Near-optimal mean estimators with respect to general norms

We study the problem of estimating the mean of a random vector in R d based on an i.i.d. sample, when the accuracy of the estimator is measured by a general norm on R d . We construct an estimator (that depends on the norm) that achieves an essentially optimal accuracy/confidence tradeoff under the only assumption that the random vector has a well-defined covariance matrix. At the heart of the argument is the construction of a uniform median-of-means estimator in a class of real valued functions. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.