### Abstract

This monograph 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. All of these are

relevant for understanding fat tails, but they are not, to our knowledge, brought together in

a single source for the target readership. Proofs that give insight are included, but for fussy

calculations the reader is referred to the excellent sources referenced in the text. Multivariate

extremes are not treated. This allows us to present material spread over hundreds of pages in

specialist texts in twenty pages. Chapter 5 develops new material on heavy tail diagnostics and

gives more mathematical detail.

Second, it presents a new measure of obesity. The most popular deﬁnitions in terms of

regular variation and subexponentiality invoke putative properties that hold at inﬁnity, and this

complicates any empirical estimate. Each deﬁnition captures some but not all of the intuitions

associated with tail heaviness. Chapter 5 studies two candidate indices of tail heaviness based

on the tendency of the mean excess plot to collapse as data are aggregated. The probability that

the largest value is more than twice the second largest has intuitive appeal but its estimator has

very poor accuracy. The Obesity index is deﬁned for a positive random 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 by bootstrapping. This index reasonably captures

intuitions of tail heaviness. Among its properties, if α > 1 then Ob(X) < Ob(X). However,

it does not completely mimic the tail index of regularly varying distributions, or the extreme

value index. A Weibull distribution with shape 1/4 is more obese than a Pareto distribution

with tail index 1, even though this Pareto has inﬁnite mean and the Weibull’s moments are all

ﬁnite. Chapter 5 explores properties of the Obesity index.

Third and most important, we hope to convince the reader that fat tail phenomena pose

real problems; they are really out there and they seriously challenge our usual ways of thinking

about historical averages, outliers, trends, regression coeﬃcients and conﬁdence bounds among

many other things. Data on ﬂood insurance claims, crop loss claims, hospital discharge bills,

precipitation and damages and fatalities from natural catastrophes drive this point home.

First it gathers mathematical material from diverse but related ﬁelds of order statistics, records,

extreme value theory, majorization, regular variation and subexponentiality. All of these are

relevant for understanding fat tails, but they are not, to our knowledge, brought together in

a single source for the target readership. Proofs that give insight are included, but for fussy

calculations the reader is referred to the excellent sources referenced in the text. Multivariate

extremes are not treated. This allows us to present material spread over hundreds of pages in

specialist texts in twenty pages. Chapter 5 develops new material on heavy tail diagnostics and

gives more mathematical detail.

Second, it presents a new measure of obesity. The most popular deﬁnitions in terms of

regular variation and subexponentiality invoke putative properties that hold at inﬁnity, and this

complicates any empirical estimate. Each deﬁnition captures some but not all of the intuitions

associated with tail heaviness. Chapter 5 studies two candidate indices of tail heaviness based

on the tendency of the mean excess plot to collapse as data are aggregated. The probability that

the largest value is more than twice the second largest has intuitive appeal but its estimator has

very poor accuracy. The Obesity index is deﬁned for a positive random 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 by bootstrapping. This index reasonably captures

intuitions of tail heaviness. Among its properties, if α > 1 then Ob(X) < Ob(X). However,

it does not completely mimic the tail index of regularly varying distributions, or the extreme

value index. A Weibull distribution with shape 1/4 is more obese than a Pareto distribution

with tail index 1, even though this Pareto has inﬁnite mean and the Weibull’s moments are all

ﬁnite. Chapter 5 explores properties of the Obesity index.

Third and most important, we hope to convince the reader that fat tail phenomena pose

real problems; they are really out there and they seriously challenge our usual ways of thinking

about historical averages, outliers, trends, regression coeﬃcients and conﬁdence bounds among

many other things. Data on ﬂood insurance claims, crop loss claims, hospital discharge bills,

precipitation and damages and fatalities from natural catastrophes drive this point home.

Original language | English |
---|---|

Place of Publication | Washington |

Number of pages | 65 |

Publication status | Published - 2011 |

### Keywords

- heavy-tailed distributions
- data
- new developments
- diagnostics

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## Cite this

Cooke, R. M., & Nieboer, D. (2011).

*Heavy-tailed distributions: data, diagnostics, and new developments*.