Beyond Gaussian Approximation: Bootstrap in Large Scale Simultaneous Inference
The Bonferroni adjustment, or the union bound, is commonly used to develop and study rate optimal statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme value theory has been proven to provide more accurate multiplicity adjustments in a number of settings, but only on ad hoc bases. Recently, Gaussian approximation was used to justify bootstrap adjustments in large scale simultaneous inference in some general settings when $n \gg (\log p)^7$, where $p$ is the multiplicity of the inference problem and $n$ is the sample size. The thrust of this theory is the validity of the Gaussian approximation for maxima of sums of independent random vectors in high-dimension. In this paper, we reduce the sample size requirement to $n \gg (\log p)^5$ for the consistency of the empirical bootstrap and the multiplier/wild bootstrap in the Kolmogorov distance, and to $n \gg \log p$ for certain approximately optimal bootstrap adjustments. New comparison and anti-concentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer directly applicable in the regime under consideration.