Optimal rates of convergence for estimating Toeplitz covariance matrices

Abstract

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 this paper, 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.

Joint Covariance Estimation With Mutual Linear Structure

“…Such matrices are typically used as approximations to Toeplitz matrices [9] which are associated with signals that obey periodic stochastic properties for example the yearly variation of temperature in a particular location. A special case of such processes are the classical stationary processes, which are ubiquitous in engineering, [16], [17]. • Toeplitz: A natural generalization of circulant are Toeplitz matrices.…”

“…, we obtain (16) For the Gaussian population the matrix is derived in [37] and reads as (17) where is the square submatrix of corresponding to the subset of indices from . The bound on the is therefore given by (18) Denote (19) To get more insight on (18) we bound it from below (20) The dependence on the model parameters here is similar to that obtained by [38] for the problem of low-rank matrix reconstruction.…”

“…Probably the most popular of them is the Toeplitz model [8], [13]- [15], [21]- [25], closely related to it are circulant matrices [16], [17]. In other settings the number of parameters can be reduced by assuming that the covariance matrix is sparse [18]- [20]. A popular sparse model is the banded covariance, which is associated with time-varying moving average models [15], [21].…”

On the statistical dependence : from nonlinearity to spatial structural estimation

“…One of the main challenge for this kind of data in R d with d ≥ 2 is that, they are not ordered as the case in R 1 such as in time series data. As a consequence, the simple structures like Toeplitz matrices Xiao and Wu [2012], Cai et al [2013] are not applicable for spatial data whose dimension is higher than one Zhu and Liu [2009].…”

“…For instance , Cai et al [2010], Cai and Yuan [2012] discuss the estimation of bandable covariance matrices, where the true covariance has high values concentrated near the diagonal. Xiao and Wu [2012], Cai et al [2013] study the Toeplitz matrices, which is closely related to stationary time series. In order to facilitate the analysis of the theoretical and empirical properties of the data, most of the aforementioned works imposes certain structural condition for the data, which result in various sparse covariance matrix models.…”