High-dimensional Gaussian and bootstrap approximations for robust means

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 approximation guarantees ``adapt'' to $m$.