Robust Statistical Estimators with Bounded Empirical Sensitivity
Valentio Iverson, Gautam Kamath, Argyris Mouzakis, Adam Smith
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.
Topics
Classified with taxonomy v2 on Wed, 2 Sept 2026.