Fitting Multivariate Regression Functions by Interaction Spline Models

SUMMARY In nonparametric multivariate regression analysis, we seek methods to reduce the dimensionality of the regression function to bypass the difficulty caused by the curse of dimensionality. The original additive models approximate a regression function by additive univariate functions, which overcome successfully the difficulty caused by the curse of dimensionality and have a clear interpretation. But since they ignore any interactions between regressors they are not sufficiently flexible to cope with the case where interactions are of concern. In this paper, we study interactive models in which the regression function is approximated by an analysis-of-variance-type function. The models retain all the advantages of the original additive models but are much more flexible. We study ways of fitting an interactive model by the smoothing spline method. We relate the well-known backfitting algorithm to a typical penalized least squares problem in the context of the smoothing spline method. We develop a non-standard method based on bootstrap methodology for model selection in the interaction spline modelling.