### Eigenvalues in multivariate random effects models

- Series
- Job Candidate Talk
- Time
- Thursday, November 30, 2017 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Zhou Fan – Stanford University – zhoufan@stanford.edu

Random effects models are commonly used to measure genetic
variance-covariance matrices of quantitative phenotypic traits. The
population eigenvalues of these matrices describe the evolutionary
response to selection. However, they may be difficult to estimate from
limited samples when the number of traits is large. In this talk, I will
present several results describing the eigenvalues of classical MANOVA
estimators of these matrices, including dispersion of the bulk
eigenvalue distribution, bias and aliasing of large "spike" eigenvalues,
and distributional limits of eigenvalues at the spectral edges. I will
then discuss a new procedure that uses these results to obtain better
estimates of the large population eigenvalues when there are many
traits, and a Tracy-Widom test for detecting true principal components
in these models. The theoretical results extend proof techniques in
random matrix theory and free probability, which I will also briefly
describe.This is joint work with Iain Johnstone, Yi Sun, Mark Blows, and Emma Hine.