Varying Coefficient Model via Adaptive Spline Fitting

The varying coefficient model is a potent dimension reduction tool for nonparametric modeling and has received extensive attention from researchers. Most existing methods for fitting this model utilize polynomial splines with equidistant knots and treat the number of knots as a hyperparameter. However, imposing equidistant knots tends to be overly rigid, and systematically determining the optimal number of knots is also challenging. In this article, we address these challenges by employing polynomial splines with adaptively selected and predictor-specific knots to fit the varying coefficients in the model. We propose an efficient dynamic programming algorithm to find the optimal solution. Numerical results demonstrate that our new method achieves significantly smaller mean squared errors for coefficient estimations compared to the equidistant spline fitting method. An implementation of our method in R is available at https://github.com/wangxf0106/vcmasf. Proofs of the theorems are provided in the online supplementary materials.