The Annals of Statistics

Projection estimation in multiple regression with application to functional ANOVA models

Jianhua Z. Huang

Full-text: Open access

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. The 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.

Article information

Source
Ann. Statist. Volume 26, Number 1 (1998), 242-272.

Dates
First available: 28 August 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1030563984

Mathematical Reviews number (MathSciNet)
MR1611780

Digital Object Identifier
doi:10.1214/aos/1030563984

Zentralblatt MATH identifier
0930.62042

Subjects
Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Keywords
ANOVA curse of dimensionality finite elements interaction least squares polynomials rate of convergence regression splines tensor product trigonometric polynomials wavelets

Citation

Huang, Jianhua Z. Projection estimation in multiple regression with application to functional ANOVA models. The Annals of Statistics 26 (1998), no. 1, 242--272. doi:10.1214/aos/1030563984. http://projecteuclid.org/euclid.aos/1030563984.


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