Uniform Estimation Beyond the Mean

TL;DR

This paper introduces a unified approach for uniform estimation beyond the mean, leveraging Gaussian averages to provide bounds for a broad class of estimation problems, including functions with bounded derivatives.

Contribution

It extends existing uniform estimation bounds to a general class of functions, beyond the empirical mean, using Gaussian averages and derivative bounds.

Findings

Provides finite sample bounds for a wide class of estimators

Recovers standard results for empirical mean and U-statistics

Extends to general estimation problems with bounded derivatives

Abstract

Finite sample bounds on the estimation error of the mean by the empirical mean, uniform over a class of functions, can often be conveniently obtained in terms of Rademacher or Gaussian averages of the class. If a function of n variables has suitably bounded partial derivatives, it can be substituted for the empirical mean, with uniform estimation again controlled by Gaussian averages. Up to a constant the result recovers standard results for the empirical mean and more recent ones about U-statistics, and extends to a general class of estimation problems.