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.

arXiv comment70 pages. Added a numerical simulation