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June, 1986 The Dimensionality Reduction Principle for Generalized Additive Models
Charles J. Stone
Ann. Statist. 14(2): 590-606 (June, 1986). DOI: 10.1214/aos/1176349940


Let $(X, Y)$ be a pair of random variables such that $X = (X_1,\cdots, X_J)$ ranges over $C = \lbrack 0, 1\rbrack^J$. The conditional distribution of $Y$ given $X = x$ is assumed to belong to a suitable exponential family having parameter $\eta \in \mathbb{R}$. Let $\eta = f(x)$ denote the dependence of $\eta$ on $x$. Let $f^\ast$ denote the additive approximation to $f$ having the maximum possible expected log-likelihood under the model. Maximum likelihood is used to fit an additive spline estimate of $f^\ast$ based on a random sample of size $n$ from the distribution of $(X, Y)$. Under suitable conditions such an estimate can be constructed which achieves the same (optimal) rate of convergence for general $J$ as for $J = 1$.


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Charles J. Stone. "The Dimensionality Reduction Principle for Generalized Additive Models." Ann. Statist. 14 (2) 590 - 606, June, 1986.


Published: June, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0603.62050
MathSciNet: MR840516
Digital Object Identifier: 10.1214/aos/1176349940

Primary: 62G20
Secondary: 62G05

Rights: Copyright © 1986 Institute of Mathematical Statistics


Vol.14 • No. 2 • June, 1986
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