User:Salix alba/sandbox

Notation	${\textrm {GEV}}(\ \mu ,\ \sigma ,\ \xi \ )$
Parameters	μ   ∈   ℝ   —   location,   σ   >   0   —   scale,   ξ   ∈   ℝ   —   shape.
Support	x   ∈   [   μ − σ  / ξ , +∞ )   when   ξ > 0 ,   x   ∈   ( −∞, +∞ )        when   ξ = 0 ,   x   ∈   ( −∞,   μ − σ  / ξ ]   when   ξ < 0  .
PDF	$={\tfrac {1}{\ \sigma \ }}\ t(x)^{\xi +1}\ e^{-t(x)}\ ,$ where   $\ t(x)\equiv {\begin{cases}\left[1+\xi \ \left(\ {\tfrac {\ x-\mu \ }{\sigma }}\right)\right]^{-{\tfrac {\ 1\ }{\xi }}}&~{\mathsf {if}}~\xi \neq 0\ ,\\{}\\\exp \left(\ -{\tfrac {\ x-\mu \ }{\sigma }}\ \right)&~{\mathsf {if}}~\xi =0\ ~.\\{}\end{cases}}$
CDF	$=e^{-t(x)}\ ,$   for   $\ x\in$ support (see above)
Mean	$={\begin{cases}{}\\\mu +{\tfrac {\ \sigma \ (g_{1}-1)\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\xi <1\ ,\\{}\\\mu +\sigma \ \gamma &~{\mathsf {if}}~\xi =0\ ,\\{}\\\infty &~{\mathsf {if}}~\xi \geq 1\ ;\\{}\end{cases}}$ where   gk ≡ Γ( 1 − k ξ ) , (see Gamma function) and   $\ \gamma \$   is Euler’s constant.
Median	$={\begin{cases}\mu +\sigma \cdot {\frac {\ (\ln 2)^{-\xi }\ -\ 1\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\\{}\\\mu -\sigma \cdot \ln \ln 2&~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}$
Mode	$={\begin{cases}\mu +\sigma \cdot {\frac {\ (1+\xi )^{-\xi }\ -\ 1\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\\\mu &~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}$
Variance	$={\begin{cases}\sigma ^{2}\cdot {\tfrac {\ g_{2}-g_{1}^{2}\ }{\xi ^{2}}}&~{\mathsf {if}}~\xi \neq 0~{\mathsf {and}}~\xi <{\frac {\ 1\ }{2}}\ ,\\{}\\\sigma ^{2}\cdot {\frac {\ \pi ^{2}\ }{6}}&~{\mathsf {if}}~\xi =0\ ,\\{}\\\infty &~{\mathsf {if}}~\xi \geq {\tfrac {\ 1\ }{2}}~.\\{}\end{cases}}$
Skewness	$={\begin{cases}\ \operatorname {sgn}(\xi )\cdot {\tfrac {\ g_{3}\ -\ 3\ g_{2}g_{1}\ +\ 2\ g_{1}^{3}\ }{\ \left(g_{2}\ -\ g_{1}^{2}\ \right)^{\tfrac {3}{2}}\ }}&~{\mathsf {if}}~\xi \neq 0~{\mathsf {and}}~\xi <{\tfrac {\ 1\ }{3}}\ ,\\{}\\{\frac {\ 12{\sqrt {6\ }}\ \zeta (3)\ }{\pi ^{3}}}&~{\mathsf {if}}~\xi =0\ ;\\{}\end{cases}}$ where $\ \operatorname {sgn}(x)\$ is the sign function and $\ \zeta (x)\$ is the Riemann zeta function
Excess kurtosis	$={\begin{cases}{}\\{\frac {\ g_{4}\ -\ 4\ g_{3}\ g_{1}\ +\ 6\ g_{1}^{2}g_{2}\ -\ 3\ g_{1}^{4}\ }{\left(g_{2}\ -\ g_{1}^{2}\right)^{2}}}&~{\mathsf {if}}~\xi \neq 0~{\mathsf {and}}~\xi <{\tfrac {\ 1\ }{4}}\ ,\\{}\\{\tfrac {12}{\ 5\ }}&~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}$
Entropy	$=\ \ln(\sigma )\ +\ \gamma \ \xi \ +\ \gamma \ +\ 1\$
MGF	see Muraleedharan, Soares & Lucas (2011)
CF	see Muraleedharan, Soares & Lucas (2011)
Expected shortfall	Failed to parse (unknown function "\begin{cases}"): {\displaystyle =\ \begin{cases} \mu + \frac{ \sigma }{\ \xi\ (1 - p )\ } \left[ \Gamma_L \left( 1 - \xi\ , \ln \left( \tfrac{\ 1\ }{ p } \right) \right) - (1 - p) \right]&\ \xi \neq 0 \\ {} \\ \mu + \tfrac{ \sigma }{\ 1 - p \} \left(\ -p\ \ln(\ -\ln(p)\ ) + li(p)\ \right) & ~\mathsf{ if }~ \xi = 0\ ; \\ {} \end{cases}} where $\ \Gamma _{L}(a,b)=\int _{0}^{b}p^{a-1}\ e^{-p}\ \mathrm {d} p\$ is the lower incomplete gamma function and $\ li(x)=\int _{0}^{\alpha }{\tfrac {1}{\ \ln p\ }}\ \mathrm {d} p\$ is the logarithmic integral function.

In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.

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 recognised three different forms outlined below. However usage of this name is sometimes restricted to mean the special case of the Gumbel distribution. The origin of the common functional form for all 3 distributions dates back to at least Jenkinson, A. F. (1955), though allegedly it could also have been given by von Mises, R. (1936).