Extremes in Random Fields A Theory and Its Applications

by Yakir, Benjamin

• ISBN13: 9781118620205
• ISBN10: 1118620208
• Edition: 1st
• Format: Hardcover
• Copyright: 2013-10-14
• Publisher: Wiley

Summary

Presents a useful new technique for analyzing the extreme-value behaviour of random fields

Modern science typically involves the analysis of increasingly complex data. The extreme values that emerge in the statistical analysis of complex data are often of particular interest. This book focuses on the analytical approximations of the statistical significance of extreme values. Several relatively complex applications of the technique to problems that emerge in practical situations are presented.  All the examples are difficult to analyze using classical methods, and as a result, the author presents a novel technique, designed to be more accessible to the user.

Extreme value analysis is widely applied in areas such as operational research, bioinformatics, computer science, finance and many other disciplines. This book will be useful for scientists, engineers and advanced graduate students who need to develop their own statistical tools for the analysis of their data. Whilst this book may not provide the reader with the specific answer it will inspire them to rethink their problem in the context of random fields, apply the method, and produce a solution.

Author Biography

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

Table of Contents

Preface

I Theory

1 Introduction

1.1 Distribution of extremes in random fields

1.2 Outline of the method

1.3 Gaussian and asymptotically Gaussian random fields

1.4 Applications

2 Basic Examples

2.1 Introduction

2.2 A power-one sequential test

2.3 A kernel-based scanning statistic

2.4 Other methods

3 Approximation of the Local Rate

3.1 Introduction

3.2 Preliminary localization and approximation

3.2.1 Localization

3.2.2 A discrete approximation

3.3 Measure transformation

3.4 Application of the localization theorem

3.5 Integration

4 From the Local to the Global

4.1 Introduction

4.2 Poisson approximation of probabilities

4.3 Average run length to false alarm

5 The Localization Theorem

5.1 Introduction

5.2 A simplifies version of the localization theorem

5.3 The Localization Theorem

5.4 A local limit theorem

5.5 Edge effects

II Applications

6 Kolmogorov-Smirnov and Peacock

6.1 Introduction

6.2 Analysis of the one-dimensional case

6.3 Peacock's test

6.4 Relations to scanning statistics

7 Copy Number Variations

7.1 Introduction

7.2 The statistical model

7.3 Analysis of statistical properties

7.4 The False Discovery Rate (FDR)

8 Sequential Monitoring of an Image

8.1 Introduction

8.2 The statistical model

8.3 Analysis of statistical properties

8.4 Optimal change-point detection

9 Buffer Overflow

9.1 Introduction

9.2 The statistical model

9.3 Analysis of statistical properties

9.4 Long-range dependence and self-similarity

10 Computing Pickands' Constants

10.1 Introduction

10.2 Representations of constants

10.3 Analysis of statistical error

10.4 Local fluctuations

Appendix A Mathematical Background

A.1 Transforms

A.2 Approximations of sum of independent random elements

A.3 Concentration inequalities

A.4 Random walks

A.5 Renewal theory

A.6 The Gaussian distribution

A.7 Large sample inference

A.8 Integration

A.9 Poisson approximation

A.10 Convexity

References

Index