1\}})$）以及费舍尔的物种问题（给定$n$个独立同分布观测值来自一个未知分布，预测接下来（未观察到的）$r \cdot n$个观测值中未见过符号的最佳预测误差是$\tilde \Theta(n^{-\min\{\frac{1}{r+1},\frac{1}{2}\}})$）。 显示英文摘要

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