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**Monte Carlo statistical methods.
2nd ed.**
*(English)*
Zbl 1096.62003

Springer Texts in Statistics. New York, NY: Springer (ISBN 0-387-21239-6/hbk). xxx, 645 p. (2004).

This second edition of ’Monte Carlo Statistical Methods’ has appeared only five years after the first one (see the review Zbl 0935.62005) and the new edition aims to incorporate recent developments. The book consists of the following 14 chapters:

1. Introduction; 2. Random Variable Generation; 3. Monte Carlo Integration; 4. Controlling Monte Carlo Variance; 5. Monte Carlo Optimization; 6. Markov Chains; 7. The Metropolis-Hastings Algorithm; 8. The Slice Sampler; 9. The Two-Stage Gibbs Sampler; 10. The Multi-Stage Gibbs Sampler; 11. Variable Dimension Models and Reversible Jump Algorithms; 12. Diagnosing Convergence; 13. Perfect Sampling; 14. Iterated and Sequential Importance Sampling. Two appendices contain material about probability distributions and notation.

Each chapter includes sections with problems and notes. From the first edition there have been considerable changes. There are five new chapters: a chapter on controlling the variance in Monte Carlo algorithms is added and the material on sampling, in particular on the Gibbs sampler, has been extended to several chapters. Also, the material on random variable generation in the second chapter has been revised and several approaches are now unified using a ‘Fundamental Theorem of Simulation’, which underlies the Accept-Reject methodology. The chapter on missing data models is omitted, the corresponding examples have been distributed over different other chapters. The style of the presentation and many carefully designed examples make the book very readable and easily accessible. It represents a comprehensive account of the topic containing valuable material for lecture courses as well as for research in this area.

1. Introduction; 2. Random Variable Generation; 3. Monte Carlo Integration; 4. Controlling Monte Carlo Variance; 5. Monte Carlo Optimization; 6. Markov Chains; 7. The Metropolis-Hastings Algorithm; 8. The Slice Sampler; 9. The Two-Stage Gibbs Sampler; 10. The Multi-Stage Gibbs Sampler; 11. Variable Dimension Models and Reversible Jump Algorithms; 12. Diagnosing Convergence; 13. Perfect Sampling; 14. Iterated and Sequential Importance Sampling. Two appendices contain material about probability distributions and notation.

Each chapter includes sections with problems and notes. From the first edition there have been considerable changes. There are five new chapters: a chapter on controlling the variance in Monte Carlo algorithms is added and the material on sampling, in particular on the Gibbs sampler, has been extended to several chapters. Also, the material on random variable generation in the second chapter has been revised and several approaches are now unified using a ‘Fundamental Theorem of Simulation’, which underlies the Accept-Reject methodology. The chapter on missing data models is omitted, the corresponding examples have been distributed over different other chapters. The style of the presentation and many carefully designed examples make the book very readable and easily accessible. It represents a comprehensive account of the topic containing valuable material for lecture courses as well as for research in this area.

Reviewer: Evelyn Buckwar (Berlin)

### MSC:

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

65C05 | Monte Carlo methods |

65C40 | Numerical analysis or methods applied to Markov chains |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

62F15 | Bayesian inference |