B1217
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. In this work, we develop a new testing framework and procedure to address this problem. In particular, we demonstrate that the procedure has an asymptotically controlled level, and that it is power consistent for certain spiked alternatives. Also, this testing procedure enables the design 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 that is supported by theoretical guarantees.
