Laws of Small Numbers: Extremes and Rare Events (eBook)

Preface to the Third Edition 5
Preface to the Second Edition 7
Preface to the First Edition 9
Contents 12
Part I The IID Case: Functional Laws of Small Numbers 15
Functional Laws of Small Numbers 16
1.1 Introduction 16
The Law of Small Numbers 17
Point Process Approximations 18
1.2 Bounds for the Functional Laws of Small Numbers 20
Markov Kernels 21
The Monotonicity Theorem 21
The Convexity Theorem 22
Why the Hellinger Distance? 23
Specifying H(V,W) 24
The Stein-Chen Method 25
1.3 Applications 27
Truncated Empirical Processes 28
A Geometric Classification 30
Examples 30
Extreme Value Distributions 34
Generalized Pareto Distributions 34
The Peaks-Over-Threshold Method 35
Extreme Value Theory 37
2.1 Domains of Attraction, von Mises Conditions 37
The Gnedenko-De Haan Theorem 37
Von Mises Conditions 39
Differentiable Tail Equivalence 40
Von Mises Condition With Remainder 41
Rates of Convergence of Extremes 44
Best Attainable Rates of Convergence 45
2.2 The ?-Neighborhood of a GPD 47
The Standard Form of GPD 47
?-Neighborhoods 48
Data Transformations 51
Joint Asymptotic Distribution of Extremes 52
Summarizing the Results 53
2.3 The Peaks-Over-Threshold Method 54
The Point Process of Exceedances 54
Approximation of Excess Distributions 55
Bounds for the Process Approximations 56
2.4 Parameter Estimation in ?-Neighborhoods of GPD 58
The Basic Approximation Lemma 59
Efficient Estimators of a0 and b0 60
Hill’s Estimator and Friends 61
The Pareto Model with Known Scale Factor 64
The Extreme Quantile Estimation 65
Confidence Intervals 66
2.5 Initial Estimation of the Class Index 67
The Pickands Estimator 67
Convex Combinations of Pickands Estimators 68
The Asymptotic Relative Efficiency 69
The Optimal Choice of p 69
Data-Driven Optimal Estimators 70
Dropping the ?-Neighborhood 72
Simulation Results 72
Notes on Competing Estimators 75
2.6 Power Normalization and p-Max Stable Laws 77
The Power-Max Stable Distributions 77
Max and Min Stable Distributions 78
The Characterization Theorem 79
Comparison of Max Domains of Attraction under Linear and Power Normalizations 81
Comparison of Max Domains of Attraction under Linear and Power Normalizations - The Multivariate Case 84
Examples 86
2.7 Heavy and Super-Heavy Tail Analysis 87
Heavy Tail Analysis 88
Regular and Slow Variation 90
Super-Heavy Tails and Slow Variation 90
Super-Heavy Tails in the Literature 92
The P-Pot Stable Distributions 94
Relations to p-max Stable Laws 96
Limiting Distributions of Exceedances 97
Domains of Attraction 98
Mixtures of Regularly Varying DFs 101
Testing for Super-Heavy Tails 103
Estimation of Conditional Curves 114
3.1 Poisson Process Approach 114
Truncated Empirical Process 115
The First-Order Poisson Approximation 116
The Second-Order Poisson Approximation 116
The POT-Approach 116
Basic Smoothness Conditions 117
Bounds for the Approximations 117
The Third-Order Poisson Approximation 118
A Unified Smoothness Condition 120
Bounds for Equal Bin Widths 121
3.2 Applications: The Non-parametric Case 122
Local Empirical Distribution Function 122
Kernel Estimator of a Regression Functional 122
The Basic Reduction Theorem 123
Examples 124
3.3 Applications: The Semiparametric Case 125
A Semiparametric Model 125
The Basic Approximation Lemma 126
The Hellinger Differentiability 127
Local Asymptotic Normality 127
The H´ajek-LeCam Convolution Theorem 128
Asymptotically Efficient Estimation 128
Exponential Families 129
Efficient Estimation Based on M?n 130
Regular Paths 130
Efficient Estimation Based on Nn 131
3.4 Extension to Several Points 131
Vectors of Processes 132
The Third-Order Poisson Approximation 132
The First-Order Poisson Approximation 134
Estimation Over Compact Intervals 134
3.5 A Nearest Neighbor Alternative 135
The Basic Representation Lemma 136
An Approximation Result 137
3.6 Application: Optimal Accuracy of Estimators 139
The Model Bias 140
Local Empirical Distribution Function 140
Asymptotic Normality of NN-Estimates 140
Optimal Accuracy 141
Part II The IID Case: Multivariate Extremes 143
