标题： 截尾均值的有限样本性质 显示英文标题

标题： Finite-sample properties of the trimmed mean

摘要： 从分布$P$中修剪的$n$个标量随机变量的均值是标准样本均值的一个变体，其中对于某个参数$k$，样本中最小的$k$个和最大的$k$个值被丢弃。 本文研究了作为$P$均值估计量的修剪均值的有限样本性质。假设有限方差，我们证明了修剪均值在某种意义上实现了围绕均值的高斯型集中，即它是“次高斯”的。 在稍强的假设下，我们证明了截尾均值的左尾和右尾满足与相应高斯尾部的强比值型逼近，即使对于一些数量级为$e^{-n^c}$的极小概率$c>0$也是如此。 在较具挑战性的弱矩假设和对抗性样本污染的情况下，我们证明了截尾均值在常数范围内是最小最大最优的。 显示英文摘要

摘要： The trimmed mean of $n$ scalar random variables from a distribution $P$ is the variant of the standard sample mean where the $k$ smallest and $k$ largest values in the sample are discarded for some parameter $k$. In this paper, we look at the finite-sample properties of the trimmed mean as an estimator for the mean of $P$. Assuming finite variance, we prove that the trimmed mean is ``sub-Gaussian'' in the sense of achieving Gaussian-type concentration around the mean. Under slightly stronger assumptions, we show the left and right tails of the trimmed mean satisfy a strong ratio-type approximation by the corresponding Gaussian tail, even for very small probabilities of the order $e^{-n^c}$ for some $c>0$. In the more challenging setting of weaker moment assumptions and adversarial sample contamination, we prove that the trimmed mean is minimax-optimal up to constants.