Multivariate mean estimation with direction-dependent accuracy

We consider the problem of estimating the mean of a random vector based on $N$ independent, identically distributed observations. We prove the existence of an estimator that has a near-optimal error in all directions in which the variance of the one dimensional marginal of the random vector is not too small: with probability $1-\delta$, the procedure returns $\wh{\mu}_N$ which satisfies that for every direction $u \in S^{d-1}$, $$\inr{\wh{\mu}_N - \mu, u}\le \frac{C}{\sqrt{N}} \left( \sigma(u)\sqrt{\log(1/\delta)} + \left(\E\|X-\EXP X\|_2^2\right)^{1/2} \right)~,$$ where $\sigma^2(u) = \var(\inr{X,u})$ and $C$ is a constant. To achieve this, we require only slightly more than the existence of the covariance matrix, in the form of a certain moment-equivalence assumption. The proof relies on novel bounds for the ratio of empirical and true probabilities that hold uniformly over certain classes of random variables.