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**Linear smoothers and additive models.**
*(English)*
Zbl 0689.62029

The authors are concerned with the application of nonparametric linear smoothers to additive models in regression analysis. In the first part of their paper they discuss various linear smoothers for a nonparametric regression function \(f(x)=E(Y| X=x)\) as e.g. running means and lines, cubic smoothing splines, kernel smoothers etc.

In particular they are concerned with the eigenvalues and the singular value decompositions of the associated matrices, since these are important for the second and main part of the paper. There, the authors use these smoothers for empirical versions of conditional expectations which are plugged in certain normal equations for the functions \(f_ 1,...,f_ p\) in the additive model \[ E(Y| X_ j,1\leq j\leq p)=\sum^{p}_{j=1}f_ j(Xj). \] The empirical normal equations are solved by the backfitting-algorithm (resp. Gauss-Seidel method). The main results give conditions which assure the existence of solutions of the empirical normal equations and the convergence of the backfitting and related methods. The interesting paper is completed by contributions of various discussants.

In particular they are concerned with the eigenvalues and the singular value decompositions of the associated matrices, since these are important for the second and main part of the paper. There, the authors use these smoothers for empirical versions of conditional expectations which are plugged in certain normal equations for the functions \(f_ 1,...,f_ p\) in the additive model \[ E(Y| X_ j,1\leq j\leq p)=\sum^{p}_{j=1}f_ j(Xj). \] The empirical normal equations are solved by the backfitting-algorithm (resp. Gauss-Seidel method). The main results give conditions which assure the existence of solutions of the empirical normal equations and the convergence of the backfitting and related methods. The interesting paper is completed by contributions of various discussants.

Reviewer: U.Stadtmüller

### MSC:

62G05 | Nonparametric estimation |

65D10 | Numerical smoothing, curve fitting |

65C99 | Probabilistic methods, stochastic differential equations |