zbMATH — the first resource for mathematics

Applied nonparametric regression. (English) Zbl 0714.62030
Econometric Society Monographs, 19. Cambridge etc.: Cambridge University Press. xv, 333 p. £30.00; $ 44.50 (1990).
The topic of the book is the nonparametric estimation of a smooth regression curve g in the model \(Y_ i=g(X_ i)+\epsilon_ i\), \(1\leq i\leq n\), where the \((\epsilon_ i)\) are observation errors and the \((Y_ i,X_ i)\) are measurements with random or deterministic variables \((X_ i)\). The book gives a survey of the state of the art in this area. The emphasis of the book is on the statistical aspects of the results and it includes computational points and programs as well. The following topics are considered:
After a discussion of the motivation for smoothing in this context, the most important smoothing techniques in regression analysis are introduced as e.g. kernel estimation, nearest-neighbor estimation, orthogonal series, smoothing splines. In the second part of the book kernel estimators are discussed in more detail, in particular: rates of convergence, confidence intervals and confidence bands based on various techniques including bootstrapping, boundary effects, choice of kernels and bias reduction techniques. Then various approaches for choosing the smoothing parameter are presented and compared. Further topics are robust smoothing, handling of correlated data and smoothing for regression functions with additional structure as e.g. monotonicity. The third part of the book is devoted to multivariate methods as e.g. regression trees, projection pursuit regression and generalized additive models.
For an interested reader with some prior knowledge in smoothing and a basic education in mathematical statistics, the book gives a good insiight into the methods of nonparametric regression analysis and a resonably good list of references.
Reviewer: U.Stadtmüller

62G07 Density estimation
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G15 Nonparametric tolerance and confidence regions
62H12 Estimation in multivariate analysis
62-04 Software, source code, etc. for problems pertaining to statistics