Group selection in high-dimensional partially linear additive models

Abstract

We consider the problem of simultaneous variable selection and estimation in partially linear additive models with a large number of grouped variables in the linear part and a large number of nonparametric components. In our problem, the number of grouped variables may be larger than the sample size, but the number of important groups is “small” relative to the sample size. We apply the adaptive group Lasso to select the important groups, using spline bases to approximate the nonparametric components and the group Lasso to obtain an initial consistent estimator. Under appropriate conditions, it is shown that, the group Lasso selects the number of groups which is comparable with the underlying important groups and is estimation consistent, the adaptive group Lasso selects the correct important groups with probability converging to one as the sample size increases and is selection consistent. The results of simulation studies show that the adaptive group Lasso procedure works well with samples of moderate size. A real example is used to illustrate the application of the proposed penalized method.