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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.
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International Standard Book Number 1-58488-411-8
Library of Congress Card Number 2003051602
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper
Library of Congress Cataloging-in-Publication Data
Séminaire européen de statistique (5th : 2001 : Gothenburg, Sweden)
Extreme values in Þnance, telecommunications, and the environment / edited by Barbel
Finkenstadt, Holger Rootzen.
p. cm. — (Monographs on statistics and applied probability ; 99)
Includes bibliographical references and index.
ISBN 1-58488-411-8 (alk. paper)
1. Extreme value theory—Congresses. I. Finkenstädt, Bärbel. II. Rootzén, Holger. III.
Title. IV. Series.
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
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Xem thêm: Extreme Values in Finance, Telecommunications, and the Environment pptx, Extreme Values in Finance, Telecommunications, and the Environment pptx, 1 Introduction: Classical analyses of annual maximum data, 4 The AR(1) model with ARCH(1) errors, 1 “Stylized facts” about financial data, 6 Classical measures of dependence: Mixing properties and correlations, 10 Do GARCH and stochastic volatility models explain the “stylized facts” of log-returns?, 3 Further implications of the simple model: is network traffic stable, 5 Does the model fit the data? How to detect heavy tails, 6 Does the model fit the data? Long range dependence, self-similarity, Hurst phenomenon, 7 Does the model fit the data? Small time scales: H¨ older exponents and multifractality