Computationally tractable nonparametric bootstrap of high-dimensional sample covariance matrices

TL;DR

This paper introduces a computationally efficient nonparametric bootstrap method for high-dimensional covariance matrices that accurately estimates spectral properties with minimal data and moment assumptions.

Contribution

It proposes a novel $(m,mp/n)$ bootstrap approach that is computationally feasible and consistent for high-dimensional eigenvalue statistics, extending bootstrap applicability.

Findings

Consistently reproduces empirical spectral measures in high dimensions.

Accurately approximates the distribution of linear spectral statistics.

Applicable under minimal moment conditions.

Abstract

We introduce a new ``$(m,mp/n)$ out of $(n,p)$'' sampling-with-replace\-ment bootstrap for eigenvalue statistics of high-dimensional sample covariance matrices based on $n$ independent $p$-dimensional random vectors. As it only uses $q=\lfloor mp/n\rfloor $ coordinates of the observations in a subsample of size $m \ll n $ from the original data, it is computationally tractable for large scale data. In the high-dimensional scenario $p/n\rightarrow c\in (0,\infty)$, this fully nonparametric bootstrap is shown to consistently reproduce the empirical spectral measure if $m/n\rightarrow 0$. If $m^2/n\rightarrow 0$, it approximates correctly the distribution of linear spectral statistics. The crucial component is a suitably defined Representative Subpopulation Condition which is shown to be verified in a large variety of situations. Our proofs are conducted under minimal moment requirements and incorporate delicate results on non-centered quadratic forms, combinatorial trace moments estimates as well as a conditional bootstrap martingale CLT which may be of independent interest.