FUNCTIONAL ANOVA MODELS FOR GENERALIZED REGRESSION

The Functional ANOVA model is considered in the context of generalized regression, which includes logistic regression, probit regression, an d Poisson regression as special cases. The multivariate predictor func tion is modeled as a specified sum of a constant term, main effects, a nd selected interaction terms. Maximum likelihood estimate is used, wh ere the maximization is taken over a suitably chosen approximating spa ce. The approximating space is constructed from virtually arbitrary li near spaces of functions and their tensor products and is compatible w ith the assumed ANOVA structure on the predictor Function. Under mild conditions, the maximum likelihood estimate is consistent and the comp onents of the estimate in an appropriately defined ANOVA decomposition are consistent in estimating the corresponding components of the pred ictor function. When the predictor Function does not, satisfy the assu med ANOVA form, the estimate converges to its best approximation of th at form relative to the expected log-likelihood. A rate of convergence result is obtained, which reinforces the intuition that low-order ANO VA modeling can achieve dimension reduction and thus overcome the curs e of dimensionality. (C) 1998 Academic Press