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Nonlinear Nonparametric Regression Models

Almost all of the current nonparametric regression methods such as smoothing splines, generalized additive models and varying coefficients models assume a linear relationship when nonparametric functions are regarded as parameters. In this article, we propose a general class of nonlinear nonparametric models that allow nonparametric functions to act nonlinearly. They arise in many fields as either theoretical or empirical models. Our new estimation methods are based on an extension of the Gauss-Newton method to infinite dimensional spaces and the backfitting procedure. We extend the generalized cross validation and the generalized maximum likelihood methods to estimate smoothing parameters. We establish connections between some nonlinear nonparametric models and nonlinear mixed effects models. Approximate Bayesian confidence intervals are derived for inference. We also develop a user friendly S-Plus function for fitting these models. We illustrate the methods with an application to ozone data and evaluate their finite-sample performance through simulations.