Extreme Values in Finance, Telecommunications, and the Environment Edited by Bärbel Finkenstädt and Holger Rootzén CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microÞlming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. 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QA273.6.S45 2001 519.2 ¢ 4—dc21 2003051602 C4118 disclaime new Page 1 Monday, June 30, 2003 2:27 PM Contents Contributors Participants Preface 1 StatisticsofExtremes,withApplicationsinEnvironment, Insurance, and Finance by Richard L. Smith 2 The Use and Misuse of Extreme Value Models in Practice by Stuart Coles 3 Risk Management with Extreme Value Theory by Claudia Kl ¨ uppelberg 4 Extremes in Economics and the Economics of Extremes by Paul Embrechts 5 Modeling Dependence and Tails of Financial Time Series by Thomas Mikosch 6 Modeling Data Networks by Sidney Resnick 7 Multivariate Extremes by Anne-Laure Foug ` eres ©2004 CRC Press LLC Contributors Stuart Coles Department of Mathematics University of Bristol Bristol, United Kingdom Stuart.Coles@bristol.ac.uk, http://www.stats.bris.ac.uk/masgc/ Paul Embrechts Department of Mathematics Eidgen¨ossische Technische Hochschule Z¨urich Z¨urich, Switzerland embrechts@math.ethz.ch, http://www.math.ethz.ch/∼embrechts Anne-Laure Foug`eres Laboratoire de Statistique et Probabilit´es Institut National des Sciences Appliqu´ees de Toulouse — Universit´ePaul Sabatier D´ept. GMM, INSA Toulouse, France fougeres@gmm.insa-tlse.fr, http://www.gmm.insa-tlse.fr/fougeres/englishal.html Claudia Kl¨uppelberg Center of Mathematical Sciences Munich University of Technology Munich, Germany cklu@mathematik.tu-muenchen.de, http://www.ma.tum.de/stat/ Thomas Mikosch Laboratory of Actuarial Mathematics University of Copenhagen Copenhagen, Denmark mikosch@math.ku.dk, http://www.math.ku.dk/∼mikosch ©2004 CRC Press LLC Sidney Resnick School of Operations Research and Industrial Engineering Cornell University Ithaca, New York sid@orie.cornell.edu, http://www.orie.cornell.edu/∼sid Richard L. Smith Department of Statistics University of North Carolina Chapel Hill, North Carolina rls@email.unc.edu, http://www.unc.edu/depts/statistics/faculty/rsmith.html ©2004 CRC Press LLC Participants Andriy Adreev, Helsinki (Finland), andriy.andreev@shh.fi A note on histogram approximation in Bayesian density estimation. Jenny Andersson, Gothenburg (Sweden), jennya@math.chalmers.se Analysis of corrosion on aluminium and magnesium by statistics of extremes. Bojan Basrak, Eindhoven (Netherlands), basrak@eurandom.tue.nl On multivariate regular variation and some time-series models. Nathana¨el Benjamin, Oxford (United Kingdom),nathanael.benjamin@centraliens.net Bound on an approximation for the distribution of the extreme fluctuations of exchange rates. Paola Bortot, Bologna (Italy), bortot@stat.unibo.it Extremes of volatile Markov chains. Natalia Botchkina, Bristol (United Kingdom), natasha.botchkina@mail.com Wavelets and extreme value theory. Leonardo Bottolo, Pavia (Italy), lbottolo@eco.unipv.it Mixture models in Bayesian risk analysis. Boris Buchmann, Munich (Germany), bbuch@mathematik.tu-muenchen.de Decompounding: an estimation problem for the compound Poisson distribution. Adam Butler, Lancaster (United Kingdom), a.butler@lancaster.ac.uk The impact of climate change upon extreme sea levels. Ana Cebrian, Louvain-La-Neuve (Belgium), cebrian@stat.ucl.ac.be Analysis of bivariate extreme dependence using copulas with applications to insurance. Ana Ferreira, Eindhoven (Netherlands), ferreira@eurandom.tue.nl Confidence intervals for the tail index. Christopher Ferro, Lancaster (United Kingdom), c.ferro@lancaster.ac.uk Aspects of modelling extremal temporal dependence. John Greenhough, Warwick (United Kingdom), greenh@astro.warwick.ac.uk Characterizing anomalous transport in accretion disks from X-ray observations. Viviane Grunert da Fonseca, Faro (Portugal), vgrunert@ualg.pt Stochastic multiobjective optimization and the attainment function. Janet Heffernan, Lancaster (United Kingdom), j.heffernan@lancaster.ac.uk A conditional approach for multivariate extreme values. ©2004 CRC Press LLC Rachel Hilliam, Birmingham (United Kingdom), rmh@for.mat.bham.ac.uk Statistical aspects of chaos-based communications modelling. Daniel Hlubinka, Prague (Czech Republic), hlubinka@karlin.mff.cuni.cz Stereology of extremes: shape factor. Marian Hristache, Bruz (France), marian.hristache@ensai.fr Structure adaptive approach for dimension