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June, 1989 Linear Smoothers and Additive Models
Andreas Buja, Trevor Hastie, Robert Tibshirani
Ann. Statist. 17(2): 453-510 (June, 1989). DOI: 10.1214/aos/1176347115


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.


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Andreas Buja. Trevor Hastie. Robert Tibshirani. "Linear Smoothers and Additive Models." Ann. Statist. 17 (2) 453 - 510, June, 1989.


Published: June, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0689.62029
MathSciNet: MR994249
Digital Object Identifier: 10.1214/aos/1176347115

Primary: 62G05
Secondary: 65D10

Keywords: Additive model , Gauss-Seidel algorithm , nonparametric , regression , semiparametric , Smoother

Rights: Copyright © 1989 Institute of Mathematical Statistics


Vol.17 • No. 2 • June, 1989
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