Theoretical Study On Variable Coefficient Single Index Model

We propose a new model: the single-index varying-coefficient model, which is a combination between single-index model and varying-coefficient model. This model considers the varying-coefficient in the linear part and a single-index structure in the nonlinear part.In the paper, we study the estimation and variable selection problem for the varying-coefficient, the nonparametric link function and the unknown single-index parameter,and we give the asymptotic results for the associated estimators and present the consistency for the variable selection and Oracle properties. For the single-index varying-coefficient model,we use the B splines to approximate the unknown varying coefficient functions, and adopt the SCAD penalty function to estimate unknown varying coefficient functions and do variable selection simultaneously. Lastly, we propose BIC criteria to select proper number of the knots and the tunning parameters in the variable selection, we also show the consistency of the BIC criteria.