Notation	${\displaystyle {\textrm {GEV}}(\ \mu ,\ \sigma ,\ \xi \ )}$
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	${\displaystyle ={\tfrac {1}{\ \sigma \ }}\ t(x)^{\xi +1}\ e^{-t(x)}\ ,}$ where   ${\displaystyle \ t(x)\equiv {\begin{cases}\left[1+\xi \ \left(\ {\tfrac {\ x-\mu \ }{\sigma }}\right)\right]^{-{\tfrac {\ 1\ }{\xi }}}&~{\mathsf {if}}~\xi \neq 0\ ,\\{}\\\exp \left(\ -{\tfrac {\ x-\mu \ }{\sigma }}\ \right)&~{\mathsf {if}}~\xi =0\ ~.\\{}\end{cases}}}$
CDF	${\displaystyle =e^{-t(x)}\ ,}$   for   ${\displaystyle \ x\in }$ 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\$ ;\\{}\end{cases}}} where   gk ≡ Γ( 1 − k ξ ) , (see Gamma function) and   ${\displaystyle \ \gamma \ }$   is Euler’s constant.
Median	${\displaystyle ={\begin{cases}\mu +\sigma \cdot {\frac {\ (\ln 2)^{-\xi }\ -\ 1\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\\{}\\\mu -\sigma \cdot \ln \ln 2&~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}}$
Mode	${\displaystyle ={\begin{cases}\mu +\sigma \cdot {\frac {\ (1+\xi )^{-\xi }\ -\ 1\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\\\mu &~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}}$
Variance	${\displaystyle ={\begin{cases}\sigma ^{2}\cdot {\tfrac {\ g_{2}-g_{1}^{2}\ }{\xi ^{2}}}&~{\mathsf {if}}~\xi \neq 0~{\mathsf {and}}~\xi <{\frac {\ 1\ }{2}}\ ,\\{}\\\sigma ^{2}\cdot {\frac {\ \pi ^{2}\ }{6}}&~{\mathsf {if}}~\xi =0\ ,\\{}\\\infty &~{\mathsf {if}}~\xi \geq {\tfrac {\ 1\ }{2}}~.\\{}\end{cases}}}$
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\$ ;\\{}\end{cases}}} where ${\displaystyle \ \operatorname {sgn}(x)\ }$ is the sign function and ${\displaystyle \ \zeta (x)\ }$ is the Riemann zeta function
Ex. kurtosis	${\displaystyle ={\begin{cases}{}\\{\frac {\ g_{4}\ -\ 4\ g_{3}\ g_{1}\ +\ 6\ g_{1}^{2}g_{2}\ -\ 3\ g_{1}^{4}\ }{\left(g_{2}\ -\ g_{1}^{2}\right)^{2}}}&~{\mathsf {if}}~\xi \neq 0~{\mathsf {and}}~\xi <{\tfrac {\ 1\ }{4}}\ ,\\{}\\{\tfrac {12}{\ 5\ }}&~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}}$
Entropy	${\displaystyle =\ \ln(\sigma )\ +\ \gamma \ \xi \ +\ \gamma \ +\ 1\ }$
MGF	see Muraleedharan, Soares & Lucas (2011)[1]
CF	see Muraleedharan, Soares & Lucas (2011)[1]
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\$ ;\\{\ }\end{cases}}} where ${\displaystyle \ \Gamma _{L}(a,b)=\int _{0}^{b}p^{a-1}\ e^{-p}\ \mathrm {d} p\ }$ is the lower incomplete gamma function and ${\displaystyle \ li(x)=\int _{0}^{\alpha }{\tfrac {1}{\ \ln p\ }}\ \mathrm {d} p\ }$ is the logarithmic integral function.[2]

In probability theory and statistics, the generalized extreme value (GEV) distribution^[3] 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.^[4] 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),^[5] though allegedly^[6] it could also have been given by von Mises, R. (1936).^[7]

