The bootstrap, covariance matrices and PCA in moderate and high-dimensions

Abstract

We consider the properties of the bootstrap as a tool for inference concerning the eigenvalues of a sample covariance matrix computed from an $n\times p$ data matrix $X$. We focus on the modern framework where $p/n$ is not close to 0 but remains bounded as $n$ and $p$ tend to infinity. Through a mix of numerical and theoretical considerations, we show that the bootstrap is not in general a reliable inferential tool in the setting we consider. However, in the case where the population covariance matrix is well-approximated by a finite rank matrix, the bootstrap performs as it does in finite dimension.

References in corpus (2)

• High Dimensional Statistical Inference and Random Matrices
• The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices

Cited by in corpus (1)

• Almost sure convergence of the largest and smallest eigenvalues of high-dimensional sample correlation matrices