On eigenvalue statistics of large sample covariance matrices and spearman correlation matrices

Abstract

The Random Matrix Theory (RMT) is a mathematical framework originating in mathematical physics that explores the properties of matrices with random elements. RMT is widely used in statistics and provides valuable insights into high-dimensional statistics. This thesis explores the eigenvalue statistics of two key random matrices: the sample covariance matrix and the Spearman correlation matrix, as well as their applications in high-dimensional statistics.

In the first part of this thesis, we derive the asymptotic normality for a large family of eigenvalue statistics of a general sample covariance matrix under the ultra-high dimensional setting, that is, when the dimension-to-sample-size ratio $p/n$ grows to infinity. Based on this central limit theorem (CLT) result, we extend the covariance matrix test problem to the new ultra-high dimensional context, and apply it to test a matrix-valued white noise. Simulation experiments are conducted for the investigation of finite-sample properties of the general asymptotic normality of eigenvalue statistics, as well as the two developed tests.

In the second part of the thesis, we study the spectral properties of the Spearman sample correlation matrix under the high-dimensional setting, where both dimension and sample size tend to infinity proportionally. Based on the theoretical result, we propose an estimator to determine the number of common factors in a high-dimensional factor model. This estimator is robust against heavy tails in either the common factors or idiosyncratic errors, and its consistency is established under mild assumptions. Extensive numerical experiments and real data analyses demonstrate the superiority of our estimator compared to existing methods.

In summary, this thesis contributes to high-dimensional statistics by investigating the eigenvalue statistics of the sample covariance matrix and Spearman correlation matrix, addressing challenges posed by ultra-high dimensionality and heavy-tailed data distributions in real-world scenarios.