Specification

Using the standardized variable ${\displaystyle \ s\equiv {\frac {\ x-\mu \ }{\sigma }}\ ,}$ where ${\displaystyle \ \mu \ ,}$ the location parameter, can be any real number, and ${\displaystyle \ \sigma >0\ }$ is the scale parameter; the cumulative distribution function of the GEV distribution is then

$${\displaystyle F(\ s;\ \xi \ )={\begin{cases}\exp \!{\Bigl (}-e^{-s}{\Bigr )}&~~{\text{ for }}~~\xi =0\ ,\\{}\\\exp \!{\Bigl (}-{\bigl (}1+\xi s{\bigr )}^{-{\tfrac {\ 1\ }{\xi }}}{\Bigr )}&~~{\text{ for }}~~\xi \neq 0~~{\text{ and }}~~\xi \ s>-1\ ,\\{}\\0&~~{\text{ for }}~~\xi >0~~{\text{ and }}~~s\leq -{\tfrac {\ 1\ }{\xi }}\ ,\\{}\\1&~~{\text{ for }}~~\xi <0~~{\text{ and }}~~s\geq {\tfrac {1}{\ |\ \xi \ |\ }}\$$ ;\end{cases}}}

where ${\displaystyle \ \xi \ ,}$ the shape parameter, can be any real number. Thus, for ${\displaystyle \ \xi >0\ ,}$ the expression is valid for ${\displaystyle \ s>-{\tfrac {\ 1\ }{\xi }}\ ,}$ while for ${\displaystyle \ \xi <0\ }$ it is valid for ${\displaystyle \ s<-{\tfrac {\ 1\ }{\xi }}~.}$ In the first case, ${\displaystyle \ -{\tfrac {\ 1\ }{\xi }}\ }$ is the negative, lower end-point, where ${\displaystyle \ F\ }$ is  0 ; in the second case, ${\displaystyle \ -{\tfrac {\ 1\ }{\xi }}\ }$ is the positive, upper end-point, where ${\displaystyle F}$ is 1. For ${\displaystyle \ \xi =0\ }$ the second expression is formally undefined and is replaced with the first expression, which is the result of taking the limit of the second, as ${\displaystyle \ \xi \to 0\ }$ in which case ${\displaystyle \ s\ }$ can be any real number.

In the special case of ${\displaystyle \ x=\mu \ ,}$ so ${\displaystyle \ s=0\ }$ and ${\displaystyle \ F(\ 0;\ \xi \ )=e^{-1}\ \approx 0.368\ }$ for whatever values ${\displaystyle \ \xi \ }$ and ${\displaystyle \ \sigma \ }$ might have.

The probability density function of the standardized distribution is

$${\displaystyle f(\ s;\ \xi \ )={\begin{cases}e^{-s}\exp \!{\Bigl (}-e^{-s}{\Bigr )}&~~{\text{ for }}~~\xi =0,\\{}\\{\Bigl (}\ 1+\xi s\ {\Bigr )}^{-\left(1+{\tfrac {\ 1\ }{\xi }}\right)}\ \exp \!{\Bigl (}-\left(1+\xi s\right)^{\tfrac {\ 1\ }{\xi }}{\Bigr )}&~~{\text{ for }}~~\xi \neq 0~~{\text{ and }}~~\xi \ s>-1\ ,\\{}\\0&~~{\text{ otherwise; }}\end{cases}}}$$

again valid for ${\displaystyle \ s>-{\tfrac {\ 1\ }{\xi }}\ }$ in the case ${\displaystyle \ \xi >0\ ,}$ and for ${\displaystyle \ s<-{\tfrac {\ 1\ }{\xi }}\ }$ in the case ${\displaystyle \ \xi <0~.}$ The density is zero outside of the relevant range. In the case ${\displaystyle \ \xi =0\ }$ the density is positive on the whole real line.

Since the cumulative distribution function is invertible, the quantile function for the GEV distribution has an explicit expression, namely

$${\displaystyle \ Q(\ p;\ \mu ,\ \sigma ,\ \xi \ )={\begin{cases}\mu -\sigma \ \log \!{\Bigl (}-\log(p)\ {\Bigr )}&~{\text{ for }}~\xi =0~{\text{ and }}~p\in (\ 0\ ,\ 1\ )\ ,\\{}\\\mu +\displaystyle {\frac {\ \sigma \ }{\ \xi \ }}\left({\Bigl (}-\log(p)\ {\Bigr )}^{-\xi }-1\right)&~{\text{ for }}~\xi >0~{\text{ and }}~p\in [\ 0\ ,\ 1\ )\ ,\\{}&~~{\text{ or }}~\,\xi <0~{\text{ and }}~p\in \ (\ 0\ ,\ 1\ ]\$$ ;\end{cases}}}

and therefore the quantile density function, ${\displaystyle \ q\equiv {\frac {\;\mathrm {d} \ Q\;}{\mathrm {d} \ p}}\ ,}$ is

