Additive regression and other nonparametric models. (English) Zbl 0605.62065

Stone, Charles J.

Ann. Stat. 13, 689-705 (1985).

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

Keywords:

closest additive approximation; spline estimates; flexibility; dimensionality; interpretability; parametric models; nonparametric and semiparametric models; dimensionality reduction principle; optimal rate of convergence; additive regression model