  This paper investigates the two-step estimation of a high dimensional
additive regression model, in which the number of nonparametric additive
components is potentially larger than the sample size but the number of
significant additive components is sufficiently small. The approach
investigated consists of two steps. The first step implements the variable
selection, typically by the group Lasso, and the second step applies the
penalized least squares estimation with Sobolev penalties to the selected
additive components. Such a procedure is computationally simple to implement
and, in our numerical experiments, works reasonably well. Despite its intuitive
nature, the theoretical properties of this two-step procedure have to be
carefully analyzed, since the effect of the first step variable selection is
random, and generally it may contain redundant additive components and at the
same time miss significant additive components. This paper derives a generic
performance bound on the two-step estimation procedure allowing for these
situations, and studies in detail the overall performance when the first step
variable selection is implemented by the group Lasso.
