EIGENVALUE DISTRIBUTIONS OF VARIANCE COMPONENTS ESTIMATORS IN HIGH-DIMENSIONAL RANDOM EFFECTS MODELS

• 成果类型: Article
• 署名作者: Fan, Zhou; Johnstone, Iain M.
• 署名单位: Yale University; Stanford University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1767
• 发表日期: 2019
• 页码: 2855-2886
• 关键词: Covariance matrices; free product; capacity; Respect; unitary

摘要

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.