A0479
Title: Tracy-Widom, Gaussian, and bootstrap: Approximations for leading eigenvalues in high-dimensional PCA
Authors: Miles Lopes - UC Davis (United States)
Nina Doernemann - Aarhus University (Denmark) [presenting]
Abstract: Under certain conditions, the largest eigenvalue of a sample covariance matrix undergoes a well-known phase transition when the sample size $n$ and data dimension $p$ diverge proportionally. In the subcritical regime, this eigenvalue has fluctuations of order $n^{-2/3}$ that can be approximated by a Tracy-Widom distribution, while in the supercritical regime, it has fluctuations of order $n^{-1/2}$ that can be approximated with a Gaussian distribution. However, the statistical problem of determining which regime underlies a given dataset is far from resolved. A new testing framework and procedure are developed to address this problem. In particular, it is demonstrated that the procedure has an asymptotically controlled level and that it is power-consistent for certain alternatives. Also, this testing procedure enables the design of a new bootstrap method for approximating the distributions of functionals of the leading sample eigenvalues within the subcritical regime - which is the first such method that is supported by theoretical guarantees.
