Event Date
Speaker: Zhenggang Wang, Post-Doctoral Scholar, Department of Statistics, University of California Davis
Title: "CLTs for the Sample Covariance Matrices and Covariance Structure Tests in the Ultra-High Dimension"
Abstract: We explore sample covariance matrices in situations where the dimension is significantly larger than the sample size, i.e., $p \approx n^\alpha$ with $\alpha>1$. Specifically, we consider the (unnormalized) sample covariance matrices $Q = \Sigma^{1/2}XX^∗\Sigma^{1/2}$, where $X = (x_{ij})$ is a $p \times n$ random matrix with centered iid entries whose variances are $(pn)^{−1/2}$, and $\Sigma$ is the deterministic population covariance matrix. Our main focus is on the behavior of linear spectral statistics (LSS) $\sum_{i=1}^pf(\lambda_i)$ of the sample covariance matrices. We've established two types of central limit theorems (CLTs) for the LSS induced by test functions on different scales: one that applies on the macroscopic scale (global CLTs) and another for the mesoscopic scale (local CLTs). These theorems describe how the LSS converge to Gaussian and further identify how their asymptotic means and variances depend on factors like $\Sigma$, $X$, the ratio $p / n$, and the test functions $f$ used. A key finding of our research is that while the global CLTs are influenced by the fourth cumulant (a measure of distribution shape) of the matrix entries $x_{ij}$, the local CLTs are not. This motivates us to propose two sets of statistical tests to examine the structure of $\Sigma$ based on the global and the local statistics. These tests show great promise, particularly the local-scale tests, which are novel in the sense that they don't depend on the fourth moment of the matrix entries of $X$. This is a significant departure from most current methods, which focus on global statistics and require prior knowledge of the fourth cumulant. Further, we conduct numerical simulations which not only demonstrate the accuracy of our method but also show that it has better power compared to some existing methods in the literature. Moreover, we also derive a two-sample test for the equality of two covariance matrices in this setting.
Seminar Date/Time: Thursday November 16, 2023, 4:10pm @ MSB 1147
Refreshments 3:30pm, MSB Courtyard