Laws of Small Numbers: Extremes and Rare Events

Authors: Falk, Michael, Hüsler, Jürg, Reiss, Rolf-Dieter

  • Third edition of the successful seminar book, with special consideration of the recent change of emphasis in extreme value theory
  • Focus on large observations defined as transcending high thresholds
  • Emphasis on multivariate generalized Pareto distributions
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About this book

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. In this third edition, the dramatic change of focus of extreme value theory has been taken into account: from concentrating on maxima of observations it has shifted to large observations, defined as exceedances over high thresholds. One emphasis of the present third edition lies on multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation. Reviews of the 2nd edition: "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." David Stirzaker, Bulletin of the London Mathematical Society "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." Holger Drees, Metrika

Reviews

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)


Table of contents (12 chapters)

  • Functional Laws of Small Numbers

    Falk, Michael (et al.)

    Pages 3-23

  • Extreme Value Theory

    Falk, Michael (et al.)

    Pages 25-101

  • Estimation of Conditional Curves

    Falk, Michael (et al.)

    Pages 103-131

  • Basic Theory of Multivariate Maxima

    Falk, Michael (et al.)

    Pages 135-169

  • Multivariate Generalized Pareto Distributions

    Falk, Michael (et al.)

    Pages 171-257

Buy this book

eBook $69.99
price for USA (gross)
  • ISBN 978-3-0348-0009-9
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $89.95
price for USA
  • ISBN 978-3-0348-0008-2
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the eBook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Laws of Small Numbers: Extremes and Rare Events
Authors
Copyright
2011
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0348-0009-9
DOI
10.1007/978-3-0348-0009-9
Softcover ISBN
978-3-0348-0008-2
Edition Number
3
Number of Pages
XVI, 509
Additional Information
Originally published as volume 23 in the series: DMV Seminar
Topics