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Extreme value theory. An introduction. (English) Zbl 1101.62002

Springer Series in Operations Research and Financial Engineering. New York, NY: Springer (ISBN 0-387-23946-4/hbk). xvi, 417 p. (2006).
According to the authors, who have provided significant contributions to this field by themselves, the “the aim of this book is to give a thorough account of the basic theory of extreme values, probabilistic and statistical, theoretical and applied”. This goal has certainly been achieved. They call it “An introduction”, since “the account is by no means exhaustive”, which is somewhat misleading, because a thorough understanding of the material and methods presented requires more than a basic knowledge of probaility and statistics. Indeed, the reader should be familiar with modern measure theory based techniques of probability theory, including point processes and weak convergence in \(~C[0,1]\), together with recent developments of statistics, in particular in empirical and quantile process theory.
Those who are well-prepared, however, are lead to the current state of affairs in extreme value statistics, up to recent research on extremes of stochastic processes, for example. A nice feature of the book is that there are three basic examples (case studies) which are taken up throughout the text whenever they may serve to illustrate the theory developed and make things concrete and applicable.
The book is divided in a natural way into three major parts. Part I on “One-dimensional observations” presents the classical topics on limit distributions and domains of attraction, extreme and intermediate order statistics, including a discussion of the extreme value condition, but also the statistical estimation of the extreme value index and testing, extreme quantile and tail estimation (quantiles, tail probabilities, endpoints), and finally some more advanced topics like tail empirical process theory, convergence of moments, speed of convergence, large deviations, weak and strong limit laws, etc.
Part II deals with higher -, but “Finite-dimensional observations” including the basic theory (limit laws, exponents and spectral measure, domains of attraction, asymptotic independence), estimation of the dependence structure (spectral measure, a dependence coefficient, tail probability, residual dependence index), and estimation of the probability of a failure set. The latter requires nonparametric techniques since the model parameters are infinite-dimensional in this case.
Finally, Part III sketches the theory for “Observations that are stochastic processes”, first providing the probabilistic theory in \(~C[0,1]~\) (limit distributions, exponent and spectral measure, domain of attraction, spectral representation and stationarity), and being followed by statistical estimation in \(~C[0,1]\), i.e., estimation of the exponent measure, index function, scale and location, and the probabiltiy of a failure set.
For the first sake of easy reference, Part IV (Appendix) contains the key results on weak convergence (Skorohod’s theorem and Vervaat’s lemma) together with the necessary preliminaries on regularly varying functions which are used throughout the text.
A series of examples is provided at the end of each subsection giving the reader an excellent opportunity to deepen her/his understanding and to get familiar with the methods and techniques involved. Finally, an up-to date list of 105 references is included, naturally chosen somewhat from the author’s personal point of view and experience in the field.
The book certainly provides a nice “introduction” into the recent developments of statistical (and probabilistic) extreme value theory. It will serve as a useful reference for all graduate students and researchers in the field.

MSC:

62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G32 Statistics of extreme values; tail inference
62F12 Asymptotic properties of parametric estimators
62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference
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