  In the low-dimensional case, the generalized additive coefficient model
(GACM) proposed by Xue and Yang [Statist. Sinica 16 (2006) 1423-1446] has been
demonstrated to be a powerful tool for studying nonlinear interaction effects
of variables. In this paper, we propose estimation and inference procedures for
the GACM when the dimension of the variables is high. Specifically, we propose
a groupwise penalization based procedure to distinguish significant covariates
for the "large $p$ small $n$" setting. The procedure is shown to be consistent
for model structure identification. Further, we construct simultaneous
confidence bands for the coefficient functions in the selected model based on a
refined two-step spline estimator. We also discuss how to choose the tuning
parameters. To estimate the standard deviation of the functional estimator, we
adopt the smoothed bootstrap method. We conduct simulation experiments to
evaluate the numerical performance of the proposed methods and analyze an
obesity data set from a genome-wide association study as an illustration.
