Spline smoothing and nonparametric regression.

*(English)*Zbl 0702.62036
Statistics: Textbooks and Monographs, 90. New York etc.: Marcel Dekker, Inc. xvii, 438 p. (1988).

This book provides an introduction to the area of spline smoothing and nonparametric regression. The author gives a fairly broad overview on the subject and an intuitive insight into the basic ideas of data smoothing.

The first chapter introduces the concept of nonparametric regression by contrasting it with methods based on fitting a finite number of parameters in some given model. The different philosophies, which underlie the parametric and the nonparametric approach to regression analysis are well described. Several measures of the goodness of estimators are defined in Chapter 2. Here the main attention is given to the average mean squared error. Since these risks depend on the unknown regression function, cross validation and generalized cross validation methods are proposed as unbiased risk estimators.

Chapters 3-7 contain a study of important classes of nonparametric regression estimators: Fourier series type estimators, kernel estimators, smoothing spline estimators and smoothing spline variants. For each of these families of estimators the author provides a mathematical motivation, some history, comments on computation, some asymptotic properties, suggestions for diagnostics and some applications. Exercises are given at the end of each chapter. Since the application of all these estimation methods requires the choice of a smoothing parameter, adaptive smoothing parameter selection procedures on the basis of the cross validation criterion are considered.

The series estimators in Chapter 3 are introduced as generalizations of polynomial regression. The Fourier series estimators can be written as kernel estimators and provide a convenient introduction to the topic of kernel estimators treated in Chapter 4. The heart of the book is Chapter 5 and 6 where smoothing splines and smoothing spline variants are studied. These estimators, which are again motivated by considering polynomials, are derived from a penalized least-squares criterion. This estimation approach incorporates the desires for smoothness and goodness- of-fit directly into the estimation process. The chapter concludes with a derivation of the smoothing spline as a Bayes estimator. Chapter 6 describes various extensions of the spline method. The connection between smoothing splines and Fourier series and kernel estimators is pointed out. Finally, in Chapter 7 a number of other estimator types are mentioned and briefly discussed.

The text is designed for second or third year students in a graduate program in statistics, but I recommend it to anyone with research interest in the area of nonparametric regression. For those who are already familiar with splines and nonparametric regression, it provides a handy reference to details, extensions and applications, while for beginners it may be used as an introduction to the subject.

The first chapter introduces the concept of nonparametric regression by contrasting it with methods based on fitting a finite number of parameters in some given model. The different philosophies, which underlie the parametric and the nonparametric approach to regression analysis are well described. Several measures of the goodness of estimators are defined in Chapter 2. Here the main attention is given to the average mean squared error. Since these risks depend on the unknown regression function, cross validation and generalized cross validation methods are proposed as unbiased risk estimators.

Chapters 3-7 contain a study of important classes of nonparametric regression estimators: Fourier series type estimators, kernel estimators, smoothing spline estimators and smoothing spline variants. For each of these families of estimators the author provides a mathematical motivation, some history, comments on computation, some asymptotic properties, suggestions for diagnostics and some applications. Exercises are given at the end of each chapter. Since the application of all these estimation methods requires the choice of a smoothing parameter, adaptive smoothing parameter selection procedures on the basis of the cross validation criterion are considered.

The series estimators in Chapter 3 are introduced as generalizations of polynomial regression. The Fourier series estimators can be written as kernel estimators and provide a convenient introduction to the topic of kernel estimators treated in Chapter 4. The heart of the book is Chapter 5 and 6 where smoothing splines and smoothing spline variants are studied. These estimators, which are again motivated by considering polynomials, are derived from a penalized least-squares criterion. This estimation approach incorporates the desires for smoothness and goodness- of-fit directly into the estimation process. The chapter concludes with a derivation of the smoothing spline as a Bayes estimator. Chapter 6 describes various extensions of the spline method. The connection between smoothing splines and Fourier series and kernel estimators is pointed out. Finally, in Chapter 7 a number of other estimator types are mentioned and briefly discussed.

The text is designed for second or third year students in a graduate program in statistics, but I recommend it to anyone with research interest in the area of nonparametric regression. For those who are already familiar with splines and nonparametric regression, it provides a handy reference to details, extensions and applications, while for beginners it may be used as an introduction to the subject.

Reviewer: H.Liero

##### MSC:

62G07 | Density estimation |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62G20 | Asymptotic properties of nonparametric inference |

62F15 | Bayesian inference |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

65D07 | Numerical computation using splines |