  We consider a nonparametric additive model of a conditional mean function in
which the number of variables and additive components may be larger than the
sample size but the number of nonzero additive components is "small" relative
to the sample size. The statistical problem is to determine which additive
components are nonzero. The additive components are approximated by truncated
series expansions with B-spline bases. With this approximation, the problem of
component selection becomes that of selecting the groups of coefficients in the
expansion. We apply the adaptive group Lasso to select nonzero components,
using the group Lasso to obtain an initial estimator and reduce the dimension
of the problem. We give conditions under which the group Lasso selects a model
whose number of components is comparable with the underlying model, and the
adaptive group Lasso selects the nonzero components correctly with probability
approaching one as the sample size increases and achieves the optimal rate of
convergence. The results of Monte Carlo experiments show that the adaptive
group Lasso procedure works well with samples of moderate size. A data example
is used to illustrate the application of the proposed method.
