Statistics of Extremes Theory and Applications,9780471976479

Statistics of Extremes Theory and Applications

by ; ; ; ; ;
Edition: 1st
ISBN13: 9780471976479
ISBN10: 0471976474
Format: Hardcover
Pub. Date: 2004-10-15
Publisher(s): WILEY
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Summary

Research in the statistical analysis of extreme values has flourished over the past decade: new probability models, inference and data analysis techniques have been introduced; and new application areas have been explored. Statistics of Extremes comprehensively covers a wide range of models and application areas, including risk and insurance: a major area of interest and relevance to extreme value theory. Case studies are introduced providing a good balance of theory and application of each model discussed, incorporating many illustrated examples and plots of data. The last part of the book covers some interesting advanced topics, including time series, regression, multivariate and Bayesian modelling of extremes, the use of which has huge potential.

Author Biography

Jan Beirlant is the author of Statistics of Extremes: Theory and Applications, published by Wiley.

Yuri Goegebeur is the author of Statistics of Extremes: Theory and Applications, published by Wiley.

Johan Segers is the author of Statistics of Extremes: Theory and Applications, published by Wiley.

Jozef L. Teugels is the author of Statistics of Extremes: Theory and Applications, published by Wiley.

Table of Contents

Preface xi
1 WHY EXTREME VALUE THEORY?
1(44)
1.1 A Simple Extreme Value Problem
1(2)
1.2 Graphical Tools for Data Analysis
3(16)
1.2.1 Quantile-quantile plots
3(11)
1.2.2 Excess plots
14(5)
1.3 Domains of Applications
19(23)
1.3.1 Hydrology
19(2)
1.3.2 Environmental research and meteorology
21(3)
1.3.3 Insurance applications
24(7)
1.3.4 Finance applications
31(1)
1.3.5 Geology and seismic analysis
32(8)
1.3.6 Metallurgy
40(2)
1.3.7 Miscellaneous applications
42(1)
1.4 Conclusion
42(3)
2 THE PROBABILISTIC SIDE OF EXTREME VALUE THEORY
45(38)
2.1 The Possible Limits
46(5)
2.2 An Example
51(5)
2.3 The Frechet-Pareto Case: γ > 0
56(9)
2.3.1 The domain of attraction condition
56(1)
2.3.2 Condition on the underlying distribution
57(1)
2.3.3 The historical approach
58(1)
2.3.4 Examples
58(3)
2.3.5 Fitting data from a Pareto-type distribution
61(4)
2.4 The (Extremal) Weibull Case: γ less than 0 65
2.4.1 The domain of attraction condition
65(2)
2.4.2 Condition on the underlying distribution
67(1)
2.4.3 The historical approach
67(1)
2.4.4 Examples
67(2)
2.5 The Gumbel Case: γ = 0
69(4)
2.5.1 The domain of attraction condition
69(3)
2.5.2 Condition on the underlying distribution
72(1)
2.5.3 The historical approach and examples
72(1)
2.6 Alternative Conditions for (Cγ)
73(2)
2.7 Further on the Historical Approach
75(1)
2.8 Summary
76(1)
2.9 Background Information
76(7)
2.9.1 Inverse of a distribution
77(1)
2.9.2 Functions of regular variation
77(2)
2.9.3 Relation between F and U
79(1)
2.9.4 Proofs for section 2.6
80(3)
3 AWAY FROM THE MAXIMUM
83(16)
3.1 Introduction
