Fat–Tailed Distributions – Data, Diagnostics and Dependence

INTRODUCTION ix
<p>CHAPTER 1. FATNESS OF TAIL 1</p>
<p>1.1. Fat tail heuristics 1</p>
<p>1.2. History and data&nbsp;4</p>
<p>1.2.1. US flood insurance claims 4</p>
<p>1.2.2. US crop loss&nbsp;5</p>
<p>1.2.3. US damages and fatalities from natural disasters 5</p>
<p>1.2.4. US hospital discharge bills&nbsp;6</p>
<p>1.2.5. G–Econ data 6</p>
<p>1.3. Diagnostics for heavy–tailed phenomena 6</p>
<p>1.3.1. Historical averages&nbsp;7</p>
<p>1.3.2. Records 8</p>
<p>1.3.3. Mean excess&nbsp;11</p>
<p>1.3.4. Sum convergence: self–similar or normal 12</p>
<p>1.3.5. Estimating the tail index 15</p>
<p>1.3.6. The obesity index 20</p>
<p>1.4. Relation to reliability theory 24</p>
<p>1.5. Conclusion and overview of the technical chapters&nbsp;25</p>
<p>CHAPTER 2. ORDER STATISTICS 27</p>
<p>2.1. Distribution of order statistics&nbsp;27</p>
<p>2.2. Conditional distribution 32</p>
<p>2.3. Representations for order statistics&nbsp;33</p>
<p>2.4. Functions of order statistics 36</p>
<p>2.4.1. Partial sums&nbsp;36</p>
<p>2.4.2. Ratio between order statistics&nbsp;37</p>
<p>CHAPTER 3. RECORDS 41</p>
<p>3.1. Standard record value processes&nbsp;41</p>
<p>3.2. Distribution of record values 42</p>
<p>3.3. Record times and related statistics&nbsp;44</p>
<p>3.4. k–records&nbsp;46</p>
<p>CHAPTER 4. REGULARLY VARYING AND SUBEXPONENTIAL DISTRIBUTIONS&nbsp;49</p>
<p>4.1. Classes of heavy–tailed distributions&nbsp;50</p>
<p>4.1.1. Regularly varying distribution functions 50</p>
<p>4.1.2. Subexponential distribution functions&nbsp;55</p>
<p>4.1.3. Related classes of heavy–tailed distributions&nbsp;58</p>
<p>4.2. Mean excess function 59</p>
<p>4.2.1. Properties of the mean excess function&nbsp;60</p>
<p>CHAPTER 5. INDICES AND DIAGNOSTICS OF TAIL HEAVINESS 65</p>
<p>5.1. Self–similarity 66</p>
<p>5.1.1. Distribution of the ratio between order statistics 69</p>
<p>5.2. The ratio as index 76</p>
<p>5.3. The obesity index&nbsp;80</p>
<p>5.3.1. Theory of majorization 85</p>
<p>5.3.2. The obesity index of selected data sets&nbsp;91</p>
<p>CHAPTER 6. DEPENDENCE 95</p>
<p>6.1. Definition and main properties&nbsp;95</p>
<p>6.2. Isotropic distributions 96</p>
<p>6.3. Pseudo–isotropic distributions 100</p>
<p>6.3.1. Covariation as a measure of dependence for essentially heavy–tail jointly pseudo–isotropic variables 104</p>
<p>6.3.2. Codifference&nbsp;109</p>
<p>6.3.3. The linear regression model for essentially heavy–tail distribution 110</p>
<p>CONCLUSIONS AND PERSPECTIVES&nbsp;115</p>
<p>BIBLIOGRAPHY 119</p>
<p>INDEX&nbsp;123</p>