A0363
Title: Tracy-Widom, Gaussian, and Bootstrap: Approximations for leading eigenvalues in high-dimensional PCA
Authors: Nina Doernemann - Aarhus University (Denmark) [presenting]
Miles Lopes - UC Davis (United States)
Abstract: The leading eigenvalues of sample covariance matrices play a fundamental role in many aspects of high-dimensional statistics. Under certain conditions, when the data dimension and sample size diverge proportionally, these eigenvalues undergo a well-known phase transition: In the sub-critical regime, the eigenvalues have Tracy-Widom fluctuations of order $n^{-2/3}$, while in the supercritical regime, they have Gaussian fluctuations of order $n^{-1/2}$. However, the statistical problem of determining which regime underlies a given dataset has remained largely unresolved. A new testing framework and procedure to address this problem is developed. In particular, the procedure is demonstrated to be at an asymptotically controlled level, and it is power-consistent for certain spiked alternatives. Also, this testing procedure enables the design of a new bootstrap method for approximating the distributions of functionals of the leading eigenvalues within the sub-critical regime, which is the first such method supported by theoretical guarantees.
