Ridge-Regularized Largest Root Test For High-Dimensional General Linear Hypotheses

Abstract

A fundamental problem in multivariate analysis is testing general linear hypotheses for regression coefficients in a multivariate linear model. This framework encompasses a wide range of well-studied tasks, including MANOVA, joint significance testing of predictors, and detection of trends or seasonal effects. Among classical approaches, Roy's largest root test is particularly effective for detecting concentrated signals, relying on the largest eigenvalue of an F matrix constructed from residual covariance matrices. However, in high-dimensional settings, these matrices often become ill-conditioned or singular, rendering the test infeasible. To address this, we propose a ridge-regularized Roy's test that stabilizes the covariance estimation via a ridge term. We establish the asymptotic Tracy-Widom distribution of the largest eigenvalue of the regularized F-matrix under a high-dimensional regime, where both the dimension and hypotheses are comparable to the sample size, assuming only finite-moment conditions. A computationally efficient procedure is developed to estimate the associated centering and scaling parameters. We further analyze the power of the test under a class of low-rank alternatives and examine the influence of the regularization parameter. The method demonstrates strong performance in simulations and is applied to data from the Human Connectome Project to assess associations between volumetric brain measurements and behavioral variables.