Statistics of Extremes: Theory and Applications

504 Pages·2004·8.5 MB·English

by  Jan Beirlant, Yuri Goegebeur, Johan Segers, Jozef Teugels, Daniel De Waal, Chris Ferro

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:
Daniel De Waal
Department of Mathematical Statistics, University of the Free State,
South Africa
Chris Ferro
Department of Meteorology, The University of Reading, UK
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Contents
Preface xi
1 WHY EXTREME VALUE THEORY? 1
1.1 A Simple Extreme Value Problem . . . . . . . . . . . . . . . . . . 1
1.2 Graphical Tools for Data Analysis . . . . . . . . . . . . . . . . . . 3
1.2.1 Quantile-quantile plots . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Excess plots . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Domains of Applications . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.1 Hydrology . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 Environmental research and meteorology . . . . . . . . . . 21
1.3.3 Insurance applications . . . . . . . . . . . . . . . . . . . . 24
1.3.4 Finance applications . . . . . . . . . . . . . . . . . . . . . 31
1.3.5 Geology and seismic analysis . . . . . . . . . . . . . . . . 32
1.3.6 Metallurgy . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.3.7 Miscellaneous applications . . . . . . . . . . . . . . . . . . 42
1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2 THE PROBABILISTIC SIDE OF EXTREME
VALUE THEORY 45
2.1 The Possible Limits. . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3 The Fre´chet-Pareto Case: γ >0 . . . . . . . . . . . . . . . . . . . 56
2.3.1 The domain of attraction condition . . . . . . . . . . . . . 56
2.3.2 Condition on the underlying distribution . . . . . . . . . . 57
2.3.3 The historical approach . . . . . . . . . . . . . . . . . . . 58
2.3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.3.5 Fitting data from a Pareto-type distribution . . . . . . . . . 61
2.4 The (Extremal) Weibull Case: γ <0 . . . . . . . . . . . . . . . . 65
2.4.1 The domain of attraction condition . . . . . . . . . . . . . 65
2.4.2 Condition on the underlying distribution . . . . . . . . . . 67
2.4.3 The historical approach . . . . . . . . . . . . . . . . . . . 67
2.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
v
vi CONTENTS
2.5 The Gumbel Case: γ =0 . . . . . . . . . . . . . . . . . . . . . . 69
2.5.1 The domain of attraction condition . . . . . . . . . . . . . 69
2.5.2 Condition on the underlying distribution . . . . . . . . . . 72
2.5.3 The historical approach and examples. . . . . . . . . . . . 72
2.6 Alternative Conditions for (C ) . . . . . . . . . . . . . . . . . . . 73
γ
2.7 Further on the Historical Approach . . . . . . . . . . . . . . . . . 75
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.9 Background Information . . . . . . . . . . . . . . . . . . . . . . . 76
2.9.1 Inverse of a distribution . . . . . . . . . . . . . . . . . . . 77
2.9.2 Functions of regular variation . . . . . . . . . . . . . . . . 77
2.9.3 Relation between F and U . . . . . . . . . . . . . . . . . 79
2.9.4 Proofs for section 2.6 . . . . . . . . . . . . . . . . . . . . 80
3 AWAY FROM THE MAXIMUM 83
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2 Order Statistics Close to the Maximum . . . . . . . . . . . . . . . 84
3.3 Second-order Theory . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.3.1 Remainder in terms of U . . . . . . . . . . . . . . . . . . 90
3.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.3.3 Remainder in terms of F . . . . . . . . . . . . . . . . . . 93
3.4 Mathematical Derivations . . . . . . . . . . . . . . . . . . . . . . 94
