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October 2019 Eigenvalue distributions of variance components estimators in high-dimensional random effects models
Zhou Fan, Iain M. Johnstone
Ann. Statist. 47(5): 2855-2886 (October 2019). DOI: 10.1214/18-AOS1767


We study the spectra of MANOVA estimators for variance component covariance matrices in multivariate random effects models. When the dimensionality of the observations is large and comparable to the number of realizations of each random effect, we show that the empirical spectra of such estimators are well approximated by deterministic laws. The Stieltjes transforms of these laws are characterized by systems of fixed-point equations, which are numerically solvable by a simple iterative procedure. Our proof uses operator-valued free probability theory, and we establish a general asymptotic freeness result for families of rectangular orthogonally invariant random matrices, which is of independent interest. Our work is motivated in part by the estimation of components of covariance between multiple phenotypic traits in quantitative genetics, and we specialize our results to common experimental designs that arise in this application.


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Zhou Fan. Iain M. Johnstone. "Eigenvalue distributions of variance components estimators in high-dimensional random effects models." Ann. Statist. 47 (5) 2855 - 2886, October 2019.


Received: 1 November 2017; Revised: 1 August 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114931
MathSciNet: MR3988775
Digital Object Identifier: 10.1214/18-AOS1767

Primary: 62E20

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 5 • October 2019
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