Bootstrap consistency for quadratic forms of sample averages with increasing dimension

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.