| In the past decade there is a significant development in estimation of various structured covariance and precision matrices. We present estimation and inference of Toeplitz covariance structure and Gaussian graphical model, paying special attention on optimality theory in this dissertation.;Toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging, target detection, speech recognition, and communications systems. In the first part of the dissertation, we consider optimal estimation of large Toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm. The properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound. The results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces. The minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent Gaussian scale model and through a careful selection of a finite collection of least favorable parameters. In addition, optimal rate of convergence for estimating the inverse of a Toeplitz covariance matrix is also established.;The Gaussian graphical model, a popular paradigm for studying relationship among variables in a wide range of applications, has attracted great attention in recent years. The second part of this dissertation considers a fundamental question: when is it possible to obtain asymptotic normality results for estimation of large Gaussian graphical model? A novel regression approach is proposed to obtain asymptotically efficient estimation of each entry under a sparseness condition. When the precision matrix is not sufficiently sparse, i.e., the sparseness condition fails, a lower bound is established to show that it is no longer possible to achieve a parametric rate estimation of each entry through a novel construction of a subset of sparse precision matrices and Le Cam's Lemma. The asymptotic normality result is applied to do support recovery, to obtain rate-optimal estimation of the precision matrix under various matrix lq norms, and to do inference and estimation for latent variable graphical models, without the irrepresentable condition and the l1 constraint of the precision matrix which are commonly required in literature, and the procedures are adaptive. Numerical results confirm our theoretical findings. The ROC curve of the proposed algorithm, Asymptotic Normal Thresholding (ANT), for support recovery significantly outperforms that of the GLasso algorithm. |
