0 Component Selection in the Additive Regression Model

Similar to variable selection in the linear regression model, selecting significant components in the popular additive regression model is of great interest. However, such components are unknown smooth functions of independent variables, which are unobservable. As such, some approximation is needed. In this paper, we suggest a combination of penalized regression spline approximation and group variable selection, called the lasso-type spline method (LSM), to handle this component selection problem with a diverging number of strongly correlated variables in each group. It is shown that the proposed method can select significant components and estimate nonparametric additive function components simultaneously with an optimal convergence rate simultaneously. To make the LSM stable in computation and able to adapt its estimators to the level of smoothness of the component functions, weighted power spline bases and projected weighted power spline bases are proposed. Their performance is examined by simulation studies across two set-ups with independent predictors and correlated predictors, respectively, and appears superior to the performance of competing methods. The proposed method is extended to a partial linear regression model analysis with real data, and gives reliable results.