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  \textbf{\Large{\textsc{STANFORD UNIVERSITY}}}\\[5pt]
  \textbf{\Large{\textsc{DEPARTMENT OF STATISTICS}}}\\[5pt]
  \Large{\textsc{DEPARTMENTAL SEMINAR}}
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  4:15 p.m., Tuesday, July 10, 2007\\
  %% Example: 4:15 p.m., Tuesday, February 13, 2007\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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  \textsl{Yonina C. Eldar} \\
  Department of Electrical Engineering\\
  Technion-Israel Institute of Technology
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  \subsection*{Improving Maximum Likelihood and the Cramer-Rao Bound}
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\noindent

One of the goals of statistical estimation theory is the
development of performance bounds when estimating parameters of
interest in a given model, as well as determining estimators that
achieve these bounds. When the parameters to be estimated are
deterministic, a popular approach is to restrict attention to
unbiased estimators and develop bounds on the smallest
mean-squared error (MSE) achievable within estimators of this
class. Although it is well-known that lower MSE can be achieved by
allowing for a bias, in applications it is typically unclear how
to choose such an appropriate bias.

In this talk we develop bounds that dominate the conventional
unbiased Cramer-Rao bound (CRB) so that the resulting MSE bound is
lower than the CRB for all values of the unknowns. When an
efficient maximum-likelihood (ML) estimator achieving the CRB
exists, we show how to construct an estimator with lower MSE
regardless of the true unknown values, by linearly transforming
the ML estimator.

We then specialize the results to estimation in linear Gaussian
models.  In particular, using an adaptive minimax MSE framework,
we derive a class of estimators that dominate least-squares and
perform better in many situations than the traditional James-Stein
type methods.

The procedures we develop are based on a saddle-point formulation
of the problem which admits the use of convex optimization tools.



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