A Bootstrap Method for Spectral Statistics in High-Dimensional Elliptical Models

Abstract: Although there is an extensive literature on the eigenvalues of high-dimensional sample covariance matrices, much of it is specialized to independent components (IC) models -- in which observations are represented as linear transformations of random vectors with independent entries. By contrast, less is known in the context of elliptical models, which violate the independence structure of IC models and exhibit quite different statistical phenomena. In particular, very little is known about the scope of bootstrap methods for doing inference with spectral statistics in high-dimensional elliptical models. To fill this gap, we show how a bootstrap approach developed previously for IC models can be extended to handle the different properties of elliptical models. Within this setting, our main theoretical result guarantees that the proposed method consistently approximates the distributions of linear spectral statistics, which play a fundamental role in multivariate analysis. We also provide empirical results showing that the proposed method performs well for a variety of nonlinear spectral statistics.