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Nonparametric and semiparametric models. (English) Zbl 1059.62032

Springer Series in Statistics. Berlin: Springer (ISBN 3-540-20722-8/hbk). xxvii, 299 p. (2004).
This is another book by Professor Wolfgang Härdle and his colleagues on nonparametric statistics and smoothing. The unique feature of this book is the inclusion of topics on semi-parametric regression models for high-dimensional data. The only other book devoted to these topics is the recent one by D. Ruppert, M. P. Wand and R. J. Carroll [Semiparametric Regression. (2003; Zbl 1038.62042)], although some of these topics have been also reviewed in the book edited by M. G. Schimek [Smoothing and Regression. Approaches, Computation and Application. (2000)]. After a brief introduction to the topics that are covered, the book is divided into two parts.
The first part deals with traditional topics in nonparametric density and regression estimation. It starts with the presentation of the histogram as a nonparametric estimate of density functions. The discussions on the shortcomings of this simple nonparametric density estimate lead naturally to the next topic in this part: the estimation of density functions by kernel methods. The statistical properties of kernel density estimates and selection of bandwidths and kernel functions are discussed in detail. The construction of confidence intervals and confidence bands for the unknown density function based on kernel estimates is also mentioned. Nonparametric regression models are dealt with next. The kernel method for univariate nonparametric regression models is treated in more detail. Other methods, such as local polynomial regression, nearest-neighbor estimator, median and spline smoothing, and estimators based on orthogonal series, are briefly introduced and their connections with the kernel method are mentioned. Selection of bandwidth through cross-validation and penalizing functions is discussed for kernel estimators. Statistical procedures for the construction of confidence intervals and bands for nonparametric regression functions and tests for hypotheses related to nonparametric regression functions are also introduced. The generalization of kernel methods to multivariate nonparametric regression functions is briefly discussed at the end of this chapter. The problem of the “curse of dimensionality” with nonparametric estimation for multivariate regression functions is addressed.
This problem and the lack of interpretability of the general nonparametric models motivated the development of semi-parametric models for high-dimensional data, which are the topic in the second part of this book. The generalized linear models (GLMs) are used as the background to introduce these models and the methods of fitting GLMs are first reviewed. The following semi-parametric models are then discussed in detail: single index models, generalized partial linear models, generalized additive models including generalized additive partial linear models. Detailed estimation procedures and algorithms are presented for the fitting of these models to the data. Procedures for the assessment of the goodness-of-fit of the semi-parametric models and comparisons of these models with simple parametric models are also described.
The first part of this book is intended for undergraduate students while the second part for master and PhD students or researchers. Minimum theory and numerical examples are covered in this book, which makes this book mostly suitable for a course in nonparametric regression to graduate students. The overheads available on the website of the book will be especially helpful to instructors who use this book to teach such a course. No descriptions of computer programs are presented in the book, although some macros in Xplore may be downloaded from the website of the book. This would make this book less appealing to researchers or practioners who are more familiar with some more popular computer software packages such as SAS, S-plus or R.
In summary, as the authors mentioned in the preface of the book, this book will be useful for readers who would like to understand the statistical and mathematical principles and basic concepts and techniques of smoothing.

MSC:

62G07 Density estimation
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G08 Nonparametric regression and quantile regression
62J12 Generalized linear models (logistic models)

Citations:

Zbl 1038.62042

Software:

R; SAS; XploRe; S-PLUS
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