Distribution with heavy margins

In probability theory , a distribution with heavy edges ( English heavy tails ) or end- heavy distribution or heavy-tailed distribution ( English Heavy-tailed distribution ) is a probability distribution whose density falls more slowly than exponentially. The term clearly means that there is more mass on the “edges” or “tails” of the distribution than, for example, in the exponential distribution . There are three main subclasses of distributions with heavy margins: the distributions with fat distribution ends ( english fat tails ), the distributions with long distribution ends ( english long tails ) and the subexponential distributions .

definition

A random variable has a distribution with heavy margins if the following applies to its distribution function: ${\displaystyle X}$

${\displaystyle \int _{-\infty }^{\infty }e^{tx}\,\mathrm {d} F(x)=\infty \quad \mathrm {f{\ddot {u}}r\;alle\;} t>0.}$

The following also applies to the subset of the sub-exponential distributions:

If random variables are independently and identically distributed , then assuming that they are distributed sub-exponentially, the distribution of the sum of the is determined asymptotically by the distribution of the maximum of . ${\displaystyle X_{n}}$ ${\displaystyle X_{n}}$ ${\displaystyle X_{n}}$

Examples

• Pareto distribution
• Logarithmic normal distribution
• Logarithmic gamma distribution
• Lévy distribution
• Cauchy distribution
• Weibull distribution with shape parameters less than 1
• t distribution

Applications

In actuarial mathematics , distributions with heavy margins are used to model large losses and extreme events. Liability lines are also referred to as so-called long-tail lines because of their long processing times. On the other hand, insurance lines such as comprehensive insurance, household contents or glass insurance are so-called short-tail lines. The settlement of claims in these short-tail lines is generally short. In the long-tail sectors, processing times of over 40 years are not uncommon.

Heavy margins are also important in the financial sector. For example, Benoit Mandelbrot and Eugene Fama showed that the returns on stocks and other speculative investments differ significantly from normal distribution and are usually end-heavy, a finding known as the Fama-French three-factor model .

literature

• Paul Embrechts, Thomas Mikosch, Claudia Klüppelberg : Modeling extremal events. Springer, Berlin 1997, ISBN 3-540-60931-8 .
• Christian Grimm, Georg Schlüchtermann: Traffic theory in IP networks . 1st edition. Hüthig, Bonn, Heidelberg 2004, ISBN 3-8266-5047-6 .

Individual evidence

1. ↑ Grimm / Schlüchtermann p. 174 f.
2. ^ S. Foss, D. Korshunov, S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions , Springer Science & Business Media, 21 May 2013
3. ↑ Wolf-Rüdiger Heilmann and Klaus Jürgen Schröter: Basic concepts of risk theory. VVW GmbH 2013
4. ↑ Mandelbrot, B. (1963), The Variation of Certain Speculative Prices, in: Journal of Business , Vol. 36, No. 4, pp. 394-419.
5. ↑ Fama, EF (1965), The Behavior of Stock-Market Prices, in: Journal of Business , Vol. 38, No. 1, pp. 34-105.