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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 1319-1378

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2010.253

\vspace{3pc} \normalsize \centerline{\large\bf MINIMAX ESTIMATION OF
LARGE COVARIANCE} \vspace{2pt} \centerline{\large\bf MATRICES UNDER
$\ell _{1}$-NORM} \vspace{.25cm}
\author{T. Tony Cai and Harrison H. Zhou}
\vspace{.4cm} \centerline{\it University of Pennsylvania and Yale
University} \vspace{.55cm}
\begin{quotation}
\noindent {\it 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.\par
\vspace{9pt} \noindent {\it Key words and phrases:} Covariance
matrix, $\ell_1$ norm, minimax lower bound, operator norm, optimal
rate of convergence, tapering, thresholding. \par
\end{quotation}

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