${\displaystyle \ q(\ p;\ \sigma ,\ \xi \ )={\frac {\sigma }{\ {\Bigl (}-\log(p)\ {\Bigr )}^{\xi +1}\ p\;}}\quad {\text{ for }}~~p\in (\ 0\ ,\ 1\ )\ ,}$

valid for ${\displaystyle \ \sigma >0\ }$ and for any real ${\displaystyle \ \xi ~.}$

Summary statistics

Some simple statistics of the distribution are:

${\displaystyle \operatorname {\mathbb {E} } (X)=\mu +(g_{1}-1){\frac {\sigma }{\xi }}}$ for ${\displaystyle \xi <1}$

${\displaystyle \operatorname {Var} (X)=(g_{2}-g_{1}^{2}){\frac {\sigma ^{2}}{\xi ^{2}}},}$

${\displaystyle \operatorname {Mode} (X)=\mu +{\frac {\sigma }{\xi }}[(1+\xi )^{-\xi }-1].}$

The skewness is for ξ>0

${\displaystyle \operatorname {skewness} (X)={\frac {g_{3}-3g_{2}g_{1}+2g_{1}^{3}}{(g_{2}-g_{1}^{2})^{3/2}}}}$

For ξ < 0, the sign of the numerator is reversed.

The excess kurtosis is:

${\displaystyle \operatorname {kurtosis\ excess} (X)={\frac {g_{4}-4g_{3}g_{1}+6g_{2}g_{1}^{2}-3g_{1}^{4}}{(g_{2}-g_{1}^{2})^{2}}}-3~.}$

where ${\displaystyle \ g_{k}=\Gamma (1-k\ \xi )\ ,}$ ${\displaystyle \ k=1,2,3,4\ ,}$ and ${\displaystyle \ \Gamma (t)\ }$ is the gamma function.

Link to Fréchet, Weibull, and Gumbel families

The shape parameter ${\displaystyle \ \xi \ }$ governs the tail behavior of the distribution. The sub-families defined by three cases: ${\displaystyle \ \xi =0\ ,}$ ${\displaystyle \ \xi >0\ ,}$ and ${\displaystyle \ \xi <0\$ ;} these correspond, respectively, to the Gumbel, Fréchet, and Weibull families, whose cumulative distribution functions are displayed below.

• Type I or Gumbel extreme value distribution, case ${\displaystyle ~\xi =0\ ,\quad }$ for all ${\displaystyle \quad x\in {\Bigl (}\ -\infty \ ,\ +\infty \ {\Bigr )}\$ :}

${\displaystyle F(\ x;\ \mu ,\ \sigma ,\ 0\ )=\exp \left(-\exp \left(-{\frac {\ x-\mu \ }{\sigma }}\right)\right)~.}$

• Type II or Fréchet extreme value distribution, case ${\displaystyle ~\xi >0\ ,\quad }$ for all ${\displaystyle \quad x\in \left(\ \mu -{\tfrac {\sigma }{\ \xi \ }}\ ,\ +\infty \ \right)\$ :}

Let ${\displaystyle \quad \alpha \equiv {\tfrac {\ 1\ }{\xi }}>0\quad }$ and ${\displaystyle \quad y\equiv 1+{\tfrac {\xi }{\sigma }}(x-\mu )\$ ;}

${\displaystyle F(\ x;\ \mu ,\ \sigma ,\ \xi \ )={\begin{cases}0&y\leq 0\quad {\mathsf {~or\ equiv.~}}\quad x\leq \mu -{\tfrac {\sigma }{\ \xi \ }}\\\exp \left(-{\frac {1}{~y^{\alpha }\ }}\right)&y>0\quad {\mathsf {~or\ equiv.~}}\quad x>\mu -{\tfrac {\sigma }{\ \xi \ }}~.\end{cases}}}$

• Type III or reversed Weibull extreme value distribution, case ${\displaystyle ~\xi <0\ ,\quad }$ for all ${\displaystyle \quad x\in \left(-\infty \ ,\ \mu +{\tfrac {\sigma }{\ |\ \xi \ |\ }}\ \right)\$ :}

Let ${\displaystyle \quad \alpha \equiv -{\tfrac {1}{\ \xi \ }}>0\quad }$ and ${\displaystyle \quad y\equiv 1-{\tfrac {\ |\ \xi \ |\ }{\sigma }}(x-\mu )\$ ;}

${\displaystyle F(\ x;\ \mu ,\ \sigma ,\ \xi \ )={\begin{cases}\exp \left(-y^{\alpha }\right)&y>0\quad {\mathsf {~or\ equiv.~}}\quad x<\mu +{\tfrac {\sigma }{\ |\ \xi \ |\ }}\\1&y\leq 0\quad {\mathsf {~or\ equiv.~}}\quad x\geq \mu +{\tfrac {\sigma }{\ |\ \xi \ |\ }}~.\end{cases}}}$

The subsections below remark on properties of these distributions.

