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March, 1991 Estimation of a Projection-Pursuit Type Regression Model
Hung Chen
Ann. Statist. 19(1): 142-157 (March, 1991). DOI: 10.1214/aos/1176347974

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 $d$.

Citation

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Hung Chen. "Estimation of a Projection-Pursuit Type Regression Model." Ann. Statist. 19 (1) 142 - 157, March, 1991. https://doi.org/10.1214/aos/1176347974

Information

Published: March, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0736.62055
MathSciNet: MR1091843
Digital Object Identifier: 10.1214/aos/1176347974

Subjects:
Primary: 62J02
Secondary: 62G05, 62G20

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.19 • No. 1 • March, 1991
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