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

Understand

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 more than $\frac{3+\sqrt{5}}{2}\approx 2.62$ moments and is absolutely continuous with respect to the Lebesgue measure.