高维高斯与自助法逼近在稳健均值估计中的应用 (High-dimensional Gaussian and bootstrap approximations for robust means)

from arxiv, Updates: we removed continuity conditions, corrected some minor typos, generalized the results to hyperrectangles, and extended certain discussions

Recent years have witnessed much progress on Gaussian and bootstrap approximations to the distribution 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 the polynomial barrier $n^{\frac{m}{2}-1}$. 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 do not require knowledge of $m$. As a consequence, the performance of the approximation guarantees ``adapts'' to $m$.

翻译：近年来，在独立随机向量和的分布的高斯逼近与自助法逼近方面取得了显著进展，其中维度$d$相对于样本量$n$较大。然而，对于求和项可能具有的任意阶数$m>2$的矩，存在某些分布使得当$d$的增长速度超过多项式界限$n^{\\frac{m}{2}-1}$时，这些逼近方法失效。本文建立了对缩尾均值与截尾均值分布的高斯逼近与自助法逼近，允许$d$以指数速率随$n$增长，只要存在$m>2$阶矩。即使在存在一定程度的对抗性污染下，这些逼近仍然有效。我们实现的缩尾均值与截尾均值方法无需预先知道$m$的值。因此，逼近的性能保证能够“自适应”于$m$。