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 develop
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ws of Small Numbers: Extremes and Rare Events Third, revised and extended edition
Michael Falk Institute of Mathematics University of W¨urzburg Am Hubland 97074 W¨urzburg Germany e-mail: [email protected]
J¨urg H¨usler Department of Mathematical Statistics and Actuarial Science University of Berne Sidlerstrasse 5 3012 Bern Switzerland e-mail: [email protected]
Rolf-Dieter Reiss Department of Mathematics University of Siegen Walter Flex Str. 3 57068 Siegen Germany e-mail: [email protected]
ISBN 978-3-0348-0008-2 e-ISBN 978-3-0348-0009-9 DOI 10.1007/978-3-0348-0009-9 Library of Congress Control Number: 2010934383 Mathematics Subject Classification (2010): 60-02, 60G70, 62-02, 62G32, 60G55, 60G15, 60G10, 62H05, 62P99 c Birkh¨auser Verlag Basel – Boston – Berlin 1994, 2004 1st and 2nd edition: c Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
Preface to the Third Edition The main focus of extreme value theory has been undergoing a dramatic change. Instead of concentrating on maxima of observations, large observations are now in the focus, defined as exceedances over high thresholds. Since the pioneering papers by Balkema and de Haan (1974) and Pickands (1975) it is well known that exceedances over high thresholds can reasonably be modeled only by a generalized Pareto distribution. But only in recent years has this fact been widely spread outside the academic world as well. Just as multivariate extreme value theory was developed roughly thirty years after its univariate basis was established, we presently see the theory of multivariate exceedances and, thus, the theory of multivariate generalized Pareto distributions under extensive investigation. For that reason, one emphasis of the third edition of the present book is given to multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation. Concerning this matter, the third edition in particular benefits from the recent PhD-theses of Ren´e Michel and Daniel Hofmann, who both made substantial contributions to the theory of multivariate generalized Pareto distributions, mainly concentrated in Section 4.4, Chapter 5 and 6. We are in particular grateful to Ren´e Michel, who coauthored these parts of the present edition with high diligence. Exceedances of stochastic processes and random fields have been further considered in recent years, since the publication of the second edition. These new developments are discussed in additional sections or paragraphs. For i
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