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Additive regression and other nonparametric models. (English) Zbl 0605.62065
The first part of the paper motivates an heuristic dimensionality reduction principle for a function f, that depends on the joint distribution of \((X_ 1,...,X_ J,Y)\). It states, that f(x) is of dimensionality \(d<J\) if \(f(x)=\sum f_ j(x)\), and all \(f_ j\) are functions of at most d components of \(x=(X_ 1,...,X_ J)\). This principle leads to suggestion of \(n^{-2r}\), \(r=(p-m)/(2p-d)\), as optimal rate of convergence.
In the second part this suggestion is shown to hold true for the additive regression model \(f(x)=\mu +\sum^{J}_{1}f_ j(x_ j)=E(Y| X=x)\), \(x\in [0,1]^ J\) under mild conditions on the distribution of X and the functions \(f_ j\). The case of approximative additivity is also dealt with.
Reviewer: R.Schlittgen

62J02 General nonlinear regression
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
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