##
**Symmetric multivariate and related distributions.**
*(English)*
Zbl 0699.62048

Monographs on Statistics and Applied Probability, 36. London etc.: Chapman and Hall. x, 220 p. £32.50 (1990).

This is a monograph on the probabilistic properties of symmetric multivariate and related distributions. The \(220+x\) pages are divided into seven chapters, as follows: Chapter 1 on preliminaries considers the construction of symmetric multivariate distributions, and the approach of a stochastic decomposition is suggested as most convenient and fruitful. Then the symmetry of the cumulative distribution, characteristic function, density, and of a stochastic representation are considered. The chapter closes with a detailed discussion of the Dirichlet distribution, “a basic distribution for the purpose of this volume”. (Page 16).

Chapter 2 considers spherical (extension of the \(N(\underset \tilde{} 0,\underset \tilde{} I))\) and elliptical (extension of the \(N({\underset \tilde{} \mu},{\underset \tilde{} \Sigma}))\) symmetric distributions. Marginal and conditional distributions, moments, densities, characteristic functions and the like are considered for both types; also mixtures of normals, log-elliptical and additive logistic elliptical, and complex elliptically symmetric distributions. A more statistical Section 2.7 deals with robust statistics and the regression model.

In Chapter 3 on some subclasses of elliptical distributions, the following types of distributions are studied: multiuniform, symmetric Kotz type, symmetric multivariate Pearson type VII (including the multivariate t) and II, and others. Chapter 4 deals with characterization problems for spherical and normal distributions. Chapter 5 introduces three main families of multivariate \(\ell_ 1\)-norm symmetric distributions, namely: the family \(L_ n\) (misprinted n on page 112) of scale mixtures of uniform distributions on the surface of the \(\ell_ 1\)-norm unit sphere, a more general family \(T_ n\) of which the survival functions are functions of \(\ell_ 1\)-norm, and a subset \(L_{n,\infty}\) of \(L_ n\) which is constructed as a scale mixture of random vectors with i.i.d. exponential components. The relationships among these three families and some applications are given.

Chapter 6 gives a detailed discussion of multivariate Liouville distributions and Chapter 7 of \(\alpha\)-symmetric distributions, defined as follows: an n-dimensional distribution whose characteristic function is \(\psi (\| \underset \tilde{} t\|_{\alpha})\), where \(\| \underset \tilde{} t\|_{\alpha}=(\sum | t_ i|^{\alpha})^{1/\alpha}\) is called \(\alpha\)-symmetric with c.f. generator \(\psi\) (\(\cdot).\)

A sizable portion of this work is related to research work by the authors, in particular of the first author alone or with co-authors. Most of this work is concentrated on the 1980’s. However, references are made to work done by many other authors, in scattered sources. This is an asset of the book; for example, the bibliography has 224 entries, of which 27 are authored or co-authored by one or more of the present authors. The preface is rather non-informative, and introductory remarks are promised for the beginning of each chapter; I find these remarks quite brief, which is perhaps due to the monographic character of the work.

Statisticians may wonder how much of statistics is properly covered in this monograph, and the answer is not much. Except for Section 2.7 mentioned above and Section 6.6 on scale invariant statistics, the statistical implications of symmetry are considered only briefly in this monograph, that is more concentrated on the basic probabilistic properties of the distributions. References to statistical work are given. The field of symmetric multivariate distributions is very active at the present, and this monograph will be a valuable addition to the sources of information for those interested in contributing to the field.

Chapter 2 considers spherical (extension of the \(N(\underset \tilde{} 0,\underset \tilde{} I))\) and elliptical (extension of the \(N({\underset \tilde{} \mu},{\underset \tilde{} \Sigma}))\) symmetric distributions. Marginal and conditional distributions, moments, densities, characteristic functions and the like are considered for both types; also mixtures of normals, log-elliptical and additive logistic elliptical, and complex elliptically symmetric distributions. A more statistical Section 2.7 deals with robust statistics and the regression model.

In Chapter 3 on some subclasses of elliptical distributions, the following types of distributions are studied: multiuniform, symmetric Kotz type, symmetric multivariate Pearson type VII (including the multivariate t) and II, and others. Chapter 4 deals with characterization problems for spherical and normal distributions. Chapter 5 introduces three main families of multivariate \(\ell_ 1\)-norm symmetric distributions, namely: the family \(L_ n\) (misprinted n on page 112) of scale mixtures of uniform distributions on the surface of the \(\ell_ 1\)-norm unit sphere, a more general family \(T_ n\) of which the survival functions are functions of \(\ell_ 1\)-norm, and a subset \(L_{n,\infty}\) of \(L_ n\) which is constructed as a scale mixture of random vectors with i.i.d. exponential components. The relationships among these three families and some applications are given.

Chapter 6 gives a detailed discussion of multivariate Liouville distributions and Chapter 7 of \(\alpha\)-symmetric distributions, defined as follows: an n-dimensional distribution whose characteristic function is \(\psi (\| \underset \tilde{} t\|_{\alpha})\), where \(\| \underset \tilde{} t\|_{\alpha}=(\sum | t_ i|^{\alpha})^{1/\alpha}\) is called \(\alpha\)-symmetric with c.f. generator \(\psi\) (\(\cdot).\)

A sizable portion of this work is related to research work by the authors, in particular of the first author alone or with co-authors. Most of this work is concentrated on the 1980’s. However, references are made to work done by many other authors, in scattered sources. This is an asset of the book; for example, the bibliography has 224 entries, of which 27 are authored or co-authored by one or more of the present authors. The preface is rather non-informative, and introductory remarks are promised for the beginning of each chapter; I find these remarks quite brief, which is perhaps due to the monographic character of the work.

Statisticians may wonder how much of statistics is properly covered in this monograph, and the answer is not much. Except for Section 2.7 mentioned above and Section 6.6 on scale invariant statistics, the statistical implications of symmetry are considered only briefly in this monograph, that is more concentrated on the basic probabilistic properties of the distributions. References to statistical work are given. The field of symmetric multivariate distributions is very active at the present, and this monograph will be a valuable addition to the sources of information for those interested in contributing to the field.

Reviewer: R.Mentz

### MSC:

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

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

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60E05 | Probability distributions: general theory |

62H10 | Multivariate distribution of statistics |