Estimation of a Projection-Pursuit Type Regression Model

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 (X, Y) be a random vector such that X = (X1, . . , Xd)T ranges over Rd. The conditional mean of Y given X = x is assumed to be the sum of no more than d general smooth functions of 87[x, where p3i E Sd-1, the unit sphere in Rd 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 (X, Y). Under appropriate conditions, the rate of convergence of the proposed estimator is independent of d.