Link to logit models (logistic regression)

Multinomial logit models, and certain other types of logistic regression, can be phrased as latent variable models with error variables distributed as Gumbel distributions (type I generalized extreme value distributions). This phrasing is common in the theory of discrete choice models, which include logit models, probit models, and various extensions of them, and derives from the fact that the difference of two type-I GEV-distributed variables follows a logistic distribution, of which the logit function is the quantile function. The type-I GEV distribution thus plays the same role in these logit models as the normal distribution does in the corresponding probit models.

Properties

The cumulative distribution function of the generalized extreme value distribution solves the stability postulate equation. The generalized extreme value distribution is a special case of a max-stable distribution, and is a transformation of a min-stable distribution.

Applications

• The GEV distribution is widely used in the treatment of "tail risks" in fields ranging from insurance to finance. In the latter case, it has been considered as a means of assessing various financial risks via metrics such as value at risk.^[8][9]

Fitted GEV probability distribution to monthly maximum one-day rainfalls in October, Surinam^[10]

• However, the resulting shape parameters have been found to lie in the range leading to undefined means and variances, which underlines the fact that reliable data analysis is often impossible.^[11]
• In hydrology the GEV distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges.^[12] The blue picture, made with CumFreq, illustrates an example of fitting the GEV distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

Related distributions

1. If ${\displaystyle \ X\sim {\textrm {GEV}}(\mu ,\,\sigma ,\,\xi )\ }$ then ${\displaystyle \ mX+b\sim {\textrm {GEV}}(m\mu +b,\ m\sigma ,\ \xi )\ }$
2. If ${\displaystyle \ X\sim {\textrm {Gumbel}}(\mu ,\ \sigma )\ }$ (Gumbel distribution) then ${\displaystyle \ X\sim {\textrm {GEV}}(\mu ,\,\sigma ,\,0)\ }$
3. If ${\displaystyle \ X\sim {\textrm {Weibull}}(\sigma ,\,\mu )\ }$ (Weibull distribution) then ${\displaystyle \ \mu \left(1-\sigma \log {\tfrac {X}{\sigma }}\right)\sim {\textrm {GEV}}(\mu ,\,\sigma ,\,0)\ }$
4. If ${\displaystyle \ X\sim {\textrm {GEV}}(\mu ,\,\sigma ,\,0)\ }$ then ${\displaystyle \ \sigma \exp(-{\tfrac {X-\mu }{\mu \sigma }})\sim {\textrm {Weibull}}(\sigma ,\,\mu )\ }$ (Weibull distribution)
5. If ${\displaystyle \ X\sim {\textrm {Exponential}}(1)\ }$ (Exponential distribution) then ${\displaystyle \ \mu -\sigma \log X\sim {\textrm {GEV}}(\mu ,\,\sigma ,\,0)\ }$
6. If ${\displaystyle \ X\sim \mathrm {Gumbel} (\alpha _{X},\beta )\ }$ and ${\displaystyle \ Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )\ }$ then ${\displaystyle \ X-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\ }$ (see Logistic distribution).
7. If ${\displaystyle \ X\ }$ and ${\displaystyle \ Y\sim \mathrm {Gumbel} (\alpha ,\beta )\ }$ then ${\displaystyle \ X+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )\ }$ (The sum is not a logistic distribution).

Note that ${\displaystyle \ \operatorname {\mathbb {E} } \{\ X+Y\ \}=2\alpha +2\beta \gamma \neq 2\alpha =\operatorname {\mathbb {E} } \left\{\ \operatorname {Logistic} (2\alpha ,\beta )\ \right\}~.}$

See also

• Extreme value theory (univariate theory)
• Fisher–Tippett–Gnedenko theorem
• Generalized Pareto distribution
• German tank problem, opposite question of population maximum given sample maximum
• Pickands–Balkema–De Haan theorem

References

Further reading

• Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas (1997). Modelling extremal events for insurance and finance. Berlin: Springer Verlag. ISBN 9783540609315.
• Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and related properties of random sequences and processes. Springer-Verlag. ISBN 0-387-90731-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Resnick, S.I. (1987). Extreme values, regular variation and point processes. Springer-Verlag. ISBN 0-387-96481-9.
• Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag. ISBN 1-85233-459-2.