Extreme value theory

The extreme value theory (English term: Extreme-event statistics ) is a mathematical discipline that deals with outliers , i.e. H. maximum and minimum values of samples , employed.

A central result is the fact that essentially only three limit distributions are possible for the maximum (and the minimum) of a sample ( regardless of the distribution ).

Formally: Let there be independently and identically distributed random variables with values in the real numbers and their maximum. Furthermore denote the distribution function of , and be a non-degenerate distribution function - i.e. not a function that can only take on one value. If there are sequences so that the convergence applies, there can only be one of the following distributions, depending on whether the tail of the distribution declines exponentially , declines polynomially , or reaches the value zero at one point: ${\ displaystyle X_ {1}, \ ldots, X_ {n}}$ ${\ displaystyle M_ {n} = \ max _ {i} X_ {i}}$ ${\ displaystyle F (x) = P (M_ {n} <x) \;}$ ${\ displaystyle M_ {n}}$ ${\ displaystyle G}$ ${\ displaystyle a_ {n}, b_ {n} \;}$ ${\ displaystyle F (b_ {n} + a_ {n} x) \ rightarrow G (x) \;}$ ${\ displaystyle G}$

Gumbel type (Type I). More precisely: If the variable has a Gumbel distribution , then it has an extreme value distribution of type I. ${\ displaystyle x}$ ${\ displaystyle \ log (x)}$

Fréchet type (type II). More precisely: If the variable has a Fréchet distribution , then it has an extreme value distribution of type II. ${\ displaystyle x}$ ${\ displaystyle 1 / x}$

Weibull type (type III). More precisely: If the variable has a Weibull distribution , it has an extreme value distribution of type III. ${\ displaystyle x}$ ${\ displaystyle -x}$

These three distributions can also be parameterized to a single class ( Jenkinson-von-Mises representation ). The (or a) generalized distribution is called the extreme value distribution . And are often used as parameters , with a type III describing distribution and a type II distribution. ${\ displaystyle K, \ sigma}$ ${\ displaystyle \ mu}$ ${\ displaystyle K <0}$ ${\ displaystyle K> 0}$

It is used, among other things, in financial mathematics and actuarial mathematics . The theory was u. a. applied to the study of record developments in athletics and climate records.

Typical questions could include:

How high should a dam be built if you want to be sure that there is only a 1% probability that it will be flooded in the next 100 years?
What is the probability of a stock market crash in the coming year that will lead to a price decline of more than 15%?

Applications: Non-normal distributions with "fat tails"

For such very rare events, e.g. In the probability theory of extreme events, e.g. very high economic profits or losses, the usual normal distributions (or superimpositions thereof) are no longer characteristic, which behave like Gaussian functions, i.e. like "standard bell curves" of width , more precisely: how which fall faster than exponentially in the edge area. Instead, distribution functions dominate that look like Gaussian functions in the central area, but only become algebraically small in the edge area, with a characteristic "fat tail" exponent, which is referred to in the physical literature as and can assume certain "universal" values. ${\ displaystyle \ sigma}$ ${\ displaystyle dx '/ {\ sqrt {2 \ pi \ sigma}} \, \ cdot e ^ {- {(x-x') ^ {2}} / (2 \ sigma)},}$ ${\ displaystyle \ sim dx '\ cdot | x-x' | ^ {- \ alpha},}$ ${\ displaystyle \ alpha}$

literature

Emil Julius Gumbel : Statistics of extremes . Columbia University Press, New York (1958).
Laurens de Haan , Ana Ferreira: Extreme Value Theory , Springer 2006

Web links

Advanced Extremal Models for Operational Risk (PDF file)
When it comes to head and neck: extreme value theory (PDF file)

Individual evidence

↑ Daniel Gembris: development of athletes set new records . In: Spektrum der Wissenschaft , No. 8, 2008, pp. 14-16.
^ Gregor Wergen, Joachim Krug and Stefan Rahmstorf: Climate records . In: Spektrum der Wissenschaft , No. 2, 2014, pp. 80–87.
^ Rosario N. Mantegna, H. Eugene Stanley : An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, Cambridge 1999, ISBN 0-521-62008-2 .

[1] Daniel Gembris: development of athletes set new records . In: Spektrum der Wissenschaft , No. 8, 2008, pp. 14-16.

[2] Gregor Wergen, Joachim Krug and Stefan Rahmstorf: Climate records . In: Spektrum der Wissenschaft , No. 2, 2014, pp. 80–87.

[3] Rosario N. Mantegna, H. Eugene Stanley : An Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, Cambridge 1999, ISBN 0-521-62008-2 .