In multivariate analysis, many core problems involve the eigen-analysis of an
\(F\)-matrix, \(\bF = \bW_1\bW_2^{-1}\), constructed from two Wishart matrices,
\(\bW_1\) E \(\bW_2\). These so-called \textit{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 \(\bF\) for
inference. Tuttavia, in high-dimensional settings, this test becomes impractical
due to the singularity or near-singularity of \(\bW_2\). To address this
challenge, we propose a ridge-regularization framework by introducing a ridge
term to \(\bW_2\). Specifically, we develop a family of ridge-regularized
largest root tests, leveraging the largest eigenvalue of \(\bF_\lambda =
\bW_1(\bW_2 + \lambda I)^{-1}\), where \(\lambda > 0\) is the regularization
parameter. Under mild assumptions, we establish the asymptotic Tracy-Widom
distribution of the largest eigenvalue of \(\bF_\lambda\) after appropriate
scaling. An efficient method for estimating the scaling parameters is proposed
using the Mar\v{C}enko-Pastur equation, and the consistency of these estimators
is proven. The proposed framework is applied to illustrative Double Wishart
problems, and simulation studies are conducted to evaluate the numerical
performance of the methods. Finally, the proposed method is applied to the
\emph{Human Connectome Project} data to test for the presence of associations
between volumetric measurements of human brain and behavioral variables.

Questo articolo esplora i giri e le loro implicazioni.

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