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\footnotesize \noindent Statistica Sinica {\bf 21} (2011), 1515-1540

\footnotesize \noindent doi:http://dx.doi.org/10.5705/ss.2009.316

\vspace{3pc} \normalsize \centerline{\large\bf VARIABLE SELECTION
AND ESTIMATION} \vspace{2pt} \centerline{\large\bf IN
HIGH-DIMENSIONAL VARYING-COEFFICIENT MODELS} \vspace{.25cm}
\author{Fengrong Wei, Jian Huang and Hongzhe Li}
\vspace{.4cm} \centerline{\it University of West Georgia,
 University
of Iowa and
 University
of Pennsylvania} \vspace{.55cm}
\begin{quotation}
\noindent {\it Abstract:} Nonparametric varying coefficient models
are useful for studying the time-dependent effects of variables.
Many procedures have been developed for estimation and variable
selection in such models. However, existing work has focused on the
case when the number of variables is fixed or smaller than the
sample size. In this paper, we consider the problem of variable
selection and estimation in varying coefficient models in sparse,
high-dimensional settings when the number of variables can be larger
than the sample size. We apply the group Lasso and basis function
expansion to simultaneously select the important variables and
estimate the nonzero varying coefficient functions. Under
appropriate conditions, we show that
  the group Lasso selects a model of the right order of dimensionality,
  selects all variables with the norms of the corresponding coefficient functions greater than certain threshold level, and is estimation
  consistent.
However, the group Lasso is in general not selection consistent
  and tends to select variables that are not important in the model. In order to improve the
  selection results, we apply the adaptive
  group Lasso. We show that, under suitable conditions, the adaptive group Lasso has the oracle selection property
  in the sense that it correctly selects important variables with
  probability converging to one. In contrast, the group Lasso does not possess such oracle property. Both approaches are evaluated using simulation and demonstrated on a data example.\\
\par
\vspace{9pt} \noindent {\it Key words and phrases:} Basis expansion,
group Lasso, high-dimensional data,  nonparametric coefficient
function, selection consistency, sparsity \par
\end{quotation}

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