Functional ANOVA  Models for Generalized Regression

by Jianhua Z. Huang

Keywords: Exponential family, interaction, maximum likelihood estimate,
rate of convergence, splines, tensor product.

Abstract:

The Functional ANOVA model is considered in the context of 
generalized regression, which includes logistic regression, 
probit regression and Poisson regression as special cases.
The multivariate predictor function is modeled as a specified 
sum of a constant term, main effects, and selected interaction terms.
Maximum likelihood estimate is used, where the maximization is 
taken over a suitably chosen approximating space. 
The approximating space is constructed from virtually arbitrary 
linear spaces of functions and their tensor products and 
is compatible with the assumed ANOVA structure on the predictor function.
Under mild conditions, the maximum likelihood estimate is consistent
and the components of the estimate in an appropriately defined ANOVA 
decomposition are consistent in estimating the corresponding 
components of the predictor function. When the predictor function does
not satisfy the assumed ANOVA form, the estimate converges to
its best approximation of that form relative to the expected log-likelihood.
A rate of convergence result is obtained, which reinforces the intuition that 
low-order ANOVA modeling can achieve dimension reduction and thus overcome 
the curse of dimensionality.