Basic Theory of Multivariate Maxima 144
4.1 Limiting Distributions of Multivariate Maxima 144
Limiting Distributions, Max-Stability 145
Weak Convergence: The IID Case 147
4.2 Representations and Dependence Functions 149
The Max-Infinite Divisibility 149
The de Haan-Resnick Representation 152
A Spectral Representation 155
The Bivariate Case 156
4.3 Pickands Representation and Dependence Function 156
The Pickands Representation 157
The Pickands Dependence Function 158
Important Properties of Pickands Dependence Functions 159
Examples of Pickands Dependence Functions 164
4.4 The D-Norm 166
Multivariate Generalized Pareto Distributions 179
5.1 The Basics 179
The Bivariate Case 180
GPD in Higher Dimensions 181
GP Functions and Quasi-Copulas 184
5.2 Multivariate Peaks-Over-Threshold Approach 185
The Cases of Independence and Complete Dependence 190
The GPD of Asymmetric Logistic Type 191
Another Representation of GPD and EVD 193
Multivariate Piecing-Together 195
Fitting a GPD-Copula to a Given Copula 197
5.3 Peaks-Over-Threshold Stability of a GPD 198
5.4 A Spectral Decomposition Based on Pickands Coordinates 202
Pickands Coordinates in Rd 203
Spectral Decompositions of Distribution Functions 203
Estimation of the Pickands Dependence Function 205
Further POT-Stability of W 206
The Best Attainable Rate of Convergence 207
5.5 Multivariate Domains of Attraction, Spectral Neighborhoods 209
The Domain of Attraction 209
The Spectral Neighborhood of a GPD 211
A Spectral ?-Neighborhood of a GPD 214
An Estimator of D 216
5.6 The Pickands Transform 217
The Pickands Transform of a GPD Random Vector 218
Differentiable ?-Neighborhoods of Pickands Transforms 221
Expansions of Pickands densities of Finite Length with Regularly Varying Functions 222
The POT Approach Based on the Pickands Transform 224
The Pickands Transform for a General EVD 226
5.7 Simulation Techniques 229
Simulation of GPD with Bounded Pickands Density 230
A Special Case: Simulation of GPD of Logistic Type 232
Simulation of Unconditional GPD 235
5.8 Testing the GPD Assumption, Threshold Selection 237
Testing for a Multivariate GPD 237
Performance of the Test 239
Simulation of the Test 242
A t-Test Based Threshold Selection in Multivariate POT Models 246
5.9 Parametric Estimation Procedures 248
The Pickands Transformation Reloaded 248
ML Estimation with the Angular Density 250
MLE with the Pickands Density 251
Estimation via Relative Frequencies 252
Comparison of the Estimation Procedures 255
5.10 Testing in Logistic GPD Models 258
The Basic Setup 258
A One-Sample Test 260
A Two-Sample Test 262
Multivariate Generalizations 264
The Pickands Approach in the Bivariate Case 266
6.1 Preliminaries 266
The Dependence Function D in the Bivariate Case 267
The Tail Dependence Parameter 268
Towards Residual Tail Dependence 269
Differentiable Spectral Expansions of Finite Length with Regularly Varying Functions 271
6.2 The Measure Generating Function M 274
Another Representation of the Pickands Dependence Function 274
Estimation of the Measure Generating Function M 276
6.3 The Pickands Transform in the Bivariate Case 279
The Distribution of the Distance C=U+V 279
The Pickands Transform 280
Marshall-Olkin GPD Random Vectors 282
Marshall-Olkin EVD Random Vectors 284
6.4 The Tail Dependence Function 287
The Tail Dependence Function 287
An Estimator of the Tail Dependence Function 290
A Characterization of the Marshall-Olkin Dependence Function 291
LAN and Efficient Estimation of ?(z) 292
6.5 Testing Tail Dependence against Residual Tail Dependence 295
Testing the Tail Dependence under Differentiable Spectral Expansions 296
A Conditional Distribution of the Distance C=X+Y 297
Test Statistic based on the Distance C 301
Simulations of p-Values 302
Testing Tail Dependence in Arbitrary Dimension 303
6.6 Estimation of the Angular Density in Bivariate Generalized Pareto Models 306
The Bivariate Angular Density 307
An Alternative Representation of the Angular Density 309
Estimation of the Pickands Density 310
Estimation of the Angular Density 311
