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Nonlinear time series. Nonparametric and parametric methods. (English) Zbl 1014.62103
Springer Series in Statistics. New York, NY: Springer. xix, 551 p. (2003).
This is a book on (modern) time series analysis, covering standard linear models, and nonlinear models, with emphasis on the latter. The approach is stated in the Preface; for each model the authors provide definitions, probability properties, statistical inference methods, numerical examples and computing considerations. Both parametric and nonparametric procedures are covered. No sophisticated mathematical details are required from the reader. The contents of the book are wide, and a short view is provided in Chapter 1. Linear models (i.e., ARMA models) are presented with little detail. Then procedures are given of the following topics:
(1) Nonlinear time series models: ARCH (autoregressive conditional heteroscedastic) and generalized ARCH, GARCH, TAR (threshold autoregressive), NARCH (nonparametric autoregressive conditional heteroscedastic) and NAR (nonparametric autoregressive).
(2) Local and global approximations: local linear modeling, global spline approximation, and goodness of fit tests. These and related topics are covered in the book.
After two preparatory chapters (2 and 3) we find chapters on parametric nonlinear time series models, nonparametric density estimation, smoothing, spectral density estimation, nonparametric models, model validation, and nonlinear predictions. Six numerical illustrations are presented in Chapter 1, and analyzed in other chapters. Simulated series are also used. In some chapters a section of “complements” collects proofs and mathematical details. Bibliographical notes are added to each chapter, but no exercises.
In summary, the authors present a useful collection of nonlinear time series models, many of which are treated in the contemporary literature. The interested reader will find a wealth of procedures and suggestions for implementation.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62-02 Research exposition (monographs, survey articles) pertaining to statistics