[Paper Review] Beyond Gaussian Approximation: Bootstrap for Maxima of Sums of Independent Random Vectors

Hang Deng, Cun‐Hui Zhang|arXiv (Cornell University)|May 26, 2017

Statistical Methods and Inference29 references19 citations

TL;DR

This paper develops a bootstrap method for maxima of sums of independent high-dimensional random vectors without relying on Gaussian approximation, reducing the required sample size from $ n \gg (\log p)^7 $ to $ n \gg (\log p)^5 $ for consistent inference. It establishes new comparison and anti-concentration theorems to enable valid bootstrap inference even when Gaussian approximation fails.

ABSTRACT

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 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 anti-concentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable.

Motivation & Objective

• To develop a bootstrap method for maxima of sums of independent random vectors in high dimensions that does not depend on Gaussian approximation.
• To reduce the sample size requirement for consistent bootstrap inference from $ n \gg (\log p)^7 $ to $ n \gg (\log p)^5 $.
• To establish new comparison and anti-concentration theorems that are valid beyond the Gaussian regime.
• To provide a theoretically grounded, non-asymptotic framework for large-scale simultaneous inference in high-dimensional statistics.

Proposed method

• Develops a novel theoretical framework using comparison inequalities to bound the Kolmogorov-Smirnov distance between the true distribution and bootstrap distributions.
• Introduces anti-concentration theorems for maxima of sums of independent random vectors that do not rely on Gaussian approximation.
• Applies the empirical bootstrap and multiplier/wild bootstrap to high-dimensional maxima under minimal moment conditions.
• Uses a chaining-type argument combined with symmetrization and moment comparison techniques to control the bootstrap error.
• Establishes consistency of the bootstrap in the Kolmogorov-Smirnov distance under the improved sample size condition $ n \gg (\log p)^5 $.
• Relies on a new anti-concentration inequality for maxima of sums, which holds even when the Gaussian approximation is invalid.

Experimental results

Research questions

• RQ1 Can the bootstrap be consistently applied to maxima of high-dimensional sums without relying on Gaussian approximation?
• RQ2 What is the minimal sample size requirement for valid bootstrap inference in high-dimensional settings?
• RQ3 How can anti-concentration properties of maxima be established in non-Gaussian regimes?
• RQ4 Can comparison theorems be developed that are robust to the failure of Gaussian approximation?

Key findings

• The empirical bootstrap and multiplier/wild bootstrap are consistent in the Kolmogorov-Smirnov distance under the condition $ n \gg (\log p)^5 $, improving upon the prior $ n \gg (\log p)^7 $ requirement.
• New anti-concentration theorems are established that are valid even when Gaussian approximation fails, enabling robust inference in non-Gaussian high-dimensional settings.
• Comparison inequalities are developed that allow control of the bootstrap error without assuming Gaussianity of the underlying random vectors.
• The theoretical framework applies to general high-dimensional inference problems where the union bound is overly conservative and extreme value theory is needed.