Bootstrapping spectral statistics in high dimensions

TL;DR

This paper introduces the Spectral Bootstrap, a new high-dimensional bootstrap method for linear spectral statistics of sample covariance matrices, providing consistency proofs and promising empirical results.

Contribution

It develops a simple, consistent bootstrap approach tailored for high-dimensional spectral statistics, extending applicability beyond linear cases.

Findings

Proves the consistency of the Spectral Bootstrap method.

Demonstrates empirical effectiveness across various high-dimensional settings.

Shows successful application to non-linear spectral statistics like the largest eigenvalue.

Abstract

Statistics derived from the eigenvalues of sample covariance matrices are called spectral statistics, and they play a central role in multivariate testing. Although bootstrap methods are an established approach to approximating the laws of spectral statistics in low-dimensional problems, these methods are relatively unexplored in the high-dimensional setting. The aim of this paper is to focus on linear spectral statistics as a class of prototypes for developing a new bootstrap in high-dimensions --- and we refer to this method as the Spectral Bootstrap. In essence, the method originates from the parametric bootstrap, and is motivated by the notion that, in high dimensions, it is difficult to obtain a non-parametric approximation to the full data-generating distribution. From a practical standpoint, the method is easy to use, and allows the user to circumvent the difficulties of complex asymptotic formulas for linear spectral statistics. In addition to proving the consistency of the proposed method, we provide encouraging empirical results in a variety of settings. Lastly, and perhaps most interestingly, we show through simulations that the method can be applied successfully to statistics outside the class of linear spectral statistics, such as the largest sample eigenvalue and others.