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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


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.


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


Download Citation

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


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

Primary: 62E20

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

Rights: Copyright © 2021 Institute of Mathematical Statistics


Vol.49 • No. 3 • June 2021
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