Nonparametric bootstrap of high-dimensional sample covariance matrices

2024/06/24 by Dette, Holger, Rohde, Angelika
#60F05 #62G09 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2406.16849

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. In the high-dimensional scenario p/n→ c∈ (0,∞), this fully nonparametric and computationally tractable bootstrap is shown to consistently reproduce the empirical spectral measure if m/n→ 0. If m²/n→ 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.

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