Extreme value theory

The extreme value theory is a mathematical discipline that deals with outliers, that is, maximum and minimum values ​​of samples employed.

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

Formal: There are independent and identically distributed random variable with values ​​in the real numbers and their maximum. Furthermore, the distribution function call of, and let G be a non- degenerate distribution function - not a function that can take only one value. If then there are consequences, so that the convergence holds, then G can only be one of the following distributions, depending on whether the foothills of the distribution fall off exponentially, fall polynomial, or reach the value zero at one point:

• Gumbel (Type I ). More specifically, when the variable has a Gumbel distribution, as an extreme value distribution is of type I.
• Fréchet - type (type II). More specifically, when the variable has a Fréchet distribution, as an extreme value distribution has Type II
• Weibull (Type III). More specifically, when the variable has a Weibull distribution, so has an extreme value distribution of type III.

These three distributions ( Jenkinson - von Mises representation ) are parameterized also to a single class. The (or a ) generalized distribution is called extreme value distribution. As parameters are often K, σ and μ used, where K <0, a Type III distribution describes and K > 0 is a type II distribution.

You will find, among other application in financial mathematics and actuarial science.

Typical questions could include:

• How high should a dam be built if you want to be sure that he will be flooded in the next 100 years with a probability of 1 %?
• What is the probability of a stock market crash next year, leading to a price decline of more than 15 %?