Statistics of Extremes Theory and Applications.pdf

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Statistics of Extremes Theory and Applications Jan Beirlant, Yuri Goegebeur, and Jozef Teugels University Center of Statistics, Katholieke Universiteit Leuven, Belgium Johan Segers Department of Econometrics, Tilburg University, The Netherlands with contributions from: Department of Mathematical Statistics, University of the Free State, Daniel De Waal South Africa Department of Meteorology, The University of Reading, UK Chris Ferro

Copyright 2004 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770620. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Ofﬁces John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Library of Congress Cataloging-in-Publication Data Statistics of extremes : theory and applications / Jan Beirlant . . . [et al.], with contributions from Daniel De Waal, Chris Ferro. p. cm.—(Wiley series in probability and statistics) Includes bibliographical references and index. ISBN 0-471-97647-4 (acid-free paper) 1. Mathematical statistics. 2. Maxima and minima. QA276.S783447 2004 519.5–dc22 I. Beirlant, Jan. II. Series. 2004051046 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0-471-97647-4 Produced from LaTeX ﬁles supplied by the authors and processed by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production.

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Quantile-quantile plots . . . 1.2.2 Excess plots . . . . . . 1.3 Domains of Applications . . . . 1 WHY EXTREME VALUE THEORY? 1.1 A Simple Extreme Value Problem . 1.2 Graphical Tools for Data Analysis . . . . . . . . . . . . . . . . . . . 1.3.1 Hydrology . . . 1.3.2 Environmental research and meteorology . . . . . . 1.3.3 . . 1.3.4 . . 1.3.5 Geology and seismic analysis . . . . . . . 1.3.6 Metallurgy . . . . . . . 1.3.7 Miscellaneous applications . . . . . . . Insurance applications . Finance applications . . 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 THE PROBABILISTIC SIDE OF EXTREME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VALUE THEORY . . 2.1 The Possible Limits . . . 2.2 An Example . . . 2.3 The Fr´echet-Pareto Case: γ > 0 . . . . . . . . . . . . . . 2.3.1 The domain of attraction condition . . . 2.3.2 Condition on the underlying distribution . . . . 2.3.3 The historical approach . . 2.3.4 Examples . . . . . . Fitting data from a Pareto-type distribution . 2.3.5 . . . . 2.4.1 The domain of attraction condition . . . 2.4.2 Condition on the underlying distribution . . . . 2.4.3 The historical approach . . 2.4.4 Examples . . . . . . 2.4 The (Extremal) Weibull Case: γ < 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 1 3 3 14 19 19 21 24 31 32 40 42 42 45 46 51 56 56 57 58 58 61 65 65 67 67 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

vi . . . . . . . . . . . 2.5 The Gumbel Case: γ = 0 . . . . . 2.5.1 The domain of attraction condition . . . 2.5.2 Condition on the underlying distribution . . . . 2.5.3 The historical approach and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Alternative Conditions for (Cγ ) . . 2.7 Further on the Historical Approach . . 2.8 Summary . . . . . . . . . . 2.9 Background Information . . . . Inverse of a distribution . . . . 2.9.1 2.9.2 Functions of regular variation . 2.9.3 Relation between F and U . . . . 2.9.4 Proofs for section 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 AWAY FROM THE MAXIMUM . . Introduction . . . . . . . . . . . . . . . . . . 3.1 3.2 Order Statistics Close to the Maximum . . 3.3 Second-order Theory . . . . . . . . . . . . 3.3.1 Remainder in terms of U . 3.3.2 Examples . . . . 3.3.3 Remainder in terms of F . . . . . . . . . . . 3.4 Mathematical Derivations Proof of (3.6) . . Proof of (3.8) . . Solution of (3.15) Solution of (3.18) . . . . . . . . . . . . . . . . . . 3.4.1 3.4.2 3.4.3 3.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 72 72 73 75 76 76 77 77 79 80 83 83 84 90 90 92 93 94 95 96 97 98 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties . . 4.1 A Naive Approach . . 4.2 The Hill Estimator . . . . 4.2.1 Construction . . 4.2.2 . . 4 TAIL ESTIMATION UNDER PARETO-TYPE MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Other Regression Estimators . . 4.4 A Representation for Log-spacings and Asymptotic Results . . . . 4.5 Reducing the Bias . . . . . . . . . . . . . . . . . . . First-order estimation of quantiles and return periods . . Second-order reﬁnements . . . . . . . . 4.6 Extreme Quantiles and Small Exceedance Probabilities . . 4.7 Adaptive Selection of the Tail Sample Fraction . . . . . 4.5.1 The quantile view . . . 4.5.2 The probability view . . . . . . . . . . . . 4.6.1 4.6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 TAIL ESTIMATION FOR ALL DOMAINS OF ATTRACTION . . . . . . 5.1 The Method of Block Maxima . . . Parameter estimation . . 5.1.1 The basic model 5.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 . . 100 . . 101 . . 101 . . 104 . . 107 . . 109 . . 113 . . 113 . . 117 . . 119 . . 119 . . 121 . . 123 131 . . 132 . . 132 . . 132

