# Eigenvalues of multivariate variance components estimates

## Speaker:

## Speaker Link:

## Institution:

## Time:

## Location:

Variance components (a.k.a. random/mixed effects) models are commonly used to determine genetic variance-covariance matrices of quantitative phenotypic traits in a population. The eigenvalue spectra of such matrices describe the evolutionary response to selection, but may be difficult to estimate from limited samples when the number of traits is large. In this talk, I will discuss the eigenvalues of classical MANOVA estimators of these matrices, including a characterization of the bulk empirical eigenvalue distribution, Tracy-Widom fluctuations at the spectral edges under a ``sphericity'' null hypothesis, the behavior of outlier eigenvalues under spiked alternatives, and a statistical procedure for estimating true population spike eigenvalues from the sample. These results are established using tools of random matrix theory and free probability.