## The Annals of Statistics

### The Dimensionality Reduction Principle for Generalized Additive Models

Charles J. Stone

#### 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

#### 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