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**Finite mixture and Markov switching models.**
*(English)*
Zbl 1108.62002

Springer Series in Statistics. Berlin: Springer (ISBN 0-387-32909-9/hbk). xix, 492 p. (2006).

Finite mixture distributions are important for many models. Therefore they constitute a very active field of research. This book gives an up to date overview over the various models of this kind. The first part (nine chapters) is on finite mixtures of uncorrelated observations while the second (four chapters) deals with time series data.

Chapter 1 introduces to finite mixture modeling, especially to finite mixture distributions and to their identifiability. Chapters 2 and 3 deal with statistical inference when the number of components is known. Different approaches are sketched, Bayesian inference is outlined in detail.

In chapters 4 and 5 model specification uncertainty is dealt with. Several informal methods for diagnosing mixtures are reviewed. Bayesian inference under model uncertainty is introduced in chapter 4 and discussed in full detail in chapter 5 with focus on computational tools for Bayesian inference.

The special case of normal components is considered next. This is continued in chapter 9, where mixtures with nonnormal components are investigated. But the two chapters before deal with with data analysis aspects. Chapter 7 deals with model based clustering, outlier modeling and robust finite mixtures. Finite mixtures of regression models are discussed at length in chapter 8. Mixed effects models and repeated measurements are also detailed. Chapter 10 establishes the bridge to time series models with mixtures by introducing finite Markov mixture distributions and statistical models based upon them.

Statistical inference for Markov switching models is investigated in chapter 11. Nonlinear time series analysis based on Markov switching models, especially the Markov switching autoregressive model, the related Markov switching dynamic regression and switching ARCH models are introduced in chapter 12. Some extensions are studied, too. The next chapter starts with a short introduction to state space modeling with and without switching. Then the main switching state space models are reviewed and the related results are presented.

The aim of this book is to impart the finite mixture and Markov switching approach to statistical modelling to a wide-ranging community. The presentation is rather informal without abandoning mathematical correctness. It is mainly written from a Bayesian point of view, but it has a broader basis as far as statistical inference is concerned. For the frequentists, it offers a good opportunity to explore the advantages of the Bayesian approach in the context of mixing models.

Chapter 1 introduces to finite mixture modeling, especially to finite mixture distributions and to their identifiability. Chapters 2 and 3 deal with statistical inference when the number of components is known. Different approaches are sketched, Bayesian inference is outlined in detail.

In chapters 4 and 5 model specification uncertainty is dealt with. Several informal methods for diagnosing mixtures are reviewed. Bayesian inference under model uncertainty is introduced in chapter 4 and discussed in full detail in chapter 5 with focus on computational tools for Bayesian inference.

The special case of normal components is considered next. This is continued in chapter 9, where mixtures with nonnormal components are investigated. But the two chapters before deal with with data analysis aspects. Chapter 7 deals with model based clustering, outlier modeling and robust finite mixtures. Finite mixtures of regression models are discussed at length in chapter 8. Mixed effects models and repeated measurements are also detailed. Chapter 10 establishes the bridge to time series models with mixtures by introducing finite Markov mixture distributions and statistical models based upon them.

Statistical inference for Markov switching models is investigated in chapter 11. Nonlinear time series analysis based on Markov switching models, especially the Markov switching autoregressive model, the related Markov switching dynamic regression and switching ARCH models are introduced in chapter 12. Some extensions are studied, too. The next chapter starts with a short introduction to state space modeling with and without switching. Then the main switching state space models are reviewed and the related results are presented.

The aim of this book is to impart the finite mixture and Markov switching approach to statistical modelling to a wide-ranging community. The presentation is rather informal without abandoning mathematical correctness. It is mainly written from a Bayesian point of view, but it has a broader basis as far as statistical inference is concerned. For the frequentists, it offers a good opportunity to explore the advantages of the Bayesian approach in the context of mixing models.

Reviewer: R. Schlittgen (Hamburg)