Robust Statistical Estimators with Bounded Empirical Sensitivity

2026-05-20

AI Summary

This study investigates the robust stability of statistical estimators under η-fraction data corruption. To this end, it introduces “empirical sensitivity” as a novel metric quantifying an estimator’s susceptibility to data perturbations, with a focus on Gaussian mean estimation. Theoretical analysis establishes, for the first time, a lower bound of Ω(η + √(ηd/n)) on empirical sensitivity and demonstrates its tightness up to logarithmic factors. By integrating the Efron–Stein inequality with recent advances in robust mean estimation algorithms, the authors construct an estimator achieving a matching upper bound, thereby confirming the attainability of this limit. This work provides a new theoretical framework and quantitative tool for analyzing the stability of robust estimators.

Abstract

We introduce a new measure of robustness for statistical estimators, which we call \emph{empirical sensitivity}. An estimator $\hat θ$ has bounded empirical sensitivity if, with high probability over a dataset $X = (X_1, \dots, X_n) \sim \mathcal{D}^{\otimes n}$, for any dataset $Y$ obtained by modifying at most $ηn$ points in $X$, we have that $\hat θ(Y)$ is close to $\hat θ(X)$. We study bounds on this quantity for the prototypical problem of Gaussian mean estimation. We prove new lower bounds, showing that for any estimator $\hat μ$ which achieves an optimal $\ell_2$-error bound of $O\left(\sqrt{d/n}\right)$, the empirical sensitivity is at least $Ω\left(η+ \sqrt{ηd/n}\right)$. The two terms arise due to obstructions on the mean and variance (via an Efron-Stein argument) of such an estimator. We show that this bound is tight up to logarithmic factors, by employing recent results for robust empirical mean estimation.

Problem

• robustness
• statistical estimators
• empirical sensitivity
• Gaussian mean estimation
• data perturbation

Innovation

• empirical sensitivity
• robust estimation
• Gaussian mean estimation
• statistical robustness
• lower bounds