CONTENTS vii . . . . . . . . . . . . . . . . . . . . Pickands estimator . 5.1.3 Estimation of extreme quantiles . 5.1.4 . Inference: conﬁdence intervals . 5.2 Quantile View—Methods Based on (Cγ ) . . . . . . . . . . . . 5.2.1 . . 5.2.2 The moment estimator . . . 5.2.3 Estimators based on the generalized quantile plot 5.3 Tail Probability View—Peaks-Over-Threshold Method . . . . . 5.4 Estimators Based on an Exponential Regression Model 5.5 Extreme Tail Probability, Large Quantile and Endpoint Estimation . . Parameter estimation . . 5.3.1 The basic model 5.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 . . 137 . . 140 . . 140 . . 142 . . 143 . . 147 . . 147 . . 149 . . 155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using Threshold Methods . 5.5.1 The quantile view . . . 5.5.2 The probability view . . 5.5.3 . . . . . . 5.6 Asymptotic Results Under (Cγ )-(C∗ γ ) . . 5.7 Reducing the Bias . . . . . . . . Inference: conﬁdence intervals . . . . . . . . . . 5.7.1 The quantile view . . . 5.7.2 Extreme quantiles and small exceedance . . 5.8 Adaptive Selection of the Tail Sample Fraction . . . 5.9 Appendices . . . . . . . . . . . 5.9.1 5.9.2 . . 5.9.3 GRV2 functions with ρ < 0 . . 5.9.4 Asymptotic mean squared errors 5.9.5 AMSE optimal k-values . . . . . Information matrix for the GEV . . Point processes . . . . . . . . . . . . . . probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 . . 156 . . 158 . . 159 . . 160 . . 165 . . 165 . . 167 . . 167 . . 169 . . 169 . . 169 . . 171 . . 172 . . 173 177 . . 177 . . 188 . . 189 . . 191 . . 195 . . 198 . . 200 209 . . 210 . . 211 . . 211 . . 212 . . 213 . . 216 6 CASE STUDIES . . . . . . . . . 6.1 The Condroz Data . . 6.2 The Secura Belgian Re Data . . . . . . . . . . . . 6.2.1 The non-parametric approach . 6.2.2 . . . . 6.2.3 Alternative extreme value methods . 6.2.4 Mixture modelling of claim sizes . . . . Pareto-type modelling . 6.3 Earthquake Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 REGRESSION ANALYSIS . . . . . . . . Introduction . 7.1 . . 7.2 The Method of Block Maxima . . . . . . . . . . . 7.2.1 Model description . . . . 7.2.2 Maximum likelihood estimation . 7.2.3 Goodness-of-ﬁt . . . . 7.2.4 Estimation of extreme conditional quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii CONTENTS 7.3 The Quantile View—Methods Based on Exponential Regression . . . . . . . . . . . Models . . . . . 7.3.1 Model description . . . . 7.3.2 Maximum likelihood estimation . 7.3.3 Goodness-of-ﬁt . . . . 7.3.4 Estimation of extreme conditional quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The Tail Probability View—Peaks Over Threshold (POT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Method . . . . . . . 7.4.1 Model description . . . . . . 7.4.2 Maximum likelihood estimation . . . . . 7.4.3 Goodness-of-ﬁt . . . . 7.4.4 Estimation of extreme conditional quantiles . . . . 7.5.1 Maximum penalized likelihood estimation . . . 7.5.2 Local polynomial maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Case Study . 7.5 Non-parametric Estimation . . . . . . 8 MULTIVARIATE EXTREME VALUE THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . 8.1 . . 8.2 Multivariate Extreme Value Distributions . . . . . . 8.2.1 Max-stability and max-inﬁnite divisibility . . . . . 8.2.2 Exponent measure . . . Spectral measure . . 8.2.3 . . Properties of max-stable distributions . . 8.2.4 . . . . . . 8.2.5 Bivariate case . . . . 8.2.6 Other choices for the margins . . . . . 8.2.7 . . . . . . . . . . . . . . . . Summary measures for extremal dependence . . . . . . . . . . . . . . . . . . . . 8.6.1 Computing spectral densities . . 8.6.2 Representations of extreme value distributions . . . 8.3 The Domain of Attraction . 8.3.1 General conditions . . . 8.3.2 Convergence of the dependence structure . . . . . . . . . . 8.4 Additional Topics . . 8.5 Summary . . . . . . 8.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 STATISTICS OF MULTIVARIATE EXTREMES Introduction . 9.1 . . 9.2 Parametric Models . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Model construction methods . . 9.2.2 . . . . . . . . . . Some parametric models . . . . 9.3.1 Non-parametric estimation . . . 9.3.2 9.3.3 Data example . . . . Parametric estimation . . . 9.3 Component-wise Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 . . 218 . . 219 . . 222 . . 223 . . 225 . . 225 . . 226 . . 229 . . 231 . . 233 . . 234 . . 238 . . 241 251 . . 251 . . 254 . . 254 . . 255 . . 258 . . 265 . . 267 . . 271 . . 273 . . 275 . . 276 . . 281 . . 287 . . 290 . . 292 . . 292 . . 293 297 . . 297 . . 300 . . 300 . . 304 . . 313 . . 314 . . 318 . . 321