3.4.1 Proof of (3.6) . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.4.2 Proof of (3.8) . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.4.3 Solution of (3.15) . . . . . . . . . . . . . . . . . . . . . . 97
3.4.4 Solution of (3.18) . . . . . . . . . . . . . . . . . . . . . . 98
4 TAIL ESTIMATION UNDER PARETO-TYPE MODELS 99
4.1 A Naive Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.2 The Hill Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Other Regression Estimators . . . . . . . . . . . . . . . . . . . . . 107
4.4 A Representation for Log-spacings and Asymptotic Results . . . . 109
4.5 Reducing the Bias . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.1 The quantile view . . . . . . . . . . . . . . . . . . . . . . 113
4.5.2 The probability view . . . . . . . . . . . . . . . . . . . . . 117
4.6 Extreme Quantiles and Small Exceedance Probabilities . . . . . . 119
4.6.1 First-order estimation of quantiles and return periods . . . 119
4.6.2 Second-order refinements . . . . . . . . . . . . . . . . . . 121
4.7 Adaptive Selection of the Tail Sample Fraction . . . . . . . . . . . 123
5 TAIL ESTIMATION FOR ALL DOMAINS OF ATTRACTION 131
5.1 The Method of Block Maxima . . . . . . . . . . . . . . . . . . . . 132
5.1.1 The basic model . . . . . . . . . . . . . . . . . . . . . . . 132
5.1.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . 132
CONTENTS vii
5.1.3 Estimation of extreme quantiles . . . . . . . . . . . . . . . 135
5.1.4 Inference: confidence intervals. . . . . . . . . . . . . . . . 137
5.2 Quantile View—Methods Based on (C ) . . . . . . . . . . . . . . 140
γ
5.2.1 Pickands estimator . . . . . . . . . . . . . . . . . . . . . . 140
5.2.2 The moment estimator . . . . . . . . . . . . . . . . . . . . 142
5.2.3 Estimators based on the generalized quantile plot . . . . . 143
5.3 Tail Probability View—Peaks-Over-Threshold Method . . . . . . . 147
5.3.1 The basic model . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . 149
5.4 Estimators Based on an Exponential Regression Model . . . . . . 155
5.5 Extreme Tail Probability, Large Quantile and Endpoint Estimation
Using Threshold Methods . . . . . . . . . . . . . . . . . . . . . . 156
5.5.1 The quantile view . . . . . . . . . . . . . . . . . . . . . . 156
5.5.2 The probability view . . . . . . . . . . . . . . . . . . . . . 158
5.5.3 Inference: confidence intervals. . . . . . . . . . . . . . . . 159
5.6 Asymptotic Results Under (C )-(C∗) . . . . . . . . . . . . . . . . 160
γ γ
5.7 Reducing the Bias . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.7.1 The quantile view . . . . . . . . . . . . . . . . . . . . . . 165
5.7.2 Extreme quantiles and small exceedance
probabilities . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.8 Adaptive Selection of the Tail Sample Fraction . . . . . . . . . . . 167
5.9 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.9.1 Information matrix for the GEV . . . . . . . . . . . . . . . 169
5.9.2 Point processes . . . . . . . . . . . . . . . . . . . . . . . . 169
5.9.3 GRV functions with ρ <0 . . . . . . . . . . . . . . . . . 171
2
5.9.4 Asymptotic mean squared errors . . . . . . . . . . . . . . 172
5.9.5 AMSE optimal k-values . . . . . . . . . . . . . . . . . . . 173
6 CASE STUDIES 177
6.1 The Condroz Data . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.2 The Secura Belgian Re Data . . . . . . . . . . . . . . . . . . . . . 188
6.2.1 The non-parametric approach . . . . . . . . . . . . . . . . 189
6.2.2 Pareto-type modelling . . . . . . . . . . . . . . . . . . . . 191
6.2.3 Alternative extreme value methods . . . . . . . . . . . . . 195
6.2.4 Mixture modelling of claim sizes . . . . . . . . . . . . . . 198
6.3 Earthquake Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7 REGRESSION ANALYSIS 209
