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© 2011

Laws of Small Numbers: Extremes and Rare Events

Book

Table of contents

  1. Front Matter
    Pages i-xv
  2. The IID Case: Functional Laws of Small Numbers

    1. Front Matter
      Pages 1-1
    2. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 3-23
    3. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 25-101
    4. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 103-131
  3. The IID Case: Multivariate Extremes

    1. Front Matter
      Pages 133-133
    2. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 135-169
    3. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 171-257
    4. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 259-309
    5. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 311-340
  4. Non-IID Observations

    1. Front Matter
      Pages 341-341
    2. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 343-356
    3. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 357-380
    4. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 381-418
    5. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 419-454
    6. Michael Falk, Jürg Hüsler, Rolf-Dieter Reiss
      Pages 455-471
  5. Back Matter
    Pages 473-509

About this book

Introduction

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

Keywords

Gaussian process Maxima Peak probability theory statistics

Authors and affiliations

  1. 1.Lehrstuhl für StatistikInstitut für MathematikWürzburgGermany
  2. 2.Inst. Mathematische Statistik und, VersicherungslehreUniversität BernBernSwitzerland
  3. 3.FB 6 MathematikUniversität SiegenSiegenGermany

Bibliographic information

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)