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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 1217-1232

\footnotesize \noindent

\vspace{3pc} \normalsize \centerline{\large\bf HIGH DIMENSIONAL
EXPONENTIAL FAMILY} \vspace{2pt} \centerline{\large\bf ESTIMATION
VIA EMPIRICAL BAYES} \vspace{.25cm}
\author{Omkar Muralidharan}
\vspace{.4cm} \centerline{\it Stanford University} \vspace{.55cm}
\begin{quotation}
\noindent {\it Abstract:} Consider estimating
$\theta_{1},\ldots,\theta_{N}$ based on data $z_{i}\sim
f_{\theta_{i}}$, where $f_{\theta}$ is a continuous natural
exponential family. One successful approach is Empirical Bayes (EB).
EB methods assume that $\theta$ come from an unknown prior, and
estimate the Bayes procedure corresponding to that prior.\par In
this paper, we propose a general form of nonparametric EB estimator
that uses estimates of the marginal density of $z$ and its
derivative. This estimator was first proposed by Zhang (1997)
%\citet{Zhang1997}
 for the normal means problem,
$z_{i}\sim\mathcal{N}\left(\theta_{i},1\right)$. We bound the regret
of our method in terms of the error in estimating the marginal
density and its derivative. As a side point, our proof yields a
lower bound on the regret of general estimators.\par We illustrate
our method in the simultaneous chi-squared problem, where $z_{i}$ is
a chi-squared random variable with scale $1/\theta_{i}$. We
specialize our theoretical results to this case, and study the
empirical performance of our method under different estimators of
the marginal and its derivative. Our method outperforms the UMVU
estimator and a conjugate prior parametric EB approach.\par
\vspace{9pt} \noindent {\it Key words and phrases:} Chi-squared
estimation, density estimation, empirical Bayes, exponential family,
regret bound, separable estimator.
\par
\end{quotation}

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