Multivariate mean estimation with direction-dependent accuracy
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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-δ, the procedure returns μ_N which satisfies that for every direction u ∈ S^d-1, μ_N - μ, u≤C/√(N)( σ(u)√(log(1/δ)) + (X- X_2^2)^1/2) , where σ^2(u) = (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.
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