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Huang, Jianhua Z.

Projection estimation in multiple regression with application to functional ANOVA models. (English) Zbl 0930.62042

Ann. Stat. 26, No. 1, 242-272 (1998).
Summary: 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 estimates 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.

Cited in 1 Review
Cited in 66 Documents

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62J10 Analysis of variance and covariance (ANOVA)

Keywords:

finite elements; interaction; polynomials; splines; tensor product; trigonometric polynomials; wavelets; rates of convergence; ANOVA; least-squares; curse of dimensionality
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References:

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[35] PHILADELPHIA, PENNSy LVANIA 19104-6302 E-MAIL: jianhua@compstat.wharton.upenn.edu
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