CONVERGENCE OF THE BACKFITTING ALGORITHM FOR ADDITIVE-MODELS

The backfitting algorithm is an iterative procedure for fitting additi ve models in which, at each step, one component is estimated keeping t he other components fixed, the algorithm proceeding component by compo nent and iterating until convergence. Convergence of the algorithm has been studied by Buja, Hastie, and Tibshirani (1989). We give a simple , but more general, geometric proof of the convergence of the backfitt ing algorithm when the additive components are estimated by penalized least squares. Our treatment covers spline smoothers and structural ti me series models, and we give a full discussion of the degenerate case . Our proof is based on Halperin's (1962) generalization of von Neuman n's alternating projection theorem.