Title: High-dimensional Gaussian and bootstrap approximations for robust means Show Chinese title

Title: 高维高斯和自助法逼近稳健均值

Abstract: Recent years have witnessed much progress on Gaussian and bootstrap approximations to the distribution of max-type statistics of sums of independent random vectors with dimension $d$ large relative to the sample size $n$. However, for any number of moments $m>2$ that the summands may possess, there exist distributions such that these approximations break down if $d$ grows faster than $n^{\frac{m}{2}-1}$. In this paper, we establish Gaussian and bootstrap approximations to the distributions of winsorized and trimmed means that allow $d$ to grow at an exponential rate in $n$ as long as $m>2$ moments exist. The approximations remain valid under some amount of adversarial contamination. Our implementations of the winsorized and trimmed means are fully data-driven and do not depend on any unknown population quantities. As a consequence, the performance of the approximation guarantees ``adapts'' to $m$. Show Chinese abstract

Abstract: 近年来，在高维独立随机向量之和的最大型统计量的分布的高斯逼近和自助法逼近方面取得了很大进展，其中随机向量的维度 $d$ 相对于样本容量 $n$ 较大。 然而，对于随机变量项可能具有的任意阶矩 $m>2$，若 $d$ 的增长速度超过 $n^{\frac{m}{2}-1}$，则这些逼近方法失效。 本文中，我们建立了高斯近似和自助法（bootstrap）近似来逼近 Winsorized 均值和修剪均值的分布，允许$d$在$n$中以指数速率增长，只要存在$m>2$阶矩。 在一定程度的对抗性污染下，这些近似仍然有效。 我们的 Winsorized 均值和修剪均值的实现完全是数据驱动的，并且不依赖于任何未知的总体量。 因此，近似的性能保证会“适应”$m$。