Laws of Small Numbers: Extremes and Rare Events / Edition 2

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Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results on about 130 additional pages.

Part II, which has been added in the second edition, discusses recent developments in multivariate extreme value theory. Particularly notable is a new spectral decomposition of multivariate distributions in univariate ones which makes multivariate questions more accessible in theory and practice. One of the most innovative and fruitful topics during the last decades was the introduction of generalized Pareto distributions in the univariate extreme value theory. Such a statistical modelling of extremes is now systematically developed in the multivariate framework.

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Editorial Reviews

From the Publisher
From the reviews of the third edition:
Review Janet Hefferman, Journal of Applied Statistics The authors claim that the book is aimed at graduate students and researchers with basic knowledge of probability theory. This claim seems slightly misleading, the material instead being written at a research level, and I suspect generally impenetrable to readers without some previous specialist knowledge of the area. Being almost exclusively theoretical in nature, the book is likely to be of little interest or indeed practical use to an applied statistician. However the book offers a useful addition to the literature of the probabilistic theory of extremes as it consolidates recent developments in the field. Review Holger Drees, Metrika The book is completed by an extensive bibliography with almost 400 references and the usual indices. Indeed, this is a big improvement over the first edition where the references were given in each section separately. The readability is further improved by a considerable number of new plots to visualize examples. Unfortunately, a distorting technical error has crept in Sect. 1.2: on page 8 the first sentence ends abruptly in the middle and nine lines which should be printed here are instead given on page 14, where they interrupt another sentence in a similar manner. In summary, Laws of Small Numbers can be highly recommended to everyone who is looking for a smooth introduction to Poisson approximations in EVT and other fields of probability theory and statistics. In particular, it offers an interesting view on multivariate EVT and on EVT for non-iid observations, which is not presented in a similar way in any other textbook. Review David Stirzaker, Bulletin of the London Mathematical Society In brief, it is clear that this will surely be a valuable resource for anyone involved in, or seeking to master, the more mathematical features of this field.
“Chapters outline the mathematical development of the basic ideas and their extensions and applications. … The material is well presented, with clear explanations and illustrations. For a graduate student with a good background in probability theory, I believe that this book can provide a strong foundation for research into a fascinating area.” (Martin Crowder, International Statistical Review, Vol. 79 (3), 2011)
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Product Details

  • ISBN-13: 9783764324162
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 9/11/2008
  • Edition description: REV
  • Edition number: 2
  • Pages: 396
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.81 (d)

Table of Contents

Pt. I The IID case : functional laws of small numbers 1
1 Functional laws of small numbers 3
2 Extreme value theory 23
3 Estimation of conditional curves 75
Pt. II The IID case : multivariate extremes 105
4 Basic theory of multivariate maxima 107
5 Multivariate extremes : the Pickands approach 131
6 The Pickands approach in the bivariate case 161
7 Multivariate extremes : supplementary concepts and results 203
Pt. III Non IID observations 233
8 Introduction to the non IID case 235
9 Extremes of random sequences 249
10 Extremes of gaussian processes 273
11 Extensions for rare events 297
12 Statistics of extremes 329
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