The Dimensionality Reduction Principle for Generalized Additive Models

Abstract

Let (X, Y) be a pair of random variables such that X = (XI,..., XJ,) ranges over C = [0, 1]". The conditional distribution of Y given X = x is assumed to belong to a suitable exponential family having parameter q E R. Let X = f(x) denote the dependence of -q on x. Let f* 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 * 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.