The Annals of Statistics

The Dimensionality Reduction Principle for Generalized Additive Models

Charles J. Stone

Full-text: Open access

Abstract

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 $J = 1$.

Article information

Source
Ann. Statist., Volume 14, Number 2 (1986), 590-606.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349940

Digital Object Identifier
doi:10.1214/aos/1176349940

Mathematical Reviews number (MathSciNet)
MR840516

Zentralblatt MATH identifier
0603.62050

JSTOR
links.jstor.org

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G05: Estimation

Keywords
Exponential family nonparametric model additivity spline maximum quasi likelihood estimate rate of convergence

Citation

Stone, Charles J. The Dimensionality Reduction Principle for Generalized Additive Models. Ann. Statist. 14 (1986), no. 2, 590--606. doi:10.1214/aos/1176349940. https://projecteuclid.org/euclid.aos/1176349940


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