Eigenvalue distributions of variance components estimators in high-dimensional random effects models

TL;DR

This paper analyzes the spectral behavior of variance component estimators in high-dimensional multivariate random effects models, providing deterministic approximations and fixed-point characterizations using free probability theory.

Contribution

It introduces a novel approach to characterize the spectra of MANOVA estimators in high dimensions via operator-valued free probability and fixed-point equations.

Findings

Spectra are well-approximated by deterministic laws in high dimensions.

Fixed-point equations characterize the Stieltjes transforms of these laws.

The methods are applicable to covariance estimation in quantitative genetics.

Abstract

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 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.