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We consider median of means (MOM) versions of the Stahel-Donoho outlyingness(SDO) [stahel 1981, donoho 1982] and of Median Absolute Deviation (MAD)functions to construct subgaussian estimators of a mean vector underadversarial contamination and heavy-tailed data. We develop a single analysisof the MOM version of the SDO which covers all cases ranging from the Gaussiancase to the L2 case. It is based on isomorphic and almost isometric propertiesof the MOM versions of SDO and MAD. This analysis also covers cases where themean does not even exist but a location parameter does; in those cases we stillrecover the same subgaussian rates and the same price for adversarialcontamination even though there is not even a first moment. These propertiesare achieved by the classical SDO median and are therefore the firstnon-asymptotic statistical bounds on the Stahel-Donoho median complementing the$\sqrt{n}$-consistency [maronna 1995] and asymptotic normality [Zuo, Cui, He,2004] of the Stahel-Donoho estimators. We also show that the MOM version of MADcan be used to construct an estimator of the covariance matrix under only aL2-moment assumption or of a scale parameter if a second moment does not exist.

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