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Generalized additive models. (English) Zbl 0645.62068
The classical linear regression model expresses the response vector Y as a function of the predictor variables \(X_ i\) through the model \(Y=\sum_{i}X_ i\beta_ i+e\), where the \(X_ i\) are observed, the \(\beta_ i\) are estimated by least squares or some other technique, e is the vector of errors. The authors replace the \(X_ i\beta_ i\) by unspecified smooth functions \(S_ i(X_ i)\), which are then estimated by a scatterplot smoother in an iterative procedure they call the local scoring algorithm, which is a generalization of the Fisher scoring procedure for computing maximum likelihood estimates.
The paper is well-written, not technically demanding, provides a general framework in which to view the estimation procedure and a general form of local scoring applicable to any likelihood-based regression model. The authors illustrate the method with binary response and survival data and include the loglinear model and Cox’s model for censored data.
The commentaries following by D. R. Brillinger, J. A. Nelder, C. J. Stone, and P. M. McCullagh are quite stimulating in terms of placing the paper in perspective and suggesting further relevant work. The comments of Brillinger and Stone are especially informative.
Reviewer: J.W.Green

62J05 Linear regression; mixed models
62F10 Point estimation
62G05 Nonparametric estimation
62J99 Linear inference, regression
62A01 Foundations and philosophical topics in statistics
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