83(1)
3.2 Order Statistics Close to the Maximum
84(6)
3.3 Second-order Theory
90(4)
3.3.1 Remainder in terms of U
90(2)
3.3.2 Examples
92(1)
3.3.3 Remainder in terms of F
93(1)
3.4 Mathematical Derivations
94(6)
3.4.1 Proof of (3.6)
95(1)
3.4.2 Proof of (3.8)
96(1)
3.4.3 Solution of (3.15)
97(1)
3.4.4 Solution of (3.18)
98(1)
4 TAIL ESTIMATION UNDER PARETO-TYPE MODELS
99(32)
4.1 A Naive Approach
100(1)
4.2 The Hill Estimator
101(6)
4.2.1 Construction
101(3)
4.2.2 Properties
104(3)
4.3 Other Regression Estimators
107(2)
4.4 A Representation for Log-spacings and Asymptotic Results
109(4)
4.5 Reducing the Bias
113(6)
4.5.1 The quantile view
113(4)
4.5.2 The probability view
117(2)
4.6 Extreme Quantiles and Small Exceedance Probabilities
119(4)
4.6.1 First-order estimation of quantiles and return periods
119(2)
4.6.2 Second-order refinements
121(2)
4.7 Adaptive Selection of the Tail Sample Fraction
123(8)
5 TAIL ESTIMATION FOR ALL DOMAINS OF ATTRACTION
131(46)
5.1 The Method of Block Maxima
132(8)
5.1.1 The basic model
132(1)
5.1.2 Parameter estimation
132(3)
5.1.3 Estimation of extreme quantiles
135(2)
5.1.4 Inference: confidence intervals
137(3)
5.2 Quantile View-Methods Based on (Cγ)
140(7)
5.2.1 Pickands estimator
140(2)
5.2.2 The moment estimator
142(1)
5.2.3 Estimators based on the generalized quantile plot
143(4)
5.3 Tail Probability View-Peaks-Over-Threshold Method
147(8)
5.3.1 The basic model
147(2)
5.3.2 Parameter estimation
149(6)
5.4 Estimators Based on an Exponential Regression Model
155(1)
5.5 Extreme Tail Probability, Large Quantile and Endpoint Estimation Using Threshold Methods
156(4)
5.5.1 The quantile view
156(2)
5.5.2 The probability view
158(1)
5.5.3 Inference: confidence intervals
159(1)
5.6 Asymptotic Results Under (Cγ)-(C*γ)
160(5)
5.7 Reducing the Bias
165(2)
5.7.1 The quantile view
165(2)
5.7.2 Extreme quantiles and small exceedance probabilities
167(1)
5.8 Adaptive Selection of the Tail Sample Fraction
167(2)
5.9 Appendices
169(8)
5.9.1 Information matrix for the GEV
169(1)
5.9.2 Point processes
169(3)
5.9.3 GRV2 functions with ρ less than 0 171
5.9.4 Asymptotic mean squared errors
172(1)
5.9.5 AMSE optimal kappa-values
173(4)
6 CASE STUDIES
177(32)
6.1 The Condroz Data
177(11)
6.2 The Secura Belgian Re Data
188(12)
6.2.1 The non-parametric approach
189(2)
6.2.2 Pareto-type modelling
191(4)
6.2.3 Alternative extreme value methods
195(3)
6.2.4 Mixture modelling of claim sizes
198(2)
6.3 Earthquake Data
200(9)
7 REGRESSION ANALYSIS
209(42)
7.1 Introduction
210(1)
7.2 The Method of Block Maxima
211(7)
7.2.1 Model description
211(1)
7.2.2 Maximum likelihood estimation
212(1)
7.2.3 Goodness-of-fit
213(3)
7.2.4 Estimation of extreme conditional quantiles
216(2)
7.3 The Quantile View-Methods Based on Exponential Regression Models
218(7)
7.3.1 Model description
218(1)
7.3.2 Maximum likelihood estimation
219(3)
7.3.3 Goodness-of-fit
222(1)
7.3.4 Estimation of extreme conditional quantiles
223(2)
7.4 The Tail Probability View-Peaks Over Threshold (POT) Method
225(8)
7.4.1 Model description
225(1)
7.4.2 Maximum likelihood estimation
