多尺度中位数摘要的自适应最优位置估计

中文摘要 AI 辅助

位置估计在不同噪声分布下表现出显著不同的有限样本行为：正则族通常产生根号n速率，而紧支撑分布可能允许更快的、由边界驱动的速率。我们质疑，一个不知道密度形状的单一估计器，是否能像知道底层位置族的预言机那样，适应于实例级最优估计速率。对于具有对称对数凹噪声密度f的已知位置族，在失败概率δ下，样本量为n的最优位置估计误差已知为Le Cam两点速率：sup{r>0:H^2(f_0, f_{2r})≲log(1/δ)/n}。当位置族未知时，我们提出了一种形状无关的估计器，该估计器在具有非递减风险率的所有对称单峰密度上同时达到此预言机基准，这一类别严格宽于对称对数凹分布。我们建立了Hellinger驱动的两点速率可以仅由二元分位数间隙的多尺度函数来表征。这一Hellinger散度与分位数几何之间的新结构联系，激发了一种简单的估计程序，该程序通过精心设计的数据相关权重聚合样本中位数摘要。所得估计器在有限样本下是实例最优的，并且仅在排序样本上运行O(log(n))时间。

英文摘要

Location estimation exhibits markedly different finite-sample behavior across noise distributions: regular families typically yield root-\(n\) rates, whereas compactly supported laws may admit faster, boundary-driven rates. We question whether a single estimator, without knowledge of the density's shape, can adapt to the instance-wise optimal estimation rate, as an oracle that knows the underlying location family can. For a known location family with symmetric log-concave noise density \(f\), the optimal location estimation error with sample size \(n\) under failure probability \(δ\) is known to be Le Cam's two-point rate: \[ \sup\left\{r>0:\mathsf{H}^2\left(f_0, f_{2r}\right)\lesssim \frac{\log(1/δ)}{n}\right\}. \] When the location family is unknown, we propose a shape-agnostic estimator that attains this oracle benchmark simultaneously over all symmetric unimodal densities with non-decreasing hazard rates, a class strictly broader than symmetric log-concave distributions. We establish that the Hellinger-driven two-point rate can be characterized solely by a multiscale function of dyadic quantile gaps. This new structural connection between Hellinger divergence and quantile geometry motivates a simple estimation procedure that aggregates sample mid-summaries with carefully designed data-dependent weights. The resulting estimator is finite-sample instance-optimal and runs in only \(O(\log(n))\) time on sorted samples.

发表机构

• Department of Statistics, University of Chicago（芝加哥大学统计学系）

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