Abstract

Covariance matrices play a central role in multivariate statistical analysis. A wide range of statistical methodologies, including clustering analysis, principal component analysis, linear and quadratic discriminant analysis, Gaussian graphical models require the knowledge of the covariance or precision structure.

The first half of the talk is presented in a chalk talk style. We talk about some motivations and challenges of estimating covariance matrices in the high-dimensional setting. Then we briefly review some of the developments in estimation of covariance and precision matrices in the past decade. Several classes of covariance and precision matrices are discussed. We pay special attention to optimality theory.

The second half of the talk focuses on one specific class: Toeplitz covariance structure, which is used in the analysis of stationary time series and a wide range of applications including radar imaging, target detection and speech recognition. We consider optimal estimation of large Toeplitz covariance matrices under the spectral norm. Minimax rate of convergence is established for two commonly used parameter spaces. The minimax upper bound is obtained by studying the properties of tapering and banding estimators. The minimax lower bound is obtained by first constructing a more informative model for which independent random variables are observed, and then deriving a lower bound for the more informative model by carefully constructing a collection of least favorable spectral densities and applying Fano’s Lemma.