Generalized extreme value distribution

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	$\frac{1}{\sigma}\,t(x)^{\xi+1}e^{-t(x)},$ where $t(x) = \begin{cases}\big(1+(\tfrac{x-\mu}{\sigma})\xi\big)^{-1/\xi} & \textrm{if}\ \xi\neq0 \\ e^{-(x-\mu)/\sigma} & \textrm{if}\ \xi=0\end{cases}$
CDF	$e^{-t(x)},\,$ for x ∈ support
Mean	$\begin{cases}\mu + \sigma\frac{\Gamma(1-\xi)-1}{\xi} & \text{if}\ \xi\neq 0,\xi<1,\\ \mu + \sigma\,\gamma & \text{if}\ \xi=0,\\ \infty & \text{if}\ \xi\geq 1,\end{cases}$ where $\gamma$ is Euler’s constant.
Median	$\begin{cases}\mu + \sigma \frac{(\ln2)^{-\xi}-1}{\xi} & \text{if}\ \xi\neq0,\\ \mu - \sigma \ln\ln2 & \text{if}\ \xi=0.\end{cases}$
Mode	$\begin{cases}\mu + \sigma \frac{(1+\xi)^{-\xi}-1}{\xi} & \text{if}\ \xi\neq0,\\ \mu & \text{if}\ \xi=0.\end{cases}$
Variance	$\begin{cases}\sigma^2\,(g_2-g_1^2)/\xi^2 & \text{if}\ \xi\neq0,\xi<\frac12,\\ \sigma^2\,\frac{\pi^2}{6} & \text{if}\ \xi=0, \\ \infty & \text{if}\ \xi\geq\frac12,\end{cases}$ where g_k = Γ(1 − kξ).
Skewness	$\begin{cases}\frac{g_3-3g_1g_2+2g_1^3}{(g_2-g_1^2)^{3/2}} & \text{if}\ \xi >0,\\ -\frac{g_3-3g_1g_2+2g_1^3}{(g_2-g_1^2)^{3/2}} & \text{if}\ \xi <0,\\ \frac{12 \sqrt{6} \zeta(3)}{\pi^3} & \text{if}\ \xi=0.\end{cases}$ where $\zeta (x)$ is Riemann zeta function
Ex. kurtosis	$\begin{cases}\frac{g_4-4g_1g_3+6g_2g_1^2-3g_1^4}{(g_2-g_1^2)^{2}}-3 & \text{if}\ \xi\neq0,\xi<\frac{1}{4},\\ \frac{12}{5} & \text{if}\ \xi=0,\\ \infty & \text{if}\ \xi\ge\frac{1}{4}.\end{cases}$
Entropy	$\log(\sigma)\,+\,\gamma\xi\,+\,(\gamma+1)$
MGF	^[1]
CF	^[1]

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 need not exist: this 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 function forms outlined below. However usage of this name is sometimes restricted to mean the special case of the Gumbel distribution.

Specification

The generalized extreme value distribution has cumulative distribution function

F(x;\mu,\sigma,\xi) = \exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}

for $1+\xi(x-\mu)/\sigma>0$ , where $\mu\in\mathbb R$ is the location parameter, $\sigma >0$ the scale parameter and $\xi\in\mathbb R$ the shape parameter. Thus for $\xi>0$ , the expression just given for the cumulative distribution function is valid for $x > \mu-\sigma/\xi$ , while for $\xi<0$ it is valid for $x < \mu+ \sigma/(-\xi)$ . In the first case, at the lower end-point it equals 0; in the second case, at the upper end-point, it equals 1. For $\xi = 0$ the expression just given for the cumulative distribution function is formally undefined and is replaced by the result obtained by taking the limit as $\xi\to 0$

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

without any restriction on x.

The density function is, consequently,

f(x;\mu,\sigma,\xi) = \frac{1}{\sigma}\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{(-1/\xi)-1}

\exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}

again, for $x > \mu-\sigma/\xi$ in the case $\xi>0$ , and for $x < \mu+\sigma/(-\xi)$ in the case $\xi<0$ . The density is zero outside of the relevant range. In the case $\xi =0$ the density is positive on the whole real line and equal to

f(x;\mu,\sigma,\xi) = \frac{1}{\sigma}\exp\left[-\left(\frac{x-\mu}{\sigma}\right)\right]

\exp\left\{-\exp\left[\left(-\frac{x-\mu}{\sigma}\right)\right]\right\}

Summary statistics

Some simple statistics of the distribution are:

\operatorname{E}(X) = \mu-\frac{\sigma}{\xi}+\frac{\sigma}{\xi}g_1 ,

\operatorname{Var}(X) = \frac{\sigma^2}{\xi^2}(g_2-g_1^2) ,

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

The skewness is for ξ>0

\operatorname{skewness}(X) = \frac{g_3-3g_1g_2+2g_1^3}{(g_2-g_1^2)^{3/2}}

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

The excess kurtosis is:

\operatorname{kurtosis\ excess}(X) = \frac{g_4-4g_1g_3+6g_2g_1^2-3g_1^4}{(g_2-g_1^2)^{2}}-3 .

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

Link to Fréchet, Weibull and Gumbel families

The shape parameter $\xi$ governs the tail behaviour of the distribution. The sub-families defined by $\xi= 0$ , $\xi>0$ and $\xi<0$ correspond, respectively, to the Gumbel, Fréchet and Weibull families, whose cumulative distribution functions are displayed below.

