Bootstrapping ${\ell }_p$-Statistics in High Dimensions

Abstract

This paper considers a new bootstrap procedure to estimate the distribution of high-dimensional ${\ell }_p$-statistics, i.e. the ${\ell }_p$-norms of the sum of $n$ independent $d$-dimensional random vectors with $d\gg n$ and $p\in [1,\infty ]$. We provide a non-asymptotic characterization of the sampling distribution of ${\ell }_p$-statistics based on Gaussian approximation and show that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance under mild conditions on the covariance structure of the data. As an application of the general theory we propose a bootstrap hypothesis test for simultaneous inference on high-dimensional mean vectors. We establish its asymptotic correctness and consistency under high-dimensional alternatives, and discuss the power of the test as well as the size of associated confidence sets. We illustrate the bootstrap and testing procedure numerically on simulated data.

arXiv comment80 pages, 6 figures. Added details to Section 3.2

References in corpus (8)

• A two-sample test for high-dimensional data with applications to gene-set testing
• High-dimensional covariance matrix estimation in approximate factor models
• Bootstrap confidence sets under model misspecification
• Stein's method in high dimensions with applications
• Explicit rates of approximation in the CLT for quadratic forms
• Detailed proof of Nazarov's inequality
• Beyond Gaussian Approximation: Bootstrap for Maxima of Sums of Independent Random Vectors
• Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles