## The Annals of Statistics

- Ann. Statist.
- Volume 13, Number 2 (1985), 689-705.

### Additive Regression and Other Nonparametric Models

#### Abstract

Let $(X, Y)$ be a pair of random variables such that $X = (X_1, \cdots, X_J)$ and let $f$ by a function that depends on the joint distribution of $(X, Y).$ A variety of parametric and nonparametric models for $f$ are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each $X_j \in \lbrack 0, 1\rbrack,$ that $Y$ is real valued with mean $\mu$ and finite variance, and that $f$ is the regression function of $Y$ on $X.$ Let $f^\ast,$ of the form $f^\ast(x_1, \cdots, x_J) = \mu + f^\ast_1(x_1) + \cdots + f^\ast_J(x_J),$ be chosen subject to the constraints $Ef^\ast_j = 0$ for $1 \leq j \leq J$ to minimize $E\lbrack(f(X) - f^\ast(X))^2\rbrack.$ Then $f^\ast$ is the closest additive approximation to $f,$ and $f^\ast = f$ if $f$ itself is additive. Spline estimates of $f^\ast_j$ and its derivatives are considered based on a random sample from the distribution of $(X, Y).$ Under a common smoothness assumption on $f^\ast_j, 1 \leq j \leq J,$ and some mild auxiliary assumptions, these estimates achieve the same (optimal) rate of convergence for general $J$ as they do for $J = 1.$

#### Article information

**Source**

Ann. Statist., Volume 13, Number 2 (1985), 689-705.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349548

**Digital Object Identifier**

doi:10.1214/aos/1176349548

**Mathematical Reviews number (MathSciNet)**

MR790566

**Zentralblatt MATH identifier**

0605.62065

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G20: Asymptotic properties

Secondary: 62G05: Estimation

**Keywords**

Nonparametric model additivity spline rate of convergence

#### Citation

Stone, Charles J. Additive Regression and Other Nonparametric Models. Ann. Statist. 13 (1985), no. 2, 689--705. doi:10.1214/aos/1176349548. https://projecteuclid.org/euclid.aos/1176349548