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 η ∈ R. Let η = f(x) denote the dependence of η 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 $J = 1$.