## The Annals of Statistics

- Ann. Statist.
- Volume 19, Number 1 (1991), 142-157.

### Estimation of a Projection-Pursuit Type Regression Model

#### Abstract

Since the pioneering work of Friedman and Stuetzle in 1981, projection-pursuit algorithms have attracted increasing attention. This is mainly due to their potential for overcoming or reducing difficulties arising in nonparametric regression models associated with the so-called curse of dimensionality, that is, the amount of data required to avoid an unacceptably large variance increasing rapidly with dimensionality. Subsequent work has, however, uncovered a dependence on dimensionality for projection-pursuit regression models. Here we propose a projection-pursuit type estimation scheme, with two additional constraints imposed, for which the rate of convergence of the estimator is shown to be independent of the dimensionality. Let $(\mathbf{X}, Y)$ be a random vector such that $\mathbf{X} = (X_1, \ldots, X_d)^T$ ranges over $R^d$. The conditional mean of $Y$ given $\mathbf{X} = \mathbf{x}$ is assumed to be the sum of no more than $d$ general smooth functions of $\beta^T_i\mathbf{x}$, where $\beta_i \in S^{d - 1}$, the unit sphere in $R^d$ centered at the origin. A least-squares polynomial spline and the final prediction error criterion are used to fit the model to a random sample of size $n$ from the distribution of $(\mathbf{X}, Y)$. Under appropriate conditions, the rate of convergence of the proposed estimator is independent of $d$.

#### Article information

**Source**

Ann. Statist., Volume 19, Number 1 (1991), 142-157.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347974

**Digital Object Identifier**

doi:10.1214/aos/1176347974

**Mathematical Reviews number (MathSciNet)**

MR1091843

**Zentralblatt MATH identifier**

0736.62055

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62J02: General nonlinear regression

Secondary: 62G20: Asymptotic properties 62G05: Estimation

**Keywords**

Additive models projection pursuit polynomial splines nonparametric regression

#### Citation

Chen, Hung. Estimation of a Projection-Pursuit Type Regression Model. Ann. Statist. 19 (1991), no. 1, 142--157. doi:10.1214/aos/1176347974. https://projecteuclid.org/euclid.aos/1176347974