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\begin{center}
   \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
   \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textbf\textsc{{BERKELEY-STANFORD JOINT COLLOQUIUM}}}
\end{center}


\begin{center}
   4:15 p.m., Tuesday, May 1st, 2007\\
   Sequoia Hall Room 200\\
   (Cookies at 3:45 in 1st Floor Lounge)
\end{center}

\begin{center}
   \textsl{Noureddine El Karoui} \\
   Department of Statistics\\
   UC Berkeley
\end{center}

\begin{center}
   \subsection*{Estimation of large dimensional covariance matrices}
\end{center}


\noindent

With data manipulation and storage increasingly easy, the size of
the datasets statisticians are analyzing is becoming very large. The
starting point of the analysis in this setting is often an $n\times
p$ data matrix $X$, with $n$ the number of observations and $p$ the
number of variables. A key feature of a number of ``modern" datasets
is that $p$ and $n$ are of the same order of magnitude, typically in
the 100's.

In this setting, it is known (from results in random matrix theory
and theoretical statistics) that the sample covariance matrix is a
very poor estimator of the population covariance. In particular, it
can be shown that it estimates population eigenvalues - let alone
eigenvectors -  very poorly. It is therefore natural to ask how we
can improve upon it.

In the first part of the talk, I will propose a method to
(consistently) estimate the distribution of the eigenvalues of the
population covariance from the eigenvalues of the sample covariance
matrix. The technique requires minimal assumptions on the structure
of the population covariance. The corresponding algorithm is fast
and performs quite well in practice.

In the second part of the talk, I will turn to the narrower class of
``sparse" covariance matrices. I will develop a notion of sparsity
for matrices that is well suited for the study of spectral
properties and reasonable from an application point of view. By
exploiting this sparsity, we can come up with very good estimators
that consistently estimate all eigenvalues and any well-separated
eigenspaces.

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