On the sample covariance matrix estimator of reduced effective rank population matrices, with applications to fPCA

In this paper we study properties of the sample covariance matrix $\Sigma_n$ as an estimator of $p \times p$ population matrices $\Sigma$ of reduced effective rank. The effective rank $r_e(\Sigma)$ of a matrix is the ratio of its trace to its largest singular value, and provides a measure of matrix complexity. Despite the very large body of work on covariance matrix estimation, the properties of $\Sigma_n$ over classes of population matrices of reduced $r_e(\Sigma)$ are largely unexplored. Our first contribution is to review and establish sharp finite sample bounds on the operator and Frobenius norm of $\Sigma_n - \Sigma$. These bounds reveal that, as long as $r_e(\Sigma) < n$, the sample covariance matrix $\Sigma_n$ can still serve as an accurate estimator of $\Sigma$, even if $p > n$. Moreover, and perhaps surprisingly, $\Sigma_n$ adapts to the unknown complexity of $\Sigma$ quantified by $r_e(\Sigma)$, without any need for further thresholding operations. Our main contribution is in employing these results for the study of the consistency of scree-plot selection procedure routinely used in PCA. For given $p$ and $n$, we quantify the largest number of the eigenvalues of $\Sigma$ that can be consistently estimated by thresholding the spectrum of $\Sigma_n$, and offer a data adaptive construction of the threshold level. We derive the finite sample rates of convergence for the selected eigenvalues. We show that the analysis of the selected eigenvectors requires further assumptions on the spectrum of $\Sigma$, and discuss their implications on the construction of thresholding levels. As an application, we consider aspects of functional principal components analysis (fPCA).