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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

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