Globally consistent model selection in semi-parametric additive coefficient models

We study a penalised polynomial spline (PPS) method for model selection in additive coefficient models. It approximates nonparametric coefficient functions by polynomial splines and minimises the sum of squared errors subject to an additive penalty on the norms of spline functions. For non-convex penalty functions such as smoothly clipped absolute deviation (SCAD) penalty, we investigate the asymptotic properties of the global solution of the non-convex objective function. We establish explicitly that the oracle estimator is the global solution with probability approaching one. Therefore, the global solution enjoys both model estimation and selection consistency. In the literature, the asymptotic properties of local solutions rather than global solutions are well-established for non-convex penalty functions. Our theoretical results broaden the traditional understanding of the PPS method. Extensive Monte Carlo simulation studies show the proposed method performs well numerically. We also illustrate the use of the proposed method by analysing a housing price data set.