Fat-Tailed Distributions
Data, Diagnostics and Dependence
Published on 11. November 2014
144 pages
978-1-119-05419-1 (ISBN)
Description
This title is written for the numerate nonspecialist, and hopesto serve three purposes. First it gathers mathematical materialfrom diverse but related fields of order statistics, records,extreme value theory, majorization, regular variation andsubexponentiality. All of these are relevant for understanding fattails, but they are not, to our knowledge, brought together in asingle source for the target readership. Proofs that give insightare included, but for most fussy calculations the reader isreferred to the excellent sources referenced in the text.Multivariate extremes are not treated. This allows us to presentmaterial spread over hundreds of pages in specialist texts intwenty pages. Chapter 5 develops new material on heavy taildiagnostics and gives more mathematical detail. Since variances andcovariances may not exist for heavy tailed joint distributions,Chapter 6 reviews dependence concepts for certain classes of heavytailed joint distributions, with a view to regressing heavy tailedvariables.
Second, it presents a new measure of obesity. The most populardefinitions in terms of regular variation and subexponentialityinvoke putative properties that hold at infinity, and thiscomplicates any empirical estimate. Each definition captures somebut not all of the intuitions associated with tail heaviness.Chapter 5 studies two candidate indices of tail heaviness based onthe tendency of the mean excess plot to collapse as data areaggregated. The probability that the largest value is more thantwice the second largest has intuitive appeal but its estimator hasvery poor accuracy. The Obesity index is defined for a positiverandom variable X as:
Ob(X) = P (X1 +X4 > X2 +X3|X1 <= X2 <= X3 <=X4), Xi independent copies of X.
For empirical distributions, obesity is defined bybootstrapping. This index reasonably captures intuitions of tailheaviness. Among its properties, if alpha > 1 then Ob(X)
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Roger M. Cooke | Daan Nieboer | Jolanta Misiewicz
Fat-Tailed Distributions
Data, Diagnostics and Dependence, Volume 1
Book
11/2014
1st Edition
Wiley-ISTE
€164.50
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Content
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 technicalchapters 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 SUBEXPONENTIALDISTRIBUTIONS 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 HEAVINESS65
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 essentiallyheavy-tail jointly pseudo-isotropic variables 104
6.3.2. Codifference 109
6.3.3. The linear regression model for essentially heavy-taildistribution 110
CONCLUSIONS AND PERSPECTIVES 115
BIBLIOGRAPHY 119
INDEX 123
