Robust and efficient mean estimation: an approach based on the properties of self-normalized sums

TL;DR

This paper introduces a new mean estimator that is robust to outliers, provides sub-Gaussian deviation guarantees without distributional assumptions, and is asymptotically efficient, combining multiple desirable properties in one method.

Contribution

The paper proposes the first estimator that simultaneously achieves robustness, sub-Gaussian deviation bounds, and asymptotic efficiency based on self-normalized sums.

Findings

Performs well in numerical simulations compared to existing methods.

Provides tight deviation guarantees without distributional assumptions.

Achieves robustness, efficiency, and sub-Gaussian bounds simultaneously.

Abstract

Let $X$ be a random variable with unknown mean and finite variance. We present a new estimator of the mean of $X$ that is robust with respect to the possible presence of outliers in the sample, provides tight sub-Gaussian deviation guarantees without any additional assumptions on the shape or tails of the distribution, and moreover is asymptotically efficient. This is the first estimator that provably combines all these qualities in one package. Our construction is inspired by robustness properties possessed by the self-normalized sums. Theoretical findings are supplemented by numerical simulations highlighting strong performance of the proposed estimator in comparison with previously known techniques.