Projection Estimation in Multiple Regression with Application
to Functional ANOVA Models
by Jianhua Huang
Abstract:
A general theory on rates of convergence of the least squares projection
estimate in multiple regression is developed. The theory is applied to
the functional ANOVA model, where the multivariate regression function
is modeled as a specified sum of a constant term, main effects (functions
of one variable), and selected interaction terms (functions of two or
more variables). The least squares projection is onto an approximating space
constructed from arbitrary linear spaces of functions and their tensor
products respecting the assumed ANOVA structure of the regression function. T
he linear spaces that serve as building blocks can be any of the ones
commonly used in practice: polynomials, trigonometric polynomials, splines,
wavelets, and finite elements. The rate of convergence result that is
obtained reinforces the intuition that low-order ANOVA modeling can achieve
dimension reduction and thus overcome the curse of dimensionality.
Moreover, the components of the projection estimate in an appropriately
defined ANOVA decomposition provide consistent estimates of the corresponding
components of the regression function. When the regression function does
not satisfy the assumed ANOVA form, the projection estimate converges to
its best approximation of that form.