BS-SIM: An effective variable selection method for high-dimensional single index model

The single index model is an intuitive extension of the linear regression model. It has become increasingly popular due to its flexibility in modeling. Similar to the linear regression model, the set of predictors for the single index model can contain a large number of irrelevant variables. Therefore, it is important to select the relevant variables when fitting the single index model. However, the problem of variable selection for high-dimensional single index model is not well settled in the literature. In this work, we combine the idea of applying cubic B-splines for estimating the single index model with the idea of using the family of the smooth integration of counting and absolute deviation (SICA) penalty functions for variable selection. We propose a new method to simultaneously perform parameter estimation and model selection for the single index model. This method is referred to as the B-spline and SICA method for the single index model, or in short, BS-SIM. We develop a coordinate descent algorithm to efficiently implement BS-SIM. We also show that under certain conditions, the proposed method can consistently estimate the true index and select the true model. Simulations with various settings and a real data analysis are conducted to demonstrate the estimation accuracy, selection consistency and computational efficiency of BS-SIM.