On the robustness to adversarial corruption and to heavy-tailed data of the Stahel–Donoho median of means

We consider median of means (MOM) versions of the Stahel–Donoho outlyingness (SDO) [23, 66] and of the Median Absolute Deviation (MAD) [30] functions to construct subgaussian estimators of a mean vector under adversarial contamination and heavy-tailed data. We develop a single analysis of the MOM version of the SDO which covers all cases ranging from the Gaussian case to the $L_2$ case. It is based on isomorphic and almost isometric properties of the MOM versions of SDO and MAD. This analysis also covers cases where the mean does not even exist but a location parameter does; in those cases we still recover the same subgaussian rates and the same price for adversarial contamination even though there is not even a first moment. These properties are achieved by the classical SDO median and are therefore the first non-asymptotic statistical bounds on the Stahel–Donoho median complementing the $\sqrt{n}$-consistency [58] and asymptotic normality [74] of the Stahel–Donoho estimators. We also show that the MOM version of MAD can be used to construct an estimator of the covariance matrix only under the existence of a second moment or of a scatter matrix if a second moment does not exist.