Gumbel or type I extreme value distribution ( $\xi =0$ )

F(x;\mu,\sigma,0)=e^{-e^{-(x-\mu)/\sigma}}\;\;\; \text{for} \;\; x\in\mathbb R.

Fréchet or type II extreme value distribution, if $\xi=\alpha^{-1}>0$ and $y = 1 + \xi (x-\mu)/\sigma$

F(x;\mu,\sigma,\xi)=\begin{cases} e^{-y^{-\alpha}} & y > 0 \\ 0 & y \leq 0. \end{cases}

Reversed Weibull or type III extreme value distribution, if $\xi=-\alpha^{-1}<0$ and $y=-\left(1+\xi (x-\mu )/\sigma \right)$

F(x;\mu,\sigma,\xi)=\begin{cases} e^{-(-y)^{\alpha}} & y<0 \\ 1 & y\geq 0 \end{cases}

Remark I: The theory here relates to maxima and the distribution being discussed is an extreme value distribution for maxima. A generalised extreme value distribution for minima can be obtained, for example by substituting (−x) for x in the distribution function, and subtracting from one: this yields a separate family of distributions.

Remark II: The ordinary Weibull distribution arises in reliability applications and is obtained from the distribution here by using the variable $t = \mu - x$ , which gives a strictly positive support - in contrast to the use in the extreme value theory here. This arises because the Weibull distribution is used in cases that deal with the minimum rather than the maximum. The distribution here has an addition parameter compared to the usual form of the Weibull distribution and, in addition, is reversed so that the distribution has an upper bound rather than a lower bound. Importantly, in applications of the GEV, the upper bound is unknown and so must be estimated while when applying the Weibull distribution the lower bound is known to be zero.

Remark III: Note the differences in the ranges of interest for the three extreme value distributions: Gumbel is unlimited, Fréchet has a lower limit, while the reversed Weibull has an upper limit. More precisely, Extreme Value Theory (Univariate Theory) describes which of the three is the limiting law according to the initial law X and in particular depending on its tail (e.g. maximum distribution of a heavy tailed law converges to Fréchet).

One can link the type I to types II and III the following way: if the cumulative distribution function of some random variable $X$ is of type II, and with the positive numbers as support, i.e. $F(x; 0, \sigma, \alpha)$ , then the cumulative distribution function of $\ln X$ is of type I, namely $F(x; \ln \sigma, 1/\alpha, 0)$ . Similarly, if the cumulative distribution function of $X$ is of type III, and with the negative numbers as support, i.e. $F(x; 0, \sigma, -\alpha)$ , then the cumulative distribution function of $\ln (-X)$ is of type I, namely $F(x; -\ln \sigma, 1/\alpha, 0)$ .

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.^[2] ^[3]

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.^[4]

Related distributions

If $X \sim \textrm{GEV}(\mu,\,\sigma,\,0)$ then $mX+b \sim \textrm{GEV}(m\mu+b,\,m\sigma,\,0)$
If $X \sim \textrm{Gumbel}(\mu,\,\sigma)$ (Gumbel distribution) then $X \sim \textrm{GEV}(\mu,\,\sigma,\,0)$
If $X \sim \textrm{Weibull}(\sigma,\,\mu)$ (Weibull distribution) then $\mu\left(1-\sigma\mathrm{log}{\tfrac{X}{\sigma}}\right) \sim \textrm{GEV}(\mu,\,\sigma,\,0)$
If $X \sim \textrm{GEV}(\mu,\,\sigma,\,0)$ then $\sigma \exp (-\tfrac{X-\mu}{\mu \sigma} ) \sim \textrm{Weibull}(\sigma,\,\mu)$ (Weibull distribution)
If $X \sim \textrm{Exponential}(1)\,$ (Exponential distribution) then $\mu - \sigma \log{X} \sim \textrm{GEV}(\mu,\,\sigma,\,0)$
If $X \sim \mathrm{GEV}(\alpha,\beta,0)\,$ and $Y \sim \mathrm{GEV}(\alpha,\beta,0)\,$ then $X-Y \sim \mathrm{Logistic}(0,\beta) \,$ (Logistic distribution)
If $X \sim \mathrm{GEV}(\alpha,\beta,0)\,$ and $Y \sim \mathrm{GEV}(\alpha,\beta,0)\,$ then $X+Y \sim \mathrm{Logistic}(2 \alpha,\beta) \,$

Notes

1 2 Muraleedharan. G, C. Guedes Soares and Cláudia Lucas (2011). "Characteristic and Moment Generating Functions of Generalised Extreme Value Distribution (GEV)". In Linda. L. Wright (Ed.), Sea Level Rise, Coastal Engineering, Shorelines and Tides, Chapter-14, pp. 269–276. Nova Science Publishers. ISBN 978-1-61728-655-1
↑ Moscadelli, Marco. "The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee." Available at SSRN 557214 (2004).
↑ Guégan, D.; Hassani, B.K. (2014), "A mathematical resurgence of risk management: an extreme modeling of expert opinions", Frontiers in Finance and Economics, 11 (1): 25–45
↑ Kjersti Aas, lecture, NTNU, Trondheim, 23 Jan 2008

References

Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas (1997). Modelling extremal events for insurance and finance. Berlin: Springer Verlag.
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.
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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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