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
7.2 The Method of Block Maxima . . . . . . . . . . . . . . . . . . . . 211
7.2.1 Model description . . . . . . . . . . . . . . . . . . . . . . 211
7.2.2 Maximum likelihood estimation . . . . . . . . . . . . . . . 212
7.2.3 Goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . . 213
7.2.4 Estimation of extreme conditional quantiles . . . . . . . . 216
viii CONTENTS
7.3 The Quantile View—Methods Based on Exponential Regression
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.3.1 Model description . . . . . . . . . . . . . . . . . . . . . . 218
7.3.2 Maximum likelihood estimation . . . . . . . . . . . . . . . 219
7.3.3 Goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . . 222
7.3.4 Estimation of extreme conditional quantiles . . . . . . . . 223
7.4 The Tail Probability View—Peaks Over Threshold (POT)
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
7.4.1 Model description . . . . . . . . . . . . . . . . . . . . . . 225
7.4.2 Maximum likelihood estimation . . . . . . . . . . . . . . . 226
7.4.3 Goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . . 229
7.4.4 Estimation of extreme conditional quantiles . . . . . . . . 231
7.5 Non-parametric Estimation . . . . . . . . . . . . . . . . . . . . . . 233
7.5.1 Maximum penalized likelihood estimation . . . . . . . . . 234
7.5.2 Local polynomial maximum likelihood estimation . . . . . 238
7.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
8 MULTIVARIATE EXTREME VALUE THEORY 251
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.2 Multivariate Extreme Value Distributions . . . . . . . . . . . . . . 254
8.2.1 Max-stability and max-infinite divisibility . . . . . . . . . 254
8.2.2 Exponent measure . . . . . . . . . . . . . . . . . . . . . . 255
8.2.3 Spectral measure . . . . . . . . . . . . . . . . . . . . . . . 258
8.2.4 Properties of max-stable distributions . . . . . . . . . . . . 265
8.2.5 Bivariate case. . . . . . . . . . . . . . . . . . . . . . . . . 267
8.2.6 Other choices for the margins . . . . . . . . . . . . . . . . 271
8.2.7 Summary measures for extremal dependence . . . . . . . . 273
8.3 The Domain of Attraction . . . . . . . . . . . . . . . . . . . . . . 275
8.3.1 General conditions . . . . . . . . . . . . . . . . . . . . . . 276
8.3.2 Convergence of the dependence structure . . . . . . . . . . 281
8.4 Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
8.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.6.1 Computing spectral densities . . . . . . . . . . . . . . . . 292
8.6.2 Representations of extreme value distributions . . . . . . . 293
9 STATISTICS OF MULTIVARIATE EXTREMES 297
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
9.2 Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.2.1 Model construction methods . . . . . . . . . . . . . . . . . 300
9.2.2 Some parametric models . . . . . . . . . . . . . . . . . . . 304
9.3 Component-wise Maxima . . . . . . . . . . . . . . . . . . . . . . 313
9.3.1 Non-parametric estimation . . . . . . . . . . . . . . . . . . 314
9.3.2 Parametric estimation . . . . . . . . . . . . . . . . . . . . 318
9.3.3 Data example . . . . . . . . . . . . . . . . . . . . . . . . . 321
CONTENTS ix
9.4 Excesses over a Threshold . . . . . . . . . . . . . . . . . . . . . . 325
9.4.1 Non-parametric estimation . . . . . . . . . . . . . . . . . . 326
9.4.2 Parametric estimation . . . . . . . . . . . . . . . . . . . . 333
9.4.3 Data example . . . . . . . . . . . . . . . . . . . . . . . . . 338
9.5 Asymptotic Independence . . . . . . . . . . . . . . . . . . . . . . 342
9.5.1 Coefficients of extremal dependence . . . . . . . . . . . . 343
