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.
@article{1176347115,
author = {Buja, Andreas and Hastie, Trevor and Tibshirani, Robert},
title = {Linear Smoothers and Additive Models},
journal = {Ann. Statist.},
volume = {17},
number = {1},
year = {1989},
pages = { 453-510},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347115}
}
Buja, Andreas; Hastie, Trevor; Tibshirani, Robert. Linear Smoothers and Additive Models. Ann. Statist., Tome 17 (1989) no. 1, pp. 453-510. http://gdmltest.u-ga.fr/item/1176347115/