CONTENTS . . . . . . . . . . . . 9.4 Excesses over a Threshold . Parametric estimation . . . . . . . 9.4.1 Non-parametric estimation . 9.4.2 . . . . 9.4.3 Data example . . 9.5 Asymptotic Independence . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Coefﬁcients of extremal dependence . . . . 9.5.2 Estimating the coefﬁcient of tail dependence . . . . 9.5.3 . . . . Joint tail modelling . . . . . 9.6 Additional Topics . . 9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Point-Process Models . . 10.1 Introduction . . . 10.2 The Sample Maximum . 10 EXTREMES OF STATIONARY TIME SERIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 The extremal limit theorem . . . . . 10.2.2 Data example . . . . . . . 10.2.3 The extremal index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 The extremal limit theorem . . . 10.5.2 The multivariate extremal index . 10.5.3 Further reading . . . . . . . . . . . 10.3.1 Clusters of extreme values . 10.3.2 Cluster statistics . . 10.3.3 Excesses over threshold . . . . 10.3.4 Statistical applications . . . . . 10.3.5 Data example . . . . 10.3.6 Additional topics . . . . . . . . . . . . 10.4 Markov-Chain Models . . . . . . . 10.4.1 The tail chain . . . . . . . . 10.4.2 Extremal index . . . . . . . 10.4.3 Cluster statistics 10.4.4 Statistical applications . . . 10.4.5 Fitting the Markov chain . . . . 10.4.6 Additional topics . . 10.4.7 Data example . . . . . . . . . . . 10.5 Multivariate Stationary Processes 10.6 Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 BAYESIAN METHODOLOGY IN EXTREME VALUE . . . . STATISTICS 11.1 Introduction . . . 11.2 The Bayes Approach . . 11.3 Prior Elicitation . . . 11.4 Bayesian Computation . 11.5 Univariate Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix . . 325 . . 326 . . 333 . . 338 . . 342 . . 343 . . 350 . . 354 . . 365 . . 366 369 . . 369 . . 371 . . 371 . . 375 . . 376 . . 382 . . 382 . . 386 . . 387 . . 389 . . 395 . . 399 . . 401 . . 401 . . 405 . . 406 . . 407 . . 408 . . 411 . . 413 . . 419 . . 419 . . 421 . . 424 . . 425 429 . . 429 . . 430 . . 431 . . 433 . . 434

x 11.5.1 Inference based on block maxima . . . . 11.5.2 Inference for Fr´echet-Pareto-type models . . 11.5.3 Inference for all domains of attractions . . . . . . . 11.6 An Environmental Application . . . . . . . . . . Bibliography Author Index Subject Index CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . 434 . . 435 . . 445 . . 452 461 479 485

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