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June 2021 Principal components in linear mixed models with general bulk
Zhou Fan, Yi Sun, Zhichao Wang
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Ann. Statist. 49(3): 1489-1513 (June 2021). DOI: 10.1214/20-AOS2010

Abstract

We study the principal components of covariance estimators in multivariate mixed-effects linear models. We show that, in high dimensions, the principal eigenvalues and eigenvectors may exhibit bias and aliasing effects that are not present in low-dimensional settings. We derive the first-order limits of the principal eigenvalue locations and eigenvector projections in a high-dimensional asymptotic framework, allowing for general population spectral distributions for the random effects and extending previous results from a more restrictive spiked model. Our analysis uses free probability techniques, and we develop two general tools of independent interest—strong asymptotic freeness of GOE and deterministic matrices and a free deterministic equivalent approximation for bilinear forms of resolvents.

Funding Statement

Y. S. was supported by a Junior Fellow award from the Simons Foundation and NSF Grant DMS-1701654.
Z. F. was supported in part by NSF Grant DMS-1916198.

Acknowledgments

We thank Camille Male and Roland Speicher for helpful pointers to the strong asymptotic freeness literature.

Citation

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Zhou Fan. Yi Sun. Zhichao Wang. "Principal components in linear mixed models with general bulk." Ann. Statist. 49 (3) 1489 - 1513, June 2021. https://doi.org/10.1214/20-AOS2010

Information

Received: 1 August 2019; Revised: 1 April 2020; Published: June 2021
First available in Project Euclid: 9 August 2021

Digital Object Identifier: 10.1214/20-AOS2010

Subjects:
Primary: 62E20

Keywords: Free probability , high-dimensional asymptotics , random effects models , Random matrix theory

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 3 • June 2021
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