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**Nonparametric smoothing and lack-of-fit tests.**
*(English)*
Zbl 0886.62043

Springer Series in Statistics. New York, NY: Springer. xii, 287 p. (1997).

The author writes in the preface that the primary aim of the book is to explore the use of nonparametric regression methodology in testing the fit of parametric regression models. Chapters 2 – 4 give a general introduction to estimation of regression curves in the case of single design variables with particular emphasis on smoothing methods. Chapter 2 is an expository introduction to basic methods of nonparametric regression. More attention is paid to kernel methods and Fourier series while splines and local polynomials are discussed only briefly. Chapter 3 studies statistical properties of kernel and Fourier series estimators. Chapter 4 is devoted to the problem of the choice of the estimators’ smoothing parameters and to introduction to several methods of data-driven smoothing.

Chapters 5 – 10 concern the problem of testing the fit of probability models. Chapter 5 reviews classical lack of fit tests, including likelihood ratio tests, reduction methods from linear models and some nonparametric tests. Chapter 6 considers more recently proposed lack of fit tests based on nonparametric linear smoothers. Chapters 7 – 10 are really the most interesting part of the monograph. The lack of fit tests based on data-driven smoothing parameters are studied. Chapters 7 – 8 contain a careful treatment of distributional properties of various “data-driven test” statistics. Chapter 7 introduces tests of “no effects” (regression function is constant). Using the same principles, test procedures for more general types of hypotheses are developed and studied in Chapters 8 and 9. Chapter 9 discusses extensions to multiple regression, testing additivity, testing homoscedasticity and time series trend detection, among others. Chapter 10 provides a number of illustrations of these tests on some data sets.

This is probably the first book dealing with tests of adequacy of parametric function estimates within a nonparametric framework reflecting recent developments in this area. Special attention is paid to tests and estimators based on Fourier coefficients. The book can be read on several levels. Following the advice of the author, the reader can focus only on the applications of estimation and testing procedures (for practitioners) or can learn more theoretical background of the procedures (for more theoretically oriented readers). The book is well written and suitable for graduate students, the material is explained without too many technical details. No excercices or problems to solve are included (this would be useful for graduate students).

Chapters 5 – 10 concern the problem of testing the fit of probability models. Chapter 5 reviews classical lack of fit tests, including likelihood ratio tests, reduction methods from linear models and some nonparametric tests. Chapter 6 considers more recently proposed lack of fit tests based on nonparametric linear smoothers. Chapters 7 – 10 are really the most interesting part of the monograph. The lack of fit tests based on data-driven smoothing parameters are studied. Chapters 7 – 8 contain a careful treatment of distributional properties of various “data-driven test” statistics. Chapter 7 introduces tests of “no effects” (regression function is constant). Using the same principles, test procedures for more general types of hypotheses are developed and studied in Chapters 8 and 9. Chapter 9 discusses extensions to multiple regression, testing additivity, testing homoscedasticity and time series trend detection, among others. Chapter 10 provides a number of illustrations of these tests on some data sets.

This is probably the first book dealing with tests of adequacy of parametric function estimates within a nonparametric framework reflecting recent developments in this area. Special attention is paid to tests and estimators based on Fourier coefficients. The book can be read on several levels. Following the advice of the author, the reader can focus only on the applications of estimation and testing procedures (for practitioners) or can learn more theoretical background of the procedures (for more theoretically oriented readers). The book is well written and suitable for graduate students, the material is explained without too many technical details. No excercices or problems to solve are included (this would be useful for graduate students).

Reviewer: M.Huškova (Praha)