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Bootstrap Consistency for Quadratic Forms of Sample Averages with Increasing Dimension

11 November 2014
Demian Pouzo
ArXiv (abs)PDFHTML
Abstract

This paper establishes consistency of the weighted bootstrap for quadratic forms (n−1/2∑i=1nZi,n)T(n−1/2∑i=1nZi,n)\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)(n−1/2∑i=1n​Zi,n​)T(n−1/2∑i=1n​Zi,n​) where (Zi,n)i=1n(Z_{i,n})_{i=1}^{n}(Zi,n​)i=1n​ are mean zero, independent Rd\mathbb{R}^{d}Rd-valued random variables and d=d(n)d=d(n)d=d(n) is allowed to grow with the sample size nnn, slower than n1/4n^{1/4}n1/4. 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.

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