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

• Introduction ixChapter 1 Fatness of Tail 11.1 Fat tail heuristics 11.2 History and data 41.2.1 US flood insurance claims 41.2.2 US crop loss 51.2.3 US damages and fatalities from natural disasters 51.2.4 US hospital discharge bills 61.2.5 G-Econ data 61.3 Diagnostics for heavy-tailed phenomena 61.3.1 Historical averages 71.3.2 Records 81.3.3 Mean excess 111.3.4 Sum convergence: self-similar or normal 121.3.5 Estimating the tail index 151.3.6 The obesity index 201.4 Relation to reliability theory 241.5. Conclusion and overview of the technical chapters 25Chapter 2 Order Statistics 272.1 Distribution of order statistics 272.2 Conditional distribution 322.3 Representations for order statistics 332.4 Functions of order statistics 362.4.1 Partial sums 362.4.2 Ratio between order statistics 37Chapter 3 Records 413.1 Standard record value processes 413.2 Distribution of record values 423.3 Record times and related statistics 443.4 k-records 46Chapter 4 Regularly Varying and Subexponential Distributions 494.1 Classes of heavy-tailed distributions 504.1.1 Regularly varying distribution functions 504.1.2 Subexponential distribution functions 554.1.3 Related classes of heavy-tailed distributions 584.2 Mean excess function 594.2.1 Properties of the mean excess function 60Chapter 5 Indices and Diagnostics of Tail Heaviness 655.1 Self-similarity 665.1.1 Distribution of the ratio between order statistics 695.2 The ratio as index 765.3 The obesity index 805.3.1 Theory of majorization 855.3.2 The obesity index of selected data sets 91Chapter 6 Dependence 956.1 Definition and main properties 956.2 Isotropic distributions 966.3 Pseudo-isotropic distributions 1006.3.1 Covariation as a measure of dependence for essentially heavy-tail jointly pseudo-isotropic variables 1046.3.2 Codifference 1096.3.3 The linear regression model for essentially heavy-tail distribution 110Conclusions and Perspectives 115Bibliography 119Index 123

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

• Introduction ixChapter 1 Fatness of Tail 11.1 Fat tail heuristics 11.2 History and data 41.2.1 US flood insurance claims 41.2.2 US crop loss 51.2.3 US damages and fatalities from natural disasters 51.2.4 US hospital discharge bills 61.2.5 G-Econ data 61.3 Diagnostics for heavy-tailed phenomena 61.3.1 Historical averages 71.3.2 Records 81.3.3 Mean excess 111.3.4 Sum convergence: self-similar or normal 121.3.5 Estimating the tail index 151.3.6 The obesity index 201.4 Relation to reliability theory 241.5. Conclusion and overview of the technical chapters 25Chapter 2 Order Statistics 272.1 Distribution of order statistics 272.2 Conditional distribution 322.3 Representations for order statistics 332.4 Functions of order statistics 362.4.1 Partial sums 362.4.2 Ratio between order statistics 37Chapter 3 Records 413.1 Standard record value processes 413.2 Distribution of record values 423.3 Record times and related statistics 443.4 k-records 46Chapter 4 Regularly Varying and Subexponential Distributions 494.1 Classes of heavy-tailed distributions 504.1.1 Regularly varying distribution functions 504.1.2 Subexponential distribution functions 554.1.3 Related classes of heavy-tailed distributions 584.2 Mean excess function 594.2.1 Properties of the mean excess function 60Chapter 5 Indices and Diagnostics of Tail Heaviness 655.1 Self-similarity 665.1.1 Distribution of the ratio between order statistics 695.2 The ratio as index 765.3 The obesity index 805.3.1 Theory of majorization 855.3.2 The obesity index of selected data sets 91Chapter 6 Dependence 956.1 Definition and main properties 956.2 Isotropic distributions 966.3 Pseudo-isotropic distributions 1006.3.1 Covariation as a measure of dependence for essentially heavy-tail jointly pseudo-isotropic variables 1046.3.2 Codifference 1096.3.3 The linear regression model for essentially heavy-tail distribution 110Conclusions and Perspectives 115Bibliography 119Index 123

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

• Introduction ixChapter 1 Fatness of Tail 11.1 Fat tail heuristics 11.2 History and data 41.2.1 US flood insurance claims 41.2.2 US crop loss 51.2.3 US damages and fatalities from natural disasters 51.2.4 US hospital discharge bills 61.2.5 G-Econ data 61.3 Diagnostics for heavy-tailed phenomena 61.3.1 Historical averages 71.3.2 Records 81.3.3 Mean excess 111.3.4 Sum convergence: self-similar or normal 121.3.5 Estimating the tail index 151.3.6 The obesity index 201.4 Relation to reliability theory 241.5. Conclusion and overview of the technical chapters 25Chapter 2 Order Statistics 272.1 Distribution of order statistics 272.2 Conditional distribution 322.3 Representations for order statistics 332.4 Functions of order statistics 362.4.1 Partial sums 362.4.2 Ratio between order statistics 37Chapter 3 Records 413.1 Standard record value processes 413.2 Distribution of record values 423.3 Record times and related statistics 443.4 k-records 46Chapter 4 Regularly Varying and Subexponential Distributions 494.1 Classes of heavy-tailed distributions 504.1.1 Regularly varying distribution functions 504.1.2 Subexponential distribution functions 554.1.3 Related classes of heavy-tailed distributions 584.2 Mean excess function 594.2.1 Properties of the mean excess function 60Chapter 5 Indices and Diagnostics of Tail Heaviness 655.1 Self-similarity 665.1.1 Distribution of the ratio between order statistics 695.2 The ratio as index 765.3 The obesity index 805.3.1 Theory of majorization 855.3.2 The obesity index of selected data sets 91Chapter 6 Dependence 956.1 Definition and main properties 956.2 Isotropic distributions 966.3 Pseudo-isotropic distributions 1006.3.1 Covariation as a measure of dependence for essentially heavy-tail jointly pseudo-isotropic variables 1046.3.2 Codifference 1096.3.3 The linear regression model for essentially heavy-tail distribution 110Conclusions and Perspectives 115Bibliography 119Index 123