Extreme value distribution

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The general extreme value distribution is a continuous probability distribution . It plays an outstanding role in extreme value theory , as it summarizes the essential possible distributions of extreme values ​​in a sample in one representation.

definition

A continuous random variable satisfies a Fisher-Tippett distribution with the parameters , and , if it is the probability density ${\displaystyle X}$ ${\displaystyle a>0}$ ${\displaystyle b>0}$ ${\displaystyle c\in \mathbb {R} }$

${\displaystyle f(x)={\frac {1}{b}}\left(1-ce^{-{\frac {x-a}{b}}}\right)^{{\frac {1}{c}}-1}e^{\left(1-c{\frac {x-a}{b}}\right)^{\frac {1}{c}}}}$

owns.

Double exponential distribution

The special case with a distribution function is used as a double exponential distribution

${\displaystyle F_{P}(x)=\exp(-\exp(-x)))}$

designated.

Relationship to other distributions

The extreme value distribution goes with the parameter to the Fisher-Tippett distribution or Gumbel distribution . ${\displaystyle c=1}$

See also

• Rossi distribution
• Fréchet distribution
• Extreme value theory

Individual evidence

1. ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .