Statistica Sinica 22 (2012), 1319-1378

doi:http://dx.doi.org/10.5705/ss.2010.253

MINIMAX ESTIMATION OF LARGE COVARIANCE

MATRICES UNDER $\ell _{1}$-NORM

T. Tony Cai and Harrison H. Zhou

University of Pennsylvania and Yale University

Abstract: Driven by a wide range of applications in high-dimensional data analysis, there has been significant recent interest in the estimation of large covariance matrices. In this paper, we consider optimal estimation of a covariance matrix as well as its inverse over several commonly used parameter spaces under the matrix $\ell _{1}$ norm. Both minimax lower and upper bounds are derived.

The lower bounds are established by using hypothesis testing arguments, where at the core are a novel construction of collections of least favorable multivariate normal distributions and the bounding of the affinities between mixture distributions. The lower bound analysis also provides insight into where the difficulties of the covariance matrix estimation problem arise. A specific thresholding estimator and tapering estimator are constructed and shown to be minimax rate optimal. The optimal rates of convergence established in the paper can serve as a benchmark for the performance of covariance matrix estimation methods.

Key words and phrases: Covariance matrix, ℓ₁ norm, minimax lower bound, operator norm, optimal rate of convergence, tapering, thresholding.