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**Dependence modeling with copulas.**
*(English)*
Zbl 1346.62001

Monographs on Statistics and Applied Probability 134. Boca Raton, FL: CRC Press (ISBN 978-1-4665-8322-1/hbk; 978-1-4665-8323-8/ebook). xviii, 462 p. (2014).

In 1997, the author published a book that soon became one of the key references in dependence models [Multivariate models and dependence concepts. London: Chapman and Hall (1997; Zbl 0990.62517)]. Since then, the research field devoted to copulas and dependence has grown considerably and a state-of-the-art monograph was long awaited. This book contains a detailed review of recent research and, although it is not a revised edition of the 1997 book, it is written with the same unique style in combining an in depth knowledge of the literature about theoretical models and constructions of copulas with the ability to provide sound procedures and practical suggestions for data analysis.

The content of the book can be summarized as follows. Chapter 1 provides an introduction to dependence models and their historical motivations, together with an account of likelihood methods in multivariate models. Chapter 2 presents basic aspects of dependence emphasizing measures of association that are particularly appropriate to describe the tail of the distribution. Chapters 3 and 4 illustrate, respectively, construction methods for copulas and parametric families. These two chapters provide an impressive quantity of material that should be taken into account by any “statistical modeler”. Chapters 5 and 6 are devoted to statistical aspects like inference and model selection, as well as computational problems (including algorithms). Examples of different applications to real data are collected in Chapter 7. Finally, Chapter 8 offers a detailed account of various mathematical results used in the book.

The main contributions of the present book are: a) a general overview of vine copulas, which represent one of the most significant advances for copula models with high-dimensional data; b) the emphasis on the various aspects of tail dependence as a way to distinguish various models; and c) the attention devoted to numerical aspects and algorithms, which are at the core of statistical estimation, especially with large amounts of data. I was also favorably impressed by the fact that the author provides free access to several codes and software related to the book (http://copula.stat.ubc.ca/).

To conclude, the book under review is most likely to become a key reference for a wide audience including PhD students, researchers and practitioners with an interest in multivariate dependence and copulas.

The content of the book can be summarized as follows. Chapter 1 provides an introduction to dependence models and their historical motivations, together with an account of likelihood methods in multivariate models. Chapter 2 presents basic aspects of dependence emphasizing measures of association that are particularly appropriate to describe the tail of the distribution. Chapters 3 and 4 illustrate, respectively, construction methods for copulas and parametric families. These two chapters provide an impressive quantity of material that should be taken into account by any “statistical modeler”. Chapters 5 and 6 are devoted to statistical aspects like inference and model selection, as well as computational problems (including algorithms). Examples of different applications to real data are collected in Chapter 7. Finally, Chapter 8 offers a detailed account of various mathematical results used in the book.

The main contributions of the present book are: a) a general overview of vine copulas, which represent one of the most significant advances for copula models with high-dimensional data; b) the emphasis on the various aspects of tail dependence as a way to distinguish various models; and c) the attention devoted to numerical aspects and algorithms, which are at the core of statistical estimation, especially with large amounts of data. I was also favorably impressed by the fact that the author provides free access to several codes and software related to the book (http://copula.stat.ubc.ca/).

To conclude, the book under review is most likely to become a key reference for a wide audience including PhD students, researchers and practitioners with an interest in multivariate dependence and copulas.

Reviewer: Fabrizio Durante (Bozen-Bolzano)

### MSC:

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

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

62H05 | Characterization and structure theory for multivariate probability distributions; copulas |

62Hxx | Multivariate analysis |

65C60 | Computational problems in statistics (MSC2010) |