Fat-Tailed Distributions (eBook)

Data, Diagnostics and Dependence, Volume 1

Roger M. Cooke, Daan Nieboer, Jolanta Misiewicz (Autoren)

eBook Download: EPUB

2014 | 1. Auflage
144 Seiten
John Wiley & Sons (Verlag)
978-1-119-05412-2 (ISBN)

Autor

Roger M. Cooke, Chauncey Starr Chair for Risk Analysis Resources for the Future, USA and Dept. Math. TU Delft, Netherlands Daan Nieboer, Erasmus Universiteit Rotterdam, Department of Public Health (MGZ), Netherlands Jolanta Misiewicz, Professor (Full), Warsaw University of Technology, Faculty of Mathematics and Information Science, Mazowieckie, Poland

Inhalt

INTRODUCTION ix

CHAPTER 1. FATNESS OF TAIL 1

1.1. Fat tail heuristics 1

1.2. History and data 4

1.2.1. US flood insurance claims 4

1.2.2. US crop loss 5

1.2.3. US damages and fatalities from natural disasters 5

1.2.4. US hospital discharge bills 6

1.2.5. G-Econ data 6

1.3. Diagnostics for heavy-tailed phenomena 6

1.3.1. Historical averages 7

1.3.2. Records 8

1.3.3. Mean excess 11

1.3.4. Sum convergence: self-similar or normal 12

1.3.5. Estimating the tail index 15

1.3.6. The obesity index 20

1.4. Relation to reliability theory 24

1.5. Conclusion and overview of the technical
chapters 25

CHAPTER 2. ORDER STATISTICS 27

2.1. Distribution of order statistics 27

2.2. Conditional distribution 32

2.3. Representations for order statistics 33

2.4. Functions of order statistics 36

2.4.1. Partial sums 36

2.4.2. Ratio between order statistics 37

CHAPTER 3. RECORDS 41

3.1. Standard record value processes 41

3.2. Distribution of record values 42

3.3. Record times and related statistics 44

3.4. k-records 46

CHAPTER 4. REGULARLY VARYING AND SUBEXPONENTIAL
DISTRIBUTIONS 49

4.1. Classes of heavy-tailed distributions 50

4.1.1. Regularly varying distribution functions 50

4.1.2. Subexponential distribution functions 55

4.1.3. Related classes of heavy-tailed distributions 58

4.2. Mean excess function 59

4.2.1. Properties of the mean excess function 60

CHAPTER 5. INDICES AND DIAGNOSTICS OF TAIL HEAVINESS
65

5.1. Self-similarity 66

5.1.1. Distribution of the ratio between order statistics 69

5.2. The ratio as index 76

5.3. The obesity index 80

5.3.1. Theory of majorization 85

5.3.2. The obesity index of selected data sets 91

CHAPTER 6. DEPENDENCE 95

6.1. Definition and main properties 95

6.2. Isotropic distributions 96

6.3. Pseudo-isotropic distributions 100

6.3.1. Covariation as a measure of dependence for essentially
heavy-tail jointly pseudo-isotropic variables 104

6.3.2. Codifference 109

6.3.3. The linear regression model for essentially heavy-tail
distribution 110

CONCLUSIONS AND PERSPECTIVES 115

BIBLIOGRAPHY 119

INDEX 123

Produkt-Details

• Erscheint lt. Verlag: 11.11.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Technik
• Schlagworte: Mathematical Modeling • Mathematics • Mathematik • Mathematische Modellierung
• ISBN-10: 1-119-05412-5 / 1119054125
• ISBN-13: 978-1-119-05412-2 / 9781119054122