This title is written for the numerate nonspecialist, and hopes to serve three purposes. First it gathers mathematical material from diverse but related fields of order statistics, records, extreme value theory, majorization, regular variation and subexponentiality.
This title is written for the numerate nonspecialist, and hopes to serve three purposes. First it gathers mathematical material from diverse but related fields of order statistics, records, extreme value theory, majorization, regular variation and subexponentiality.
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
Inhaltsangabe
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
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
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