Back To Index Previous Article Next Article Full Text

Statistica Sinica 33 (2023), 1461-1481

RATES OF BOOTSTRAP APPROXIMATION FOR
EIGENVALUES IN HIGH-DIMENSIONAL PCA

Junwen Yao and Miles E. Lopes

University of California, Davis

Abstract: In the context of principal components analysis (PCA), the bootstrap is commonly applied to solve a variety of inference problems, such as constructing confidence intervals for the eigenvalues of the population covariance matrix Σ. However, when the data are high-dimensional, there are relatively few theoretical guarantees that quantify the performance of the bootstrap. Our aim in this paper is to analyze how well the bootstrap can approximate the joint distribution of the leading eigenvalues of the sample covariance matrix align="AbsBottom", and we establish non-asymptotic rates of approximation with respect to the multivariate Kolmogorov metric. Under certain assumptions, we show that the bootstrap can achieve a dimension-free rate of r(Σ)/align="AbsBottom" up to logarithmic factors, where r(Σ) is the effective rank of Σ, and n is the sample size. From a methodological standpoint, we show that applying a transformation to the eigenvalues of align="AbsBottom" before bootstrapping is an important consideration in high-dimensional settings.

Key words and phrases: Bootstrap, covariance matrices, high-dimensional statistics, principal components analysis.

Back To Index Previous Article Next Article Full Text