Generalized extreme value distribution

Notation
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	; where
CDF	for support (see above)
Mean	; where gk ≡ Γ( 1 − k ξ ) , (see Gamma function); and is Euler’s constant.
Median
Mode
Variance
Skewness	; where is the sign function ; and is the Riemann zeta function
Ex. kurtosis
Entropy
MGF	see Muraleedharan, Soares & Lucas (2011)
CF	see Muraleedharan, Soares & Lucas (2011)
Expected shortfall	; where is the lower incomplete gamma function and 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).

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Notation	${\textrm {GEV}}(\ \mu ,\ \sigma ,\ \xi \ )$
Parameters	$μ \in ℝ$ — location, $σ > 0$ — scale, $ξ \in ℝ$ — shape.
Support	$x ∈ [ .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px} μ − σ / ξ , +∞ )$ when $ξ > 0$ , $x \in ( -\infty, +\infty )$ when $ξ = 0$ , $x \in ( -\infty, μ - σ / ξ]$ 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	${\displaystyle ={\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\$ where $g k \equiv Γ ( 1 - k ξ)$ , (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	${\displaystyle ={\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\$ where $\ \operatorname {sgn}(x)\$ is the sign function and $\ \zeta (x)\$ is the Riemann zeta function
Ex. 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	${\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\$ 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.