Laws of Small Numbers: Extremes and Rare Events

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

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)

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)

The mathematical theory and applications of extremes and rare events have developed much since this book was first published in 1994. This third edition includes details of multivariate generalized Pareto distributions and their testing and estimation.

Preface to the Third Edition.- Preface to the Second Edition.- Preface to the First Edition.- Part I. The IID Case: Functional Laws Of Small Numbers.- 1 Functional Laws of Small Numbers.- 2 Extreme Value Theory.- 3 Estimation of Conditional Curves.- Part II. The IID Case: Multivariate Extremes.- 4 Basic Theory of Multivariate Maxima.- 5 Multivariate Generalized Pareto Distributions.- 6 The Pickands Approach in the Bivariate Case.- 7 Multivariate Extremes: Supplementary Concepts and Results.- Part III. Non-IID Observations.- 8 Introduction to the Non-IID Case.- 9 Extremes of Random Sequences.- 10 Extremes of Gaussian Processes.- 11 Extensions for Rare Events.- 12 Statistics of Extremes.- Author Index.- Subject Index.- Bibliography.