Multivariate mean estimation with direction-dependent accuracy
(2010.11921)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 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|_22\right){1/2} \right)~, ] where $\sigma2(u) = \var(\inr{X,u})$ 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.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.