2020/10/22 by Gábor Lugosi, Shahar Mendelson, Lugosi, Gabor +1
Mathematics · #Statistical Methods and Inference #Markov Chains and Monte Carlo Methods #Statistical Methods and Bayesian Inference
paper · pdf · doi:10.48550/arxiv.2010.11921
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 \whμN which satisfies that for every direction u ∈ Sd-1, \inr\whμN - μ, u≤ (C)/(√(N)) ( σ(u)√(log(1/δ)) + (\E‖X-\EXP X‖22)1/2 )~, where σ2(u) = \var(\inrX,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.