reduction. P¨ar Johannesson, Gothenburg (Sweden), par.johannesson@fcc.chalmers.se Crossings of intervals in fatigue of materials. Joachim Johansson, Gothenburg (Sweden), joachimj@math.chalmers.se A semi-parametric estimator of the mean of heavy-tailed distributions. Elisabeth Joossens, Leuven (Belgium), bettie.joossens@wis.kuleuven.ac.be On the estimation of the largest inclusions in a piece of steel using extreme value analysis. Vadim Kuzmin, St. Petersburg (Russia), kuzmin@rw.ru Stochastic forecasting of extreme flood transformation. Fabrizio Laurini, Padova (Italy), flaurini@stat.unipd.it Estimating the extremal index in financial time series. Tao Lin, Rotterdam (Netherlands), lin@few.eur.nl Statistics of extremes in C[0,1]. Alexander Lindner, Munich (Germany), lindner@ma.tum.de Angles and linear reconstruction of missing data. Owen Lyne, Nottingham (United Kingdom), owen.lyne@nottingham.ac.uk Statistical inference for multitype households SIR epidemics. Hans Malmsten, Stockholm (Sweden), hans.malmsten@hhs.se Moment structure of a family of first-order exponential GARCH models. Alex Morton, Warwick (United Kingdom), a.morton@warwick.ac.uk Anew class of models for irregularly sampled time series. Natalie Neumeyer, Bochum (Germany), natalie.neumeyer@ruhr-uni-bochum.de Nonparametric comparison of regression functions—an empirical process approach. Paul Northrop, Oxford (United Kingdom), northrop@stats.ox.ac.uk An empirical Bayes approach to flood estimation. Gr´egory Nuel, Evry (France), gnuel@maths.univ-evry.fr Unusual word frequencies in Markov chains: the large deviations approach. Fehmi ¨ Ozkan, Freiburg im Breisgau (Germany), oezkan@fdm.uni-freiburg.de The defaultable L´evy term structure: ratings and restructuring. Francesco Pauli, Trieste (Italy), francescopauli@interfree.it A multivariate model for extremes. Olivier Perrin, Toulouse (France), perrin@cict.fr On a time deformation reducing stochastic processes to local stationarity. ©2004 CRC Press LLC Martin Schlather, Bayreuth (Germany), martin.schlather@uni-bayreuth.de A dependence measure for extreme values. Manuel Scotto, Algueir˜ao (Portugal), arima@mail.telepac.pt Extremal behaviour of certain transformations of time series. Scott Sisson, Bristol (United Kingdom), scott.sisson@bristol.ac.uk An application involving uncertain asymptotic temporal dependence in the extremes of time series. Alwin Stegeman, Groningen (Netherlands), stegeman@math.rug.nl Long-range dependence in computer network traffic: theory and practice. Scherbakov Vadim, Glasgow (United Kingdom), vadim@stats.gla.ac.uk Voter model with mean-field interaction. Yingcun Xia, Cambridge (United Kingdom), ycxia@zoo.cam.ac.uk A childhood epidemic model with birthrate-dependent transmission. ©2004 CRC Press LLC Preface The chapters in this volume are the invited papers presented at the fifth S´eminaire Europ´een de Statistique (SemStat) on extreme value theory and applications, held un- der the auspices of Chalmers and Gothenburg University at the Nordic Folk Academy in Gothenburg, 10–16 December, 2001. The volume is thus the most recent in a sequence of conference volumes that have appeared as a result of each S´eminaire Europ´een de Statistique. The first of these workshops took place in 1992 at Sandbjerg Manor in the southern part of Denmark. The topic was statistical aspects of chaos and neural networks. A second meeting on time series models in econometrics, finance, and other fields was held in Oxford in December, 1994. The third meeting on stochastic geometry: likelihood and com- putation took place in Toulouse, 1996, and a fourth meeting, on complex stochastic systems, was held at EURANDOM, Eindhoven, 1999. Since August, 1996, SemStat has been under the auspices of the European Regional Committee of the Bernoulli Society for Mathematical Statistics and Probability. The aim of the S´eminaire Europ´een de Statistique is to provide young scientists with an opportunity to get quickly to the forefront of knowledge and research in areas of statistical science which are of current major interest. About 40 young re- searchers from various European countries participated in the 2001 s´eminaire. Each of them presented his or her work either by giving a seminar talk or contributing to a poster session. A list of the invited contributors and the young attendants of the s´eminaire, along with the titles of their presentations, can be found on the preceding pages. The