High-dimensional Gaussian and bootstrap approximations for robust means

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 $m$.