Multivariate mean estimation with direction-dependent accuracy

Abstract

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 $1-\delta$, the procedure returns ${\hat{\mu }}_N$ which satisfies, for every direction $u\in S^{d-1}$, $⟨{\hat{\mu }}_N-\mu ,u⟩\le \frac{C}{\sqrt{N}}(\sigma (u)\sqrt{\log (1/\delta )}+(\mathbb{E}\Vert X-\mathbb{E}X{\Vert }^2{)}^{1/2}),$ where ${\sigma }^2(u)=\mathrm{Var}(⟨X,u⟩)$ 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.