Statistics of Extremes Theory and Applications

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

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ISBN: 9780471976479 | 0471976474
Cover: Hardcover
Copyright: 10/22/2004

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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.

Preface

1 WHY EXTREME VALUE THEORY?

(44)

1.1 A Simple Extreme Value Problem

(2)

1.2 Graphical Tools for Data Analysis

(16)

1.2.1 Quantile-quantile plots

(11)

1.2.2 Excess plots

(5)

1.3 Domains of Applications

(23)

1.3.1 Hydrology

(2)

1.3.2 Environmental research and meteorology

(3)

1.3.3 Insurance applications

(7)

1.3.4 Finance applications

(1)

1.3.5 Geology and seismic analysis

(8)

1.3.6 Metallurgy

(2)

1.3.7 Miscellaneous applications

(1)

1.4 Conclusion

(3)

2 THE PROBABILISTIC SIDE OF EXTREME VALUE THEORY

(38)

2.1 The Possible Limits

(5)

2.2 An Example

(5)

2.3 The Frechet-Pareto Case: γ > 0

(9)

2.3.1 The domain of attraction condition

(1)

2.3.2 Condition on the underlying distribution

(1)

2.3.3 The historical approach

(1)

2.3.4 Examples

(3)

2.3.5 Fitting data from a Pareto-type distribution

(4)

2.4 The (Extremal) Weibull Case: γ less than 0 65

2.4.1 The domain of attraction condition

(2)

2.4.2 Condition on the underlying distribution

(1)

2.4.3 The historical approach

(1)

2.4.4 Examples

(2)

2.5 The Gumbel Case: γ = 0

(4)

2.5.1 The domain of attraction condition

(3)

2.5.2 Condition on the underlying distribution

(1)

2.5.3 The historical approach and examples

(1)

2.6 Alternative Conditions for (Cγ)

(2)

2.7 Further on the Historical Approach

(1)

2.8 Summary

(1)

2.9 Background Information

(7)

2.9.1 Inverse of a distribution

(1)

2.9.2 Functions of regular variation

(2)

2.9.3 Relation between F and U

(1)

2.9.4 Proofs for section 2.6

(3)

3 AWAY FROM THE MAXIMUM

(16)

3.1 Introduction

(1)

3.2 Order Statistics Close to the Maximum

(6)

3.3 Second-order Theory

(4)

3.3.1 Remainder in terms of U

(2)

3.3.2 Examples

(1)

3.3.3 Remainder in terms of F

(1)

3.4 Mathematical Derivations

(6)

3.4.1 Proof of (3.6)

(1)

3.4.2 Proof of (3.8)

(1)

3.4.3 Solution of (3.15)

(1)

3.4.4 Solution of (3.18)

(1)

4 TAIL ESTIMATION UNDER PARETO-TYPE MODELS

(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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Statistics of Extremes Theory and Applications

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