Extreme value distributions. Theory and applications.

*(English)*Zbl 0960.62051
London: Imperial College Press (ISBN 1-86094-224-5/hbk; 978-1-86094-402-4/ebook). vii, 185 p. (2000).

From the preface: “This monograph attempts to describe in an organized manner the central ideas and results of probabilistic extreme-value theory and related models stemming from pioneering contributions of E.J. Gumbel in the early forties of this century.” “It is a book about extreme value distributions – both univariate and multivariate – and their applications, supplemented by an up-to-data extensive bibliography, aimed mainly at a novice in the field; hopefully a specialist may find therein some useful information as well.” “In our opinion, the extreme value theory – as described in this book – is a most important and successful example of applicability of mathematics to modern engineering, empirical and environmental problems of great significance…”.

The contents are as follows: Chapter 1, Univariate extreme value distributions: 1.1, Historical survey; 1.2, The three types of extreme value distributions; 1.3, Limiting distributions and domains of attraction; 1.4, Distribution functions and moments of type 1 distributions; 1.5, Order statistics, record values and characterizations; 1.6, Generation, tables, probability paper, plots and goodness of fit; 1.7, Methods of inference; 1.8, Distributions related to the classical extremal distributions; 1.9, Applications of the classical extreme value distributions.

Chapter 2, Generalized extreme value distributions: 2.1, Basic properties; 2.2, Statistical inference (Classical approach); 2.3, Bayesian inference, 2.4, Robust estimation; 2.5, Zempléni’s test of hypothesis for GEV distributions; 2.6, Estimation of the tail index of a distribution; 2.7, Other forms of generalized extreme value distributions; 2.8, Some selected applications.

Chapter 3, Multivariate extreme value distributions: 3.1, Limit laws for multivariate extremes; 3.2, Characterizations of the domain of attraction; 3.3, Characterizations of multivariate extreme value distributions; 3.4, Parametric families of bivariate extreme value distributions; 3.5, Parametric families of multivariate extreme value distributions; 3.6, Statistical estimation; 3.7, Simulations; 3.8, A selective survey of applications of multivariate extreme value distributions; Bibliography; Subject Index.

In summary, this is a solid mathematical treatment of some topics of extreme value theory addressed to professionals in the field of statistical distributions and statistical inference as well as to people interested in the applications of extreme value distributions. The book can be used very constructively as a textbook or a consulting book.

The contents are as follows: Chapter 1, Univariate extreme value distributions: 1.1, Historical survey; 1.2, The three types of extreme value distributions; 1.3, Limiting distributions and domains of attraction; 1.4, Distribution functions and moments of type 1 distributions; 1.5, Order statistics, record values and characterizations; 1.6, Generation, tables, probability paper, plots and goodness of fit; 1.7, Methods of inference; 1.8, Distributions related to the classical extremal distributions; 1.9, Applications of the classical extreme value distributions.

Chapter 2, Generalized extreme value distributions: 2.1, Basic properties; 2.2, Statistical inference (Classical approach); 2.3, Bayesian inference, 2.4, Robust estimation; 2.5, Zempléni’s test of hypothesis for GEV distributions; 2.6, Estimation of the tail index of a distribution; 2.7, Other forms of generalized extreme value distributions; 2.8, Some selected applications.

Chapter 3, Multivariate extreme value distributions: 3.1, Limit laws for multivariate extremes; 3.2, Characterizations of the domain of attraction; 3.3, Characterizations of multivariate extreme value distributions; 3.4, Parametric families of bivariate extreme value distributions; 3.5, Parametric families of multivariate extreme value distributions; 3.6, Statistical estimation; 3.7, Simulations; 3.8, A selective survey of applications of multivariate extreme value distributions; Bibliography; Subject Index.

In summary, this is a solid mathematical treatment of some topics of extreme value theory addressed to professionals in the field of statistical distributions and statistical inference as well as to people interested in the applications of extreme value distributions. The book can be used very constructively as a textbook or a consulting book.

Reviewer: Wiesław Dziubdziela (Kielce)

##### MSC:

62G32 | Statistics of extreme values; tail inference |

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

60G70 | Extreme value theory; extremal stochastic processes |

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

62G30 | Order statistics; empirical distribution functions |