Bootstrap Consistency for Quadratic Forms of Sample Averages with Increasing Dimension

TL;DR

This paper proves the consistency of the weighted bootstrap for quadratic forms of sample averages in high dimensions where the dimension grows with the sample size, using Gaussian approximation techniques.

Contribution

It introduces a novel bootstrap consistency result for quadratic forms with increasing dimension, extending previous fixed-dimension theories.

Findings

Bootstrap is consistent for quadratic forms when dimension grows slower than n^{1/4}.

The proof uses an adapted Lindeberg interpolation technique for Gaussian approximation.

Applications include model-specification testing with growing moments.

Abstract

This paper establishes consistency of the weighted bootstrap for quadratic forms $\left( n^{-1/2} \sum_{i=1}^{n} Z_{i,n} \right)^{T}\left( n^{-1/2} \sum_{i=1}^{n} Z_{i,n} \right)$ where $(Z_{i,n})_{i=1}^{n}$ are mean zero, independent $\mathbb{R}^{d}$-valued random variables and $d=d(n)$ is allowed to grow with the sample size $n$, slower than $n^{1/4}$. The proof relies on an adaptation of Lindeberg interpolation technique whereby we simplify the original problem to a Gaussian approximation problem. We apply our bootstrap results to model-specification testing problems when the number of moments is allowed to grow with the sample size.