Multivariate mean estimation with direction-dependent accuracy

Lugosi, Gábor; Mendelson, Shahar

doi:http://dx.doi.org/10.4171/JEMS/1321

Multivariate mean estimation with direction-dependent accuracy

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Citació

Lugosi G, Mendelson S. Multivariate mean estimation with direction-dependent accuracy. J. Eur. Math. Soc. 2024;26(6):2211-47. DOI: 10.4171/JEMS/1321

Resum

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 μN which satisfies, for every direction u∈Sd−1, ⟨μN−μ,u⟩≤NC(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.