Generalized additive models. (English) Zbl 0645.62068

Hastie, Trevor; Tibshirani, Robert

Stat. Sci. 1, 297-318 (1986).

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

MSC:

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

Keywords:

generalized linear models; smoothing; nonparametric regression; partial residuals; nonlinearity; scatterplot smoother; iterative procedure; local scoring algorithm; generalization of the Fisher scoring procedure; likelihood-based regression model; binary response; survival data; loglinear model; Cox’s model; censored data