The Dimensionality Reduction Principle for Generalized Additive Models

Let (X, Y) be a pair of random variables such that X = (X₁, , XJ) ranges over C = 0, 1J. The conditional distribution of Y given X = x is assumed to belong to a suitable exponential family having parameter R. Let = f (x) denote the dependence of 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.