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.