9.5.2 Estimating the coefficient of tail dependence . . . . . . . . 350
9.5.3 Joint tail modelling . . . . . . . . . . . . . . . . . . . . . . 354
9.6 Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
10 EXTREMES OF STATIONARY TIME SERIES 369
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
10.2 The Sample Maximum . . . . . . . . . . . . . . . . . . . . . . . . 371
10.2.1 The extremal limit theorem . . . . . . . . . . . . . . . . . 371
10.2.2 Data example . . . . . . . . . . . . . . . . . . . . . . . . . 375
10.2.3 The extremal index . . . . . . . . . . . . . . . . . . . . . . 376
10.3 Point-Process Models . . . . . . . . . . . . . . . . . . . . . . . . . 382
10.3.1 Clusters of extreme values . . . . . . . . . . . . . . . . . . 382
10.3.2 Cluster statistics . . . . . . . . . . . . . . . . . . . . . . . 386
10.3.3 Excesses over threshold . . . . . . . . . . . . . . . . . . . 387
10.3.4 Statistical applications . . . . . . . . . . . . . . . . . . . . 389
10.3.5 Data example . . . . . . . . . . . . . . . . . . . . . . . . . 395
10.3.6 Additional topics . . . . . . . . . . . . . . . . . . . . . . . 399
10.4 Markov-Chain Models . . . . . . . . . . . . . . . . . . . . . . . . 401
10.4.1 The tail chain . . . . . . . . . . . . . . . . . . . . . . . . . 401
10.4.2 Extremal index . . . . . . . . . . . . . . . . . . . . . . . . 405
10.4.3 Cluster statistics . . . . . . . . . . . . . . . . . . . . . . . 406
10.4.4 Statistical applications . . . . . . . . . . . . . . . . . . . . 407
10.4.5 Fitting the Markov chain. . . . . . . . . . . . . . . . . . . 408
10.4.6 Additional topics . . . . . . . . . . . . . . . . . . . . . . . 411
10.4.7 Data example . . . . . . . . . . . . . . . . . . . . . . . . . 413
10.5 Multivariate Stationary Processes . . . . . . . . . . . . . . . . . . 419
10.5.1 The extremal limit theorem . . . . . . . . . . . . . . . . . 419
10.5.2 The multivariate extremal index . . . . . . . . . . . . . . . 421
10.5.3 Further reading . . . . . . . . . . . . . . . . . . . . . . . . 424
10.6 Additional Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
11 BAYESIAN METHODOLOGY IN EXTREME VALUE
STATISTICS 429
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
11.2 The Bayes Approach . . . . . . . . . . . . . . . . . . . . . . . . . 430
11.3 Prior Elicitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
11.4 Bayesian Computation . . . . . . . . . . . . . . . . . . . . . . . . 433
11.5 Univariate Inference . . . . . . . . . . . . . . . . . . . . . . . . . 434
x CONTENTS
11.5.1 Inference based on block maxima . . . . . . . . . . . . . . 434
11.5.2 Inference for Fre´chet-Pareto-type models . . . . . . . . . . 435
11.5.3 Inference for all domains of attractions . . . . . . . . . . . 445
11.6 An Environmental Application . . . . . . . . . . . . . . . . . . . . 452
Bibliography 461
Author Index 479
Subject Index 485
Preface
The key result obtained by Fisher and Tippett in 1928 on the possible limit laws
of the sample maximum has seemingly created the idea that extreme value theory
was something rather special, very different from classical central limit theory. In
fact, the number of publications dealing with statistical aspects of extremes dated
before1970isatmostadozen.ThebookbyE. J. Gumbel,publishedbyColumbia
UniversityPressin1958,hasforalongtimebeenconsideredasthemainreferential
workforapplicationsofextremevaluetheoryinengineeringsubjects.Acloselook
at this seminal publication shows that in the early stages one tried to approach
extreme value theory via central limit machinery. During the decade following
its appearance, no change occurred in the lack of interest among probabilists and
statisticians who contributed only a very limited number of relevant papers.
