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