The Problem of Generalization to the Trivariate Case 313
An Alternative Estimation for the Multivariate Case 315
Multivariate Extremes: Supplementary Concepts and Results 317
7.1 Strong Approximation of Exceedances 317
Approximation Technique 318
First-Order Poisson Approximation 318
Pathwise Exceedances 319
Generalized Pareto Distributions 320
Vectors of Exceedances 321
Random Thresholds 322
Multivariate Maxima 323
7.2 Further Concepts of Extremes 323
Vertices of the Convex Hull 324
Pareto Points 324
7.3 Thinned Empirical Processes 326
Introduction 326
Truncated Empirical Processes 327
A Very Brief Sketch of Some LAN Theory 328
The Basic Representation Lemma 329
The Model Assumptions 330
A Crucial Condition 331
Further Regularity Conditions 332
The Main Result 333
Example: Right-Censored Data 334
An Efficient Estimator 336
Application to Fuzzy Set Density Estimation 337
The Estimator ?n is Actually Efficient 339
Asymptotically Biased Density Estimators 340
7.4 Max-Stable Stochastic Processes 341
Max-Stability 341
Max-Stable Extremal Processes 342
Generation of Max-Stable Processes by Convolutions 343
A Max-Stable Process Corresponding to Brownian Motion 344
Maxima of Independent Brownian Motions 345
Theoretical Results 346
PART III Non-IID Observations 347
Introduction to the Non-IID Case 348
8.1 Definitions 348
8.2 Stationary Random Sequences 349
8.3 Independent Random Sequences 351
8.4 Non-stationary Random Sequences 356
8.5 Triangular Arrays of Discrete Random Variables 357
Extremes of Random Sequences 362
9.1 Introduction and General Theory 362
Long Range Dependence 365
Local Dependence 366
Point Process of Exceedances 368
Point Process of Upcrossings 369
Point Process of Clusters 370
9.2 Applications: Stationary Sequences 373
9.3 Applications: Independent Sequences 377
9.4 Applications: Non-stationary Sequences 379
Convergence Rate of the Poisson Approximation 380
9.5 Extensions: Random Fields 384
Extremes of Gaussian Processes 386
10.1 Introduction 386
10.2 Stationary Gaussian Processes 387
Local Behavior of Extremes 388
Limit Behavior of Extremes 390
Maxima of Discrete and Continuous Processes 391
Crossings or Level Sets of Smooth Gaussian Processes 392
10.3 Non-stationary Gaussian Processes 393
Locally Stationary Gaussian Processes 395
Constant Boundaries 397
Multifractional processes 398
Excursions above Very High Boundaries 400
Boundaries with a Unique Point of Minimal Value 400
More Pickands Constants 403
Random Variance or Random Boundary 405
Approximation of a Stationary Gaussian Process 410
Ruin Probability and Gaussian Processes with Drift 413
Extremes of Storage Models 416
Change-Point Regression and Boundary Crossing 418
10.4 Application: Empirical Characteristic Functions 419
The First Zero 419
Convergence Result 420
10.5 Extensions: Maxima of Gaussian Fields 421
Extensions for Rare Events 424
11.1 Rare Events of Random Sequences 424
Convergence of the Point Processes 426
11.2 The Point Process of Exceedances 428
Long Range Dependence 429
Extended Compound Poisson Process 430
Stationary Random Measures 431
Point Processes 432
11.3 Application to Peaks-over-Threshold 433
11.4 Application to Rare Events 436
Random Thinning of a Stationary Sequence or Process 439
Random Search for an Optimal Value 442
11.5 Triangular Arrays of Rare Events 444
Arrays of Rare Events 444
Point Processes of Rare Events 446
Weaker Restrictions on Local Dependence 447
Applications 450
Triangular Array of Rare Events Based on Gaussian Sequences 451
11.6 Multivariate Extremes of Non-IID Sequences 452
Stationary Multivariate Sequences 454
Independent Multivariate Sequences 456
General Multivariate Sequences 457
Statistics of Extremes 460
12.1 Introduction 460
12.2 Estimation of ? and ?(·) 461
Choice of Threshold 462
12.3 Application to Ecological Data 463
Clusters of Runs 463
Clusters given by Blocks 465
Other Estimates 467
12.4 Frost Data: An Application 467
Asymptotic Approximation 471
Bonferroni Inequalities 472
Slepian Inequality 473
Simulation 474
Comparison 475
Author Index 477
Subject Index 481
Bibliography 487