central paradigm of extreme value theory is semiparametric: you cannot trust standard statistical modeling by normal, lognormal, Weibull, or other distributions all the way out into extreme tails and maxima. On the other hand, nonparametric methods cannot be used either, because interest centers on more extreme events than those one already has encountered. The solution to this dilemma is semiparametric models which only specify the distributional shapes of maxima, as the extreme value distributions, or of extreme tails, as the generalized Pareto distributions. The rationales for these models are very basic limit and stability arguments. The first chapter, written by Richard Smith, gives a survey of how this paradigm answers a variety of questions of interest to an applied scientist in climatology, in- surance, and finance. The chapter also reviews parts of univariate extreme value theory and discusses estimation, diagnostics, multivariate extremes, and max-stable processes. ©2004 CRC Press LLC In the second chapter, Stuart Coles focuses on the particularly extreme event of the 1999 rainfall in Venezuela that caused widespread distruction and loss of life. He demonstrates that the probability for such an event would have been miscalculated even by the standard extreme value models, and discusses the use of various options available for extension in order to achieve a more satisfactory analysis. The next three chapters consider applications of extreme value theory to risk man- agement in finance and economics. First, in Chapter 3, Claudia Kl¨uppelberg reviews aspects of Value-at-Risk (VaR) and its estimation based on extreme value theory. She presents results of a comprehensive investigation of the extremal behavior of some of the most important continuous and discrete time series models that are of current interest in finance. Her discussions are followed by an historic overview of financial risk management given by Paul Embrechts in Chapter 4. In Chapter 5, Thomas Mikosch introduces the stylized facts of financial time series, in par- ticular the heavy tails exhibited by log-returns. He studies, in depth, their connec- tion with standard econometric models such as the GARCH and stochastic volatil- ity processes. The reader is also introduced to the mathematical concept of regular variation. Another important area where extreme value theory plays a significant role is data network modelling. In Chapter 6, Sidney Resnick reviews issues to consider for data network modelling, some of the basic models and statistical techniques for fitting these models. Finally, in Chapter 7, Anne-Laure Foug`eres gives an overview of multivariate extreme value distributions and the problem of measuring extremal dependence. The order in which the chapters are compiled approximately follows the order in which they were presented at the conference. Naturally it is not possible to cover all aspects of this interesting and exciting research area in a single conference volume. The most important omission may be the extensive use of extreme value theory in reliability theory. This includes modelling of extreme wind and wave loads on structures, of strength of materials, and of metal corrosion and fatigue. In addition to methods discussed in this volume, these areas use the deep and interesting theory of extremes of Gaussian processes. Nevertheless it is our hope that the coverage provided by this volume will help the readers to acquaint themselves speedily with current research issues and techniques in extreme value theory. The scientific programme of the fifth S´eminaire Europ´een de Statistique was orga- nized by the steering group, which, at the time of the conference, consisted of O.E. Barndorff-Nielsen (Aarhus University), B. Finkenst¨adt (University of Warwick), W.S. Kendall (University of Warwick), C. Kl¨uppelberg (Munich University of Technol- ogy), D. Picard (Paris VII), H. Rootz´en (Chalmers University Gothenburg), and A. van der Vaart (Free University Amsterdam). The local organization of the s´eminaire wasinthe hands of H. Rootz´en and the smooth running was to a large part due to Johan Segers (Tilburg University), Jenny Andersson, and Jacques de Mar´e (both at Chalmers University Gothenburg). The fifth S´eminaire Europ´een de Statistique was supported by the TMR-network in statistical and computational methods for the analysis of spatial data, the Stochastic Centre in Gothenburg, the Swedish Institute of Applied Mathematics, the Swedish ©2004 CRC Press LLC [...]