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-δ, the procedure returns whμN which satisfies that for every direction u ∈ Sd-1, [ inrwhμN - μ, u≤ fracC√N left( σ(u)√log(1/δ) + left(E|X-EXP X|22right)1/2 right)~, ] where σ2(u) = var(inrX,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.