Generalized Extreme Value
The extreme value theory was developed in order to estimate the probability of the occurrence of the extreme and rare events. It allows to extrapolate the behavior of the probability distribution tails using a large observations. The generalized extreme value distribution includes the Gumbel distribution, Fréchet distribution and Weibull distribution. In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after Ronald Fisher and L. H. C. Tippett who recognized three function forms outlined below. The can give some statistics of random variable such as:
![E(X) = \mu +\frac{\sigma}{\xi} \left( g_1 - 1 \right)\\ \quad\\ Var(X) = \frac{\sigma^2}{\xi^2} (g_2 - g_1 ^{2})\\ \quad\\ Mode(X) = \mu + \frac{\sigma}{\xi} [(1 + \xi)^{- \xi} -1]](../res/eq9.png)
Where
,
is the gamma function.
The skewness is for
:
For
, the sign of the numerator ir reversed.
The kurtosis is:
