Multivariate \(t\) distributions and their applications.

*(English)*Zbl 1100.62059
Cambridge University Press (ISBN 0-521-82654-3/hbk). xii, 272 p. (2004).

The book is a collection of all the results existing in the literature on multivariate \(t\) distributions over the last 50 years. The compiled results equally emphasize theoretical aspects of probabilistic nature and statistical aspects, including estimation. These are supplemented with various generalizations and applications. The material presented is structured in 12 chapters.

In chapter 1, the multivariate \(t\) distribution is defined through a direct generalization of its univariate version in a manner analogous to that relating the univariate and multivariate versions of the normal distribution. Various representations of a \(p\)-dimensional \(t\) vector are given and several structural properties of the distribution are presented including characterizations. Exact expressions for the densities of the unconditional and conditional marginals as well as for the moments of the distribution are given. Finally, brief discussions of the concepts of association, entropy, Kullback-Leibler and Rényi information are presented. The chapter concludes with a brief mention of special cases of multivariate \(t\) distributions studied in the literature with great detail.

Chapter 2 is devoted to the characteristic function of the multivariate \(t\) distribution providing a series representation of it as well as expressions in terms of the McDonald function. The chapter includes a section on the infinite divisibility of the distribution and the corresponding Lévy representation.

Chapter 3 discusses the distributions of linear combinations, products and ratios of components of a \(p\)-variate \(t\) vector providing explicit expressions for their densities whenever possible. Chapter 4 surveys specific bivariate distributions that contain Student’s \(t\) components. Modifications and extensions of the standard form of the multivariate \(t\) distribution as introduced in Chapter 1 are discussed in chapter 5.

Chapters 6 and 7 provide an anthology of results on probability integrals and probability inequalities for multivariate \(t\) distributions. These conclude the exposition of results of theoretical and probabilistic nature. The remaining chapters look into aspects of multivariate \(t\) distributions that emphasize the statistical aspects.

So, chapter 8 presents several approaches to the evaluation of percentage points. Some of them are of historical interest rather than of practical importance, but still worth in the book as they provide a better perspective of the developments in the area. Chapter 9 discusses sampling distributions of statistics associated with the multivariate \(t\) distribution, while Chapter 10 looks into the estimation of it presenting estimation approaches that include missing data imputation methods and discussions on the estimation of the correlation matrix and its trace. The chapter concludes with mentioning of methods for simulation from multivariate \(t\) distributions.

Chapter 11 presents regression models with error terms jointly distributed according to the multivariate \(t\) distribution including classical and general linear models, Bayesian and indexed linear models as well as nonlinear models. Finally, chapter 12 provides applications of the multivariate \(t\) distribution in the framework of exploratory projection pursuit, portfolio optimization and multiple decision problems.

Overall, the book is an excellent, well and up-to-date referenced source of information on results existing in the literature on multivariate \(t\) distributions over the last 50 years. It is well written with a user-friendly presentation. The book is the first instance where results on the multivariate \(t\) distribution have been put together in an organized manner, thus offering a useful tool to researchers and practitioners in statistics and economics.

In chapter 1, the multivariate \(t\) distribution is defined through a direct generalization of its univariate version in a manner analogous to that relating the univariate and multivariate versions of the normal distribution. Various representations of a \(p\)-dimensional \(t\) vector are given and several structural properties of the distribution are presented including characterizations. Exact expressions for the densities of the unconditional and conditional marginals as well as for the moments of the distribution are given. Finally, brief discussions of the concepts of association, entropy, Kullback-Leibler and Rényi information are presented. The chapter concludes with a brief mention of special cases of multivariate \(t\) distributions studied in the literature with great detail.

Chapter 2 is devoted to the characteristic function of the multivariate \(t\) distribution providing a series representation of it as well as expressions in terms of the McDonald function. The chapter includes a section on the infinite divisibility of the distribution and the corresponding Lévy representation.

Chapter 3 discusses the distributions of linear combinations, products and ratios of components of a \(p\)-variate \(t\) vector providing explicit expressions for their densities whenever possible. Chapter 4 surveys specific bivariate distributions that contain Student’s \(t\) components. Modifications and extensions of the standard form of the multivariate \(t\) distribution as introduced in Chapter 1 are discussed in chapter 5.

Chapters 6 and 7 provide an anthology of results on probability integrals and probability inequalities for multivariate \(t\) distributions. These conclude the exposition of results of theoretical and probabilistic nature. The remaining chapters look into aspects of multivariate \(t\) distributions that emphasize the statistical aspects.

So, chapter 8 presents several approaches to the evaluation of percentage points. Some of them are of historical interest rather than of practical importance, but still worth in the book as they provide a better perspective of the developments in the area. Chapter 9 discusses sampling distributions of statistics associated with the multivariate \(t\) distribution, while Chapter 10 looks into the estimation of it presenting estimation approaches that include missing data imputation methods and discussions on the estimation of the correlation matrix and its trace. The chapter concludes with mentioning of methods for simulation from multivariate \(t\) distributions.

Chapter 11 presents regression models with error terms jointly distributed according to the multivariate \(t\) distribution including classical and general linear models, Bayesian and indexed linear models as well as nonlinear models. Finally, chapter 12 provides applications of the multivariate \(t\) distribution in the framework of exploratory projection pursuit, portfolio optimization and multiple decision problems.

Overall, the book is an excellent, well and up-to-date referenced source of information on results existing in the literature on multivariate \(t\) distributions over the last 50 years. It is well written with a user-friendly presentation. The book is the first instance where results on the multivariate \(t\) distribution have been put together in an organized manner, thus offering a useful tool to researchers and practitioners in statistics and economics.

Reviewer: Evdokia Xekalaki (Athens)

##### MSC:

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

62-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to statistics |

62H10 | Multivariate distribution of statistics |

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

60E07 | Infinitely divisible distributions; stable distributions |