Monte Carlo statistical methods.

*(English)*Zbl 0935.62005
Springer Texts in Statistics. New York, NY: Springer. xxi, 507 p. (1999).

Monte Carlo statistical methods, particularly those based on Markov chains, have now matured to be part of the standard set of techniques used by statisticians. This book is intended to bring these techniques into the classrooms, being a self-contained logical development of the subject with all concepts being explained in detail, and all theorems etc. having detailed proofs. There is also an abundance of examples and problems, relating the concepts with statistical practice and enhancing primarily the applications of simulation techniques to statistical problems of various difficulties. As a textbook it is intended for graduate courses.

Chapters 1-3 are introductory. Chapter 1 is a review of various statistical methodologies and of corresponding computational problems. Chapters 2 and 3 contain the basics of random variable generation and Monte Carlo integration. Chapter 4, which is certainly the most theoretical in this book, is an introduction to Markov chains, covering enough theory to allow the reader to understand the working and to evaluate the performance of Markov chain Monte Carlo (MCMC) methods. Section 4.1 is provided for the reader who is already familiar with Markov chains, but needs a refresher, especially in the application of Markov chain theory to Monte Carlo calculations. Chapter 5 covers optimization and provides the first application of Markov chains to simulation methods. Chapters 6 and 7 cover the heart of MCMC methodology, the Metropolis-Hastings algorithm and the Gibbs sampler. Finally, Chapter 8 presents the state-of-art methods for monitoring convergence of the MCMC methods and Chapter 9 shows how these methods apply to some statistical settings which cannot be processed otherwise, namely the missing data models.

Each chapter concludes with a section of notes that serve to enhance the discussion in the chapters, describe alternate or more advanced methods and point the reader to further work that has been done, as well as to current research trends in the area. The level and rigor of the notes are variable, with some of the material being advanced.

The book can be used at several levels and can be presented in several ways. For example, Chapters 1-3 and most of the Chapter 5 cover standard simulation theory, and hence serve as a basic introduction to this topic. Chapters 6-9 are totally concerned with the MCMC methodology. Thus, one can easily imagine both one- and two-semester courses based on this book.

Contents: List of tables; List of figures 1. Introduction; 2. Random variable generation; 3. Monte Carlo integration; 4. Markov chains; 5. Monte Carlo integration; 6. The Metropolis-Hastings algorithm; 7. The Gibbs sampler; 8. Diagnosing convergence; 9. Implementation in missing data models. A. Probability distributions; B. Notation; C. References; Subject index; Author index.

Chapters 1-3 are introductory. Chapter 1 is a review of various statistical methodologies and of corresponding computational problems. Chapters 2 and 3 contain the basics of random variable generation and Monte Carlo integration. Chapter 4, which is certainly the most theoretical in this book, is an introduction to Markov chains, covering enough theory to allow the reader to understand the working and to evaluate the performance of Markov chain Monte Carlo (MCMC) methods. Section 4.1 is provided for the reader who is already familiar with Markov chains, but needs a refresher, especially in the application of Markov chain theory to Monte Carlo calculations. Chapter 5 covers optimization and provides the first application of Markov chains to simulation methods. Chapters 6 and 7 cover the heart of MCMC methodology, the Metropolis-Hastings algorithm and the Gibbs sampler. Finally, Chapter 8 presents the state-of-art methods for monitoring convergence of the MCMC methods and Chapter 9 shows how these methods apply to some statistical settings which cannot be processed otherwise, namely the missing data models.

Each chapter concludes with a section of notes that serve to enhance the discussion in the chapters, describe alternate or more advanced methods and point the reader to further work that has been done, as well as to current research trends in the area. The level and rigor of the notes are variable, with some of the material being advanced.

The book can be used at several levels and can be presented in several ways. For example, Chapters 1-3 and most of the Chapter 5 cover standard simulation theory, and hence serve as a basic introduction to this topic. Chapters 6-9 are totally concerned with the MCMC methodology. Thus, one can easily imagine both one- and two-semester courses based on this book.

Contents: List of tables; List of figures 1. Introduction; 2. Random variable generation; 3. Monte Carlo integration; 4. Markov chains; 5. Monte Carlo integration; 6. The Metropolis-Hastings algorithm; 7. The Gibbs sampler; 8. Diagnosing convergence; 9. Implementation in missing data models. A. Probability distributions; B. Notation; C. References; Subject index; Author index.

Reviewer: J.Antoch (Praha)

##### MSC:

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

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

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |