Principal components in linear mixed models with general bulk

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.

References in corpus (5)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Limit theorems for sample eigenvalues in a generalized spiked population model
• Spectral Moments of Correlated Wishart Matrices
• Free probability of type B and asymptotics of finite-rank perturbations of random matrices
• Spiked covariances and principal components analysis in high-dimensional random effects models