Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors

The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of 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 an ad hoc basis. Recently, Gaussian approximation has been 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–Smirnov distance, possibly in the regime where the Gaussian approximation is not available. New comparison and anticoncentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable or strong enough to produce desired results.The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical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 an ad hoc basis. Recently, Gaussian approximation has been 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–Smirnov distance, possibly in the regime where the Gaussian approximation is not available. New comparison and anticoncentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable or strong enough to produce desired results.

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