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\footnotesize \noindent Statistica Sinica {\bf 22} (2012), 629-653

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2010.027

\vspace{3pc} \normalsize \centerline{\large\bf LARGE SAMPLE
PROPERTIES OF THE SCAD-PENALIZED} \vspace{2pt} \centerline{\large\bf
MAXIMUM LIKELIHOOD ESTIMATION} \vspace{2pt} \centerline{\large\bf ON
HIGH DIMENSIONS} \vspace{.25cm}
\author{Sunghoon Kwon and Yongdai
Kim} \vspace{.4cm} \centerline{\it University of Minnesota and Seoul
National University} \vspace{.55cm} 
\begin{quotation}
\noindent {\it Abstract:} In this paper, we study large sample
properties of smoothly clipped absolute deviation (SCAD) penalized
maximum likelihood estimation for high-dimensional parameters.
First, we prove that the oracle maximum likelihood estimator (MLE)
asymptotically becomes a local maximizer of the SCAD-penalized
log-likelihood, even when the number of parameters is much larger
than the sample size; the oracle MLE is an ideal non-penalized MLE
obtained by deleting all irrelevant parameters in advance. Second,
we prove that if the log-likelihood is strictly concave, the oracle
MLE asymptotically becomes the global maximizer of the
SCAD-penalized log-likelihood with a diverging number of parameters
that is less than the sample size. Various numerical experiments on
simulated data sets are presented to verify the theoretical results,
and two data examples are analyzed.\par \vspace{9pt} \noindent {\it
Key words and phrases:}
 High dimension,
maximum likelihood estimator, MLE, oracle property, SCAD, smoothly
clipped absolute deviation, variable selection.\par
\end{quotation}

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