Optimal Rates of Convergence for Estimating Toeplitz Covariance Matrices

Tony Cai, Zhao Ren, and Harrison Zhou

---

• 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.

• Paper: pdf file.

• Other related papers:

  Cai, T., Zhang, C.-H. & Zhou, H. (2010).
  Optimal rates of convergence for covariance matrix estimation
  The Annals of Statistics 38, 2118-2144.

  Cai, T., Liu, W. & Luo, X. (2011).
  A constrained l₁ minimization approach to sparse precision matrix estimation
  J. American Statistical Association 494, 594-607.

  Cai, T. & Liu, W. (2011).
  Adaptive thresholding for sparse covariance matrix estimation.
  J. American Statistical Association 106, 672-684.

  Cai, T. & Zhou, H. (2011).
  Minimax estimation of large covariance matrices under l₁ norm (with discussion)
  Statistica Sinica , to appear.

  Cai, T. & Zhou, H. (2012).
  Optimal rates of convergence for sparse covariance matrix estimation.
  The Annals of Statistics, to appear.

  Cai, T. & Yuan, M. (2012).
  Adaptive covariance matrix estimation through block thresholding.
  The Annals of Statistics, to appear.