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. 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 are the new deviation inequalities for the U-statistics of order that is allowed to grow with the sample size, a result that could be of independent interest.

翻译：本文针对以下的问题:如果给i. id. 随机变量样本,且有一定差异,那么,人们能否构建一个几乎和数据正常分布的未知值的估测符?实现此目标的最受欢迎的例子之一是手段估测器的中位值。然而,由于结果界限的常数低于最佳值,因此效率低下。我们发现,对手段估测器的中位值进行变异性修改后,如果基本分布超过$\frac{3 ⁇ sqrt{5 ⁇ 2 ⁇ 2 ⁇ approx 2.62美元,那么其偏差保证值可高达1美元+o(1)美元,如果基本分布器拥有超过$\frac{3 ⁇ sqrt{5 ⁇ 2 ⁇ approx 2.62美元时段,且对Lebesgue测量具有绝对持续性,则其结果可能会使依赖手段估测器的中位值作为建筑块的各种算法产生潜在的改进。我们的论点的核心是,允许以样本大小增长的U- Statist秩序的新的偏差不平等性,结果可能是独立的兴趣。