Improved Concentration for Mean Estimators via Shrinkage

TL;DR

This paper introduces a robust mean estimation method that adaptively shrinks the influence of outliers, achieving stronger concentration bounds and faster convergence than traditional estimators, with theoretical guarantees and practical benefits.

Contribution

It proposes a unified framework for robust mean estimation using adaptive shrinkage, extending existing methods and providing sub-Gaussian concentration guarantees.

Findings

Achieves sub-Gaussian concentration bounds for the proposed estimators.

Demonstrates faster concentration in numerical experiments with small samples.

Unifies and extends several existing robust mean estimation approaches.

Abstract

We study a class of robust mean estimators $\widehat{\mu}$ obtained by adaptively shrinking the weights of sample points far from a base estimator $\widehat{\kappa}$. Given a data-dependent scaling factor $\widehat{\alpha}$ and a weighting function $w:[0, \infty) \to [0,1]$, we let $\widehat{\mu} = \widehat{\kappa} + \frac{1}{n}\sum_{i=1}^n(X_i - \widehat{\kappa})w(\widehat{\alpha}|X_i-\widehat{\kappa}|) $. We prove that, under mild assumptions over $w$, these estimators achieve stronger concentration bounds than the base estimate $\widehat{\kappa}$, including sub-Gaussian guarantees. This framework unifies and extends several existing approaches to robust mean estimation in $\mathbb{R}$. Through numerical experiments, we show that our shrinking approach translates to faster concentration, even for small sample sizes.