The Annals of Statistics

Linear Smoothers and Additive Models

Andreas Buja, Trevor Hastie, and Robert Tibshirani

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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 information

Ann. Statist., Volume 17, Number 2 (1989), 453-510.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 65D10: Smoothing, curve fitting

Smoother additive model nonparametric semiparametric regression Gauss-Seidel algorithm


Buja, Andreas; Hastie, Trevor; Tibshirani, Robert. Linear Smoothers and Additive Models. Ann. Statist. 17 (1989), no. 2, 453--510. doi:10.1214/aos/1176347115.

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