# Multivariate mean estimation with direction-dependent accuracy

### Gábor Lugosi

Pompeu Fabra University, Barcelona; ICREA, Barcelona; Barcelona School of Economics, Spain### Shahar Mendelson

University of Warwick, Coventry, UK; Australian National University, Canberra, Australia

## 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 $1−δ$, the procedure returns $μ _{N}$ which satisfies, for every direction $u∈S_{d−1}$, $⟨μ _{N}−μ,u⟩≤N C (σ(u)g(1/δ) +(E∥X−EX∥_{2})_{1/2}),$ where $σ_{2}(u)=Var(⟨X,u⟩)$ 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.

## Cite this article

Gábor Lugosi, Shahar Mendelson, Multivariate mean estimation with direction-dependent accuracy. J. Eur. Math. Soc. 26 (2024), no. 6, pp. 2211–2247

DOI 10.4171/JEMS/1321