Extremes in Random Fields

Benjamin Yakir, Department of Statistics, The Hebrew University of Jerusalem, Israel

• Preface xiAcknowledgments xvPart I THEORY 11 Introduction 31.1 Distribution of extremes in random fields 31.2 Outline of the method 71.3 Gaussian and asymptotically Gaussian random fields 91.4 Applications 112 Basic examples 152.1 Introduction 152.2 A power-one sequential test 152.3 A kernel-based scanning statistic 242.4 Other methods 383 Approximation of the local rate 413.1 Introduction 413.2 Preliminary localization and approximation 433.3 Measure transformation 513.4 Application of the localization theorem 553.5 Integration4 From the local to the global 714.1 Introduction 714.2 Poisson approximation of probabilities 724.3 Average run length to false alarm 785 The localization theorem 875.1 Introduction 875.2 A simplified version of the localization theorem 885.3 The localization theorem 905.4 A local limit theorem 955.5 Edge effects and higher order approximations 100Part II APPLICATIONS 1036 Nonparametric tests: Kolmogorov–Smirnov and Peacock 1056.1 Introduction 1056.2 Analysis of the one-dimensional case 1096.3 Peacock’s test 1206.4 Relations to scanning statistics 1237 Copy number variations 1257.1 Introduction 1257.2 The statistical model 1277.3 Analysis of statistical properties 1317.4 The false discovery rate 1408 Sequential monitoring of an image 1438.1 Introduction 1438.2 The statistical model 1468.3 Analysis of statistical properties 1488.4 Optimal change-point detection 1619 Buffer overflow 1659.1 Introduction 1659.2 The statistical model 1699.3 Analysis of statistical properties 1729.4 Heavy tail distribution, long-range dependence, and self-similarity 18610 Computing Pickands’ constants 19110.1 Introduction 19110.2 Representations of constants 19610.3 Analysis of statistical error 19910.4 Enumerating the effect of local fluctuations 204Appendix: Mathematical background 209A.1 Transforms 209A.2 Approximations of sum of independent random elements 211A.3 Concentration inequalities 214A.4 Random walks 215A.5 Renewal theory 215A.6 The Gaussian distribution 216A.7 Large sample inference 217A.8 Integration 218A.9 Poisson approximation 219A.10 Convexity 220References 221Index 223

“Whilst the book does not (nor does it claim to) provide a general account of the many different techniques used in the field of extreme value theory, it offers the reader an interesting and inspiring approach that has proved to be fruitful for many statistical problems.”  (Zentralblatt MATH,  1 October 2015)