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 nm/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.