From the theoretical point of view, the 1970 doctoral dissertation by L. de
HaanOnRegularVariationanditsApplicationstotheWeakConvergenceofSample
Extremes seems to be the starting point for theoretical developments in extreme
valuetheory.Forthefirsttime,theprobabilisticandstochasticpropertiesofsample
extremes were developed into a coherent and attractive theory, comparable to the
theoryofsumsofrandomvariables.Thestatisticalaspectshadtowaitevenlonger
before they received the necessary attention.
In Chapter 1, we illustrate why and how one should look at extreme values in
a data set. Many of these examples will reappear as illustrations and even as case
studiesinthe sequel.The nextfive chaptersdealwiththeunivariate theoryfor the
caseofindependentandidenticallydistributedrandomvariables.Chapter 2covers
the probabilistic limiting problem for determining the possible limits of sample
extremes together with the connected domain of attraction problem. The extremal
domainofattractionconditionis,however,tooweaktousetofullydevelopuseful
statistical theories of estimation, construction of confidence intervals, bias reduc-
tion, and so on. The need for second order information is illustrated in Chapter 3.
Armed with this information, we attack the tail estimation problem for the Pareto
caseinChapter 4andforthegeneralcaseinChapter 5.Allthemethodsdeveloped
so far are then illustrated by a number of case studies in Chapter 6.
The last five chapters deal with topics that are still in full and vigorous devel-
opment.Wecanonlytrytogiveapicturethatisascompleteaspossibleatthetime
of writing. To broaden the statistical machinery in the univariate case, Chapter 7
treatsavarietyofalternativemethodsunderacommonumbrellaofregression-type
xi
xii PREFACE
methods.Chapters 8and9dealwithmultivariate extremesandrepeatsomeofthe
methodology of previous chapters, in more than one dimension. In the first of
these two chapters, we deal with the probabilistic aspects of multivariate extreme
value theory by including the possible limits and their domains of attraction; the
next chapter is then devoted to the statistical features of this important subject.
Chapter 9givesanalmostself-containedsurveyofextremevaluemethods intime
series analysis, an area where the importance of extremes has already long been
recognized. We finish with a separate and tentative chapter on Bayesian methods,
a topic in need of further and deep study.
We are aware that it is a daring act to write a book with the title Statistics of
Extremes, the same as that of the first main treatise on extremes. What is even
more daring is our attempt to cope with the incredible speed at which statistical
extreme value theory has been exploding. More than half of the references in this
book appeared over the last ten years. However, it is our sincere conviction that
over the last two decades extreme value theory has matured and that it should
become part of any in-depth education in statistics or its applications. We hope
that this slight attempt of ours gets extreme value theory duly recognized.
Here are some of the main features of the book.
1. The probabilistic aspects in the first few chapters are streamlined to quickly
arrive at the key conditions needed to understand the behaviour of sample
extremes.Itwouldhavebeenpossibletowriteamorecompleteandrigorous
text that would automatically be much more mathematical. We felt that, for
practical purposes, we could safely restrict ourselves to the case where the
underlyingrandomvariablesaresufficientlycontinuous.Whilemoregeneral
conditions would be possible, there is little to gain with a more formal
approach.
2. Under this extra condition, the mathematical intricacies of the subject are
usually quite tractable. Wherever possible, we provide insight into why and
how the mathematical operations lead to otherwise peculiar conditions. To
keepasmoothflowinthedevelopment,technicaldetailswithinachapterare
deferredtothelastsectionofthatchapter.However,statementsoftheorems
are always given in their fullest generality.
3. Becauseofthelivelyspeedatwhichextremevaluetheoryhasbeendevelop-
ing, thoroughly different approaches are possible when solving a statistical
problem. To avoid single-handedness,we therefore included alternative pro-
cedures that boast sufficient theoretical and practical underpinning.
4. Beingstrongbelieversingraphicalprocedures,weillustrateconcepts,deriva-
tions and results by graphical tools. It is hard to overestimate the role of the
latter in getting a quick but reliable impression of the kind and quality of
data.
5. Examples and case studies are amply scattered over the manuscript, some
of them reappearing to illustrate how a more advanced technique results