Laws of Small Numbers

Since the publication of the first edition of this seminar book in 1994, the theory and applications of extremes and rare events have enjoyed an enormous and still increasing interest. The intention of the book is to give a mathematically oriented development of the theory of rare events underlying various applications. This characteristic of the book was strengthened in the second edition by incorporating various new results. In this third edition, the dramatic change of focus of extreme value theory has been taken into account: from concentrating on maxima of observations it has shifted to large observations, defined as exceedances over high thresholds. One emphasis of the present third edition lies on multivariate generalized Pareto distributions, their representations, properties such as their peaks-over-threshold stability, simulation, testing and estimation.Reviews of the 2nd edition:"In brief, it is clear that this will surely be a valuable resource for anyone involved in, or seeking to master, the more mathematical features of this field."David Stirzaker, Bulletin of the London Mathematical Society"Laws of Small Numbers can be highly recommended to everyone who is looking for a smooth introduction to Poisson approximations in EVT and other fields of probability theory and statistics. In particular, it offers an interesting view on multivariate EVT and on EVT for non-iid observations, which is not presented in a similar way in any other textbook."Holger Drees, Metrika

Preface to the Third Edition	p. v
Preface to the Second Edition	p. vii
Preface to the First Edition	p. ix
The IID Case: Functional Laws of Small Numbers	p. 1
Functional Laws of Small Numbers	p. 3
Introduction	p. 3
Bounds for the Functional Laws of Small Numbers	p. 7
Applications	p. 14
Extreme Value Theory	p. 25
Domains of Attraction, von Mises Conditions	p. 25
The ¿-Neighborhood of a GPD	p. 35
The Peaks-Over-Threshold Method	p. 42
Parameter Estimation in ¿-Neighborhoods of GPD	p. 46
Initial Estimation of the Class Index	p. 55
Power Normalization and p-Max Stable Laws	p. 65
Heavy and Super-Heavy Tail Analysis	p. 75
Estimation of Conditional Curves	p. 103
Poisson Process Approach	p. 103
Applications: The Non-parametric Case	p. 111
Applications: The Semiparametric Case	p. 114
Extension to Several Points	p. 120
A Nearest Neighbor Alternative	p. 124
Application: Optimal Accuracy of Estimators	p. 128
The IID Case: Multivariate Extremes	p. 133
Basic Theory of Multivariate Maxima	p. 135
Limiting Distributions of Multivariate Maxima	p. 135
Representations and Dependence Functions	p. 140
Pickands Representation and Dependence Function	p. 147
The D-Norm	p. 157
Multivariate Generalized Pareto Distributions	p. 171
The Basics	p. 171
Multivariate Peaks-Over-Threshold Approach	p. 177
Peaks-Over-Threshold Stability of a GPD	p. 190
A Spectral Decomposition Based on Pickands Coordinates	p. 194
Multivariate Domains of Attraction, Spectral Neighborhoods	p. 201
The Pickands Transform	p. 209
Simulation Techniques	p. 221
Testing the GPD Assumption, Threshold Selection	p. 229
Parametric Estimation Procedures	p. 240
Testing in Logistic GPD Models	p. 250
The Pickands Approach in the Bivariate Case	p. 259
Preliminaries	p. 259
The Measure Generating Function M	p. 267
The Pickands Transform in the Bivariate Case	p. 272
The Tail Dependence Function	p. 280
Testing Tail Dependence against Residual Tail Dependence	p. 288
Estimation of the Angular Density in Bivariate GP Models	p. 299
Multivariate Extremes: Supplementary Concepts and Results	p. 311
Strong Approximation of Exceedances	p. 311
Further Concepts of Extremes	p. 317
Thinned Empirical Processes	p. 320
Max-Stable Stochastic Processes	p. 335
Non-IID Observations	p. 341
Introduction to the Non-IID Case	p. 343
Definitions	p. 343
Stationary Random Sequences	p. 344
Independent Random Sequences	p. 346
Non-stationary Random Sequences	p. 351
Triangular Arrays of Discrete Random Variables	p. 352
Extremes of Random Sequences	p. 357
Introduction and General Theory	p. 357
Applications: Stationary Sequences	p. 368
Applications: Independent Sequences	p. 372
Applications: Non-stationary Sequences	p. 374
Extensions: Random Fields	p. 379
Extremes of Gaussian Processes	p. 381
Introduction	p. 381
Stationary Gaussian Processes	p. 382
Non-stationary Gaussian Processes	p. 388
Application: Empirical Characteristic Functions	p. 414
Extensions: Maxima of Gaussian fields	p. 416
Extensions for Rare Events	p. 419
Rare Events of Random Sequences	p. 419
The Point Process of Exceedances	p. 423
Application to Peaks-over-Threshold	p. 428
Application to Rare Events	p. 431
Triangular Arrays of Rare Events	p. 439
Multivariate Extremes of Non-IID Sequences	p. 447
Statistics of Extremes	p. 455
Introduction	p. 455
Estimation of ¿ and ¿( ·)	p. 456
Application to Ecological Data	p. 458
Frost Data: An Application	p. 462
Author Index	p. 473
Subject Index	p. 477
Bibliography	p. 483
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