226(3)
7.4.3 Goodness-of-fit
229(2)
7.4.4 Estimation of extreme conditional quantiles
231(2)
7.5 Non-parametric Estimation
233(8)
7.5.1 Maximum penalized likelihood estimation
234(4)
7.5.2 Local polynomial maximum likelihood estimation
238(3)
7.6 Case Study
241(10)
8 MULTIVARIATE EXTREME VALUE THEORY
251(46)
8.1 Introduction
251(3)
8.2 Multivariate Extreme Value Distributions
254(21)
8.2.1 Max-stability and max-infinite divisibility
254(1)
8.2.2 Exponent measure
255(3)
8.2.3 Spectral measure
258(7)
8.2.4 Properties of max-stable distributions
265(2)
8.2.5 Bivariate case
267(4)
8.2.6 Other choices for the margins
271(2)
8.2.7 Summary measures for extremal dependence
273(2)
8.3 The Domain of Attraction
275(12)
8.3.1 General conditions
276(5)
8.3.2 Convergence of the dependence structure
281(6)
8.4 Additional Topics
287(3)
8.5 Summary
290(2)
8.6 Appendix
292(5)
8.6.1 Computing spectral densities
292(1)
8.6.2 Representations of extreme value distributions
293(4)
9 STATISTICS OF MULTIVARIATE EXTREMES
297(72)
9.1 Introduction
297(3)
9.2 Parametric Models
300(13)
9.2.1 Model construction methods
300(4)
9.2.2 Some parametric models
304(9)
9.3 Component-wise Maxima
313(12)
9.3.1 Non-parametric estimation
314(4)
9.3.2 Parametric estimation
318(3)
9.3.3 Data example
321(4)
9.4 Excesses over a Threshold
325(17)
9.4.1 Non-parametric estimation
326(7)
9.4.2 Parametric estimation
333(5)
9.4.3 Data example
338(4)
9.5 Asymptotic Independence
342(23)
9.5.1 Coefficients of extremal dependence
343(7)
9.5.2 Estimating the coefficient of tail dependence
350(4)
9.5.3 Joint tail modelling
354(11)
9.6 Additional Topics
365(1)
9.7 Summary
366(3)
10 EXTREMES OF STATIONARY TIME SERIES 369(60)
10.1 Introduction
369(2)
10.2 The Sample Maximum
371(1)
10.2.1 The extremal limit theorem
371(1)
10.2.2 Data example
375(1)
10.2.3 The extremal index
376(6)
10.3 Point-Process Models
382(1)
10.3.1 Clusters of extreme values
382(1)
10.3.2 Cluster statistics
386(1)
10.3.3 Excesses over threshold
387(1)
10.3.4 Statistical applications
389(1)
10.3.5 Data example
395(1)
10.3.6 Additional topics
399(2)
10.4 Markov-Chain Models
401(1)
10.4.1 The tail chain
401(1)
10.4.2 Extremal index
405(1)
10.4.3 Cluster statistics
406(1)
10.4.4 Statistical applications
407(1)
10.4.5 Fitting the Markov chain
408(1)
10.4.6 Additional topics
411(1)
10.4.7 Data example
413(6)
10.5 Multivariate Stationary Processes
419(1)
10.5.1 The extremal limit theorem
419(1)
10.5.2 The multivariate extremal index
421(1)
10.5.3 Further reading
424(1)
10.6 Additional Topics
425(4)
11 BAYESIAN METHODOLOGY IN EXTREME VALUE STATISTICS 429(32)
11.1 Introduction
429(1)
11.2 The Bayes Approach
430(1)
11.3 Prior Elicitation
431(2)
11.4 Bayesian Computation
433(1)
11.5 Univariate Inference
434(1)
11.5.1 Inference based on block maxima
434(1)
11.5.2 Inference for Fréchet-Pareto-type models
435(1)
11.5.3 Inference for all domains of attractions
445(7)
11.6 An Environmental Application
452(9)
Bibliography 461(18)
Author Index 479(6)
Subject Index 485

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