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Assume that $X_{1}, \ldots, X_{N}$ is an $\varepsilon$-contaminated sample of$N$ independent Gaussian vectors in $\mathbb{R}^d$ with mean $\mu$ andcovariance $\Sigma$. In the strong $\varepsilon$-contamination model we assumethat the adversary replaced an $\varepsilon$ fraction of vectors in theoriginal Gaussian sample by any other vectors. We show that there is anestimator $\widehat \mu$ of the mean satisfying, with probability at least $1 -\delta$, a bound of the form \[ \|\widehat{\mu} - \mu\|_2 \lec\left(\sqrt{\frac{\operatorname{Tr}(\Sigma)}{N}} +\sqrt{\frac{\|\Sigma\|\log(1/\delta)}{N}} +\varepsilon\sqrt{\|\Sigma\|}\right), \] where $c > 0$ is an absolute constantand $\|\Sigma\|$ denotes the operator norm of $\Sigma$. In the samecontaminated Gaussian setup, we construct an estimator $\widehat \Sigma$ of thecovariance matrix $\Sigma$ that satisfies, with probability at least $1 -\delta$, \[ \left\|\widehat{\Sigma} - \Sigma\right\| \lec\left(\sqrt{\frac{\|\Sigma\|\operatorname{Tr}(\Sigma)}{N}} +\|\Sigma\|\sqrt{\frac{\log(1/\delta)}{N}} + \varepsilon\|\Sigma\|\right). \]Both results are optimal up to multiplicative constant factors. Despite therecent significant interest in robust statistics, achieving both dimension-freebounds in the canonical Gaussian case remained open. In fact, severalpreviously known results were either dimension-dependent and required $\Sigma$to be close to identity, or had a sub-optimal dependence on the contaminationlevel $\varepsilon$.  As a part of the analysis, we derive sharp concentration inequalities forcentral order statistics of Gaussian, folded normal, and chi-squareddistributions.

Statistically Optimal Robust Mean and Covariance Estimation for Anisotropic Gaussians