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 nearoptimal 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 μ ​N​ which satisfies, for every direction u∈Sd−1, ⟨μ ​N​−μ,u⟩≤N ​C​(σ(u)log(1/δ) ​+(E∥X−EX∥2)1/2), where σ2(u)=Var(⟨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.