We study linear smoothers and their use in building nonparametric regression models. In the first part of this paper we examine certain aspects of linear smoothers for scatterplots; examples of these are the running-mean and running-line, kernel and cubic spline smoothers. The eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother, and several other topics such as the number of degrees of freedom of a smoother are discussed. In the second part of the paper we describe how linear smoothers can be used to estimate the additive model, a powerful nonparametric regression model, using the "back-fitting algorithm." We show that backfitting is the Gauss-Seidel iterative method for solving a set of normal equations associated with the additive model. We provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers.
"Linear Smoothers and Additive Models." Ann. Statist. 17 (2) 453 - 510, June, 1989. https://doi.org/10.1214/aos/1176347115