U-statistics of growing order and sub-Gaussian mean estimators with sharp constants

This paper addresses the following question: given a sample of i.i.d. random variables with finite variance, can one construct an estimator of the unknown mean that performs nearly as well as if the data were normally distributed? One of the most popular examples achieving this goal is the median of means estimator. However, it is inefficient in a sense that the constants in the resulting bounds are suboptimal. We show that a permutation-invariant modification of the median of means estimator admits deviation guarantees that are sharp up to $1+o(1)$ factor if the underlying distribution possesses $3+p$ moments for some $p>0$ and is absolutely continuous with respect to the Lebesgue measure. This result yields potential improvements for a variety of algorithms that rely on the median of means estimator as a building block. At the core of our argument is a new deviation inequality for the U-statistics of order that is allowed to grow with the sample size, a result that could be of independent interest. Finally, we demonstrate that a hybrid of the median of means and Catoni's estimator is capable of achieving sub-Gaussian deviation guarantees with nearly optimal constants assuming just the existence of the second moment.