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Non-linear time series. A dynamical system approach. (English) Zbl 0716.62085

Oxford Statistical Science Series, 6. Oxford: Clarendon Press. xvi, 564 p. £50.00 (1990).
The book has 7 chapters and 3 appendices. The first chapter describes some fundamental concepts, especially stationary processes and linear Gaussian models. The description of some limitation of ARMA models is a natural step to nonlinear processes. An introduction to dynamical systems is given in the second chapter. Roughly speaking, the author analyzes the part of nonlinear models which does not contain random elements. One of the most important concepts is a limit cycle, which is studied in detail in the case of nonlinear difference equations. The dependence on initial values is characterized by introducing Lyapunov exponents. Basic elements of the stability theory for difference equations are also presented.
Some nonlinear time series models are given in the third chapter. The author describes nonlinear autoregression, threshold models, exponential autoregressive models, fractional autoregressive models, product autoregressive models, random coefficient autoregressive models, bilinear models, nonlinear MA models, ARCH models, doubly stochastic models and some others. Their probability structure is investigated in the fourth chapter. After dealing with deterministic stability, stochastic stability and ergodicity, some stationary distributions are calculated and a Markovian representation is derived. Chapter 5 presents statistical aspects of nonlinear models. Attention is mainly paid to tests for linearity, model selection and diagnostics.
Nonlinear least-squares prediction can be found in the 6th chapter. Some procedures are based on recursive formulas for calculating conditional densities. The problem of prediction is investigated for several important nonlinear models. A few case studies are reported in the 7 th chapter. Two of the most popular sets of time series data, namely the Canadian lynx data and the sunspot numbers, are also included.
Appendix 1, written by K. S. Chan, contains theoretical background for deterministic stability, stochastic stability and ergodicity. Appendix 2 summarizes briefly martingale limit theory and appendix 3 contains data, the analysis of which is presented in the text to illustrate theoretical results. Each chapter is endowed with bibliographical notes and with a section containing exercises and complements.
The book gives an up-to-date account of the nonlinear time series models. It is not easy to present such models within a unified theory, since special models need different parts of mathematics. The author succeeded in giving all important information about the most popular nonlinear models including references to orginal papers. The analysis of real data is very convincing. There are many pictures where the results of statistical investigation are graphically presented. This valuable book can be recommended to general applied statisticians as well as to specialists in time series analysis.
Reviewer: J.Anděl

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
62M20 Inference from stochastic processes and prediction
62-02 Research exposition (monographs, survey articles) pertaining to statistics
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
39A11 Stability of difference equations (MSC2000)
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