... Describing dependence among the extremes of different series, and using this description in the problem of managing a portfolio of investments; and 3 Modeling extremes in the presence of volatility — like all financial time series, those in Figure 1.2 show periods when the variability or volatility in the series is high, and others where it is much lower, but simple theories of extreme values in independent... of the prior distribution, nor of the MCMC method of computation, though the precise numerical result is sensitive to these The main reason for the contrasting results lies in the change of emphasis from estimating a parameter of the model — for which the information in the data is rather diffuse, resulting in wide confidence intervals — to predicting a specific quantity, for which much more precise information... outlier In Figure 1.10(b), the same points are plotted (including the final data point), but for the purpose of estimating the parameters, the final observation was omitted In this case, the plot seems to stick very closely to the straight line, except for the final data point, which is an even more obvious outlier than in Figure 1.10(a) Taken together, the two plots show that the largest data point is... corresponding plot where time is the total cumulative number of days since the start of the series in 1934 Plot (a) is a visual diagnostic for seasonality in the series and shows, not surprisingly, that there are very few exceedances during the summer months Plot (b) may be used as a diagnostic for overall trends in the series; in this case, there are three large values at the right-hand end of the series... straight line over most of its range is an indicator that the GPD fits the data reasonably well Of the other plots in Figure 1.1, plot (b) shows no visual evidence of a trend in the frequency of claims, while in (c), there is a sharp rise in the cumulative total of claims during year 7, but this arises largely because the two largest claims in the whole series were both in the same year, which raises the. .. through the origin Figure 1.10 illustrates this idea for the Ivigtut data of Figure 1.9(b) In Figure 1.10(a), the GEV distribution is fitted by maximum likelihood to the whole data set, and a QQ plot is drawn The shape of the plot — with several points below the straight line at the right-hand end of the plot, except for the final data point which is well above — supports the treatment of the final data point... the question of whether these two claims should be treated as outliers, and therefore analyzed separately from the rest of the data The case for doing this is strengthened by the fact that these were the only two claims in the entire data set that resulted from the total loss of a facility We shall return to these issues when the data are analysed in detail in Section 1.6, but for the moment, we list... (1987) contain much information on point-process viewpoints of extreme value theory In this approach, instead of considering the times at which high-threshold exceedances occur and the excess values over the threshold as two separate processes, they are combined into one process based on a two-dimensional plot of exceedance times and exceedance values The asymptotic theory of threshold exceedances shows... ends of the spectrum The substantial overlap between these two posterior densities underlines the inherent variability of the procedure In summary, the main messages of this example are: 1 The point estimates (maximum likelihood or Bayes) are quite sensitive to the chosen threshold, and in the absence of a generally agreed criterion for choosing the threshold, this is an admitted difficulty of the approach... dollars and adjusted for in ation to 1994 cost equivalents As a preliminary to the detailed analysis, two further preprocessing steps were performed: (i) the data were multiplied by a common but unspecified scaling factor — this has the effect of concealing the precise sums of money involved, without in any other way changing the characteristics of the data set, and (ii) simultaneous claims of the same . Extreme Values in Finance, Telecommunications, and the Environment Edited by Bärbel Finkenstädt and Holger Rootzén CHAPMAN &. Describing dependence among the extremes of different series, and using this description in the problem of managing a portfolio of investments; and 3. Modeling