DMS Statistics and Data Science Seminar

Dr. Haoran Li (Auburn University)

Title: Tracy-Widom Law of High Dimensional Ridge-Regularized F-matrix. 

Abstract: In multivariate analysis, many core problems involve the eigen-analysis of an F matrix,  constructed from two Wishart matrices.  These so-called Double Wishart problems arise in contexts such as MANOVA, covariance matrix equality testing, and hypothesis testing in multivariate linear regression. A prominent classical approach, Roy's largest root test, relies on the largest eigenvalue of the F matrix for inference. However, in high-dimensional settings, this test becomes impractical due to the singularity or near-singularity of the Wishart matrix. To address this challenge, we propose a ridge-regularization framework by introducing a ridge term. Specifically, we develop a family of ridge-regularized largest root tests, leveraging the largest eigenvalue of the ridge-regularized F matrix. Under mild assumptions, we establish the asymptotic Tracy-Widom distribution of the regularized largest root after appropriate scaling. An efficient method for estimating the scaling parameters is proposed using the Marčenko-Pastur equation.