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.

翻译：多元分析中的一个基本问题是检验多元线性模型中回归系数的一般线性假设。该框架涵盖了大量经过充分研究的任务，包括多元方差分析、预测变量的联合显著性检验以及趋势或季节效应的检测。在经典方法中，Roy最大根检验依赖于由残差协方差矩阵构建的F矩阵的最大特征值，在检测集中信号方面尤为有效。然而，在高维场景下，这些矩阵往往变得病态或奇异，使得该检验无法实施。为解决这一问题，我们提出了一种岭正则化的Roy检验，通过岭项稳定协方差估计。我们在高维框架下建立了正则化F矩阵最大特征值的渐近Tracy-Widom分布，其中维度和假设均与样本量可比，且仅假设有限矩条件。我们开发了一种计算高效的程序来估计相关的居中和缩放参数。我们进一步分析了该检验在一类低秩备择假设下的势，并考察了正则化参数的影响。该方法在模拟中展现出强大性能，并应用于人类连接组项目的数据，以评估脑体积测量与行为变量之间的关联。