Generalized extreme value distribution

In probability theory and statistics, the generalized extreme value (GEV) distribution[2] 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.[3] 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.

Parameters μRlocation,
σ > 0 — scale,
Support x ∈ [ μσ / ξ, +∞)   when ξ > 0,
x ∈ (−∞, +∞)   when ξ = 0,
x ∈ (−∞, μσ / ξ ]   when ξ < 0.


CDF   for x ∈ support

where gk = Γ(1 − ),
and is Euler’s constant.
Variance .
where is the sign function
and is the Riemann zeta function
Ex. kurtosis
MGF [1]
CF [1]

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),[4] though allegedly[5] it could also have been given by Mises, R. (1936).[6]


Using the standardized variable   where   the location parameter, can be any real number, and   is the scale parameter; the cumulative distribution function of the GEV distribution is then


where   the shape parameter, can be any real number. Thus for  , the expression is valid for   while for   it is valid for   In the first case,   is the negative, lower end-point, where   is 0; in the second case,   is the positive, upper end-point, where   is 1. For   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   in which case   can be any real number.

In the special case of the mean   so   and    for whatever values   and   might have.

The probability density function of the standardized distribution is


again valid for   in the case   and for   in the case   The density is zero outside of the relevant range. In the case   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


and therefore the quantile density function   is


valid for   and for any real  


Summary statisticsEdit

Some simple statistics of the distribution are:[citation needed]


The skewness is for ξ>0


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

The excess kurtosis is:


where  , k=1,2,3,4, and   is the gamma function.

Link to Fréchet, Weibull and Gumbel familiesEdit

The shape parameter   governs the tail behavior of the distribution. The sub-families defined by  ,   and   correspond, respectively, to the Gumbel, Fréchet and Weibull families, whose cumulative distribution functions are displayed below.

  • Gumbel or type I extreme value distribution ( )
  • Fréchet or type II extreme value distribution, if   and  
  • Reversed Weibull or type III extreme value distribution, if   and  

The subsections below remark on properties of these distributions.

Modification for minima rather than maximaEdit

The theory here relates to data maxima and the distribution being discussed is an extreme value distribution for maxima. A generalised extreme value distribution for data 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.

Alternative convention for the Weibull distributionEdit

The ordinary Weibull distribution arises in reliability applications and is obtained from the distribution here by using the variable  , which gives a strictly positive support - in contrast to the use in the extreme value theory here. This arises because the ordinary Weibull distribution is used in cases that deal with data minima rather than data maxima. 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 ordinary Weibull distribution in reliability applications the lower bound is usually known to be zero.

Ranges of the distributionsEdit

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.

Distribution of log variablesEdit

One can link the type I to types II and III the following way: if the cumulative distribution function of some random variable   is of type II, and with the positive numbers as support, i.e.  , then the cumulative distribution function of   is of type I, namely  . Similarly, if the cumulative distribution function of   is of type III, and with the negative numbers as support, i.e.  , then the cumulative distribution function of   is of type I, namely  .

Link to logit models (logistic regression)Edit

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.


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


  • 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.[7][8]
Fitted GEV probability distribution to monthly maximum one-day rainfalls in October, Surinam[9]
  • 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.[10]

Example for Normally distributed variablesEdit

Let   be i.i.d. normally distributed random variables with mean 0 and variance 1. The Fisher–Tippett–Gnedenko theorem tells us that  , where


This allow us to estimate e.g. the mean of   from the mean of the GEV distribution:

  where   is the Euler–Mascheroni constant.

Related distributionsEdit

  1. If   then  
  2. If   (Gumbel distribution) then  
  3. If   (Weibull distribution) then  
  4. If   then   (Weibull distribution)
  5. If   (Exponential distribution) then  
  6. If   and   then   (see Logistic_distribution).
  7. If   and   then   (The sum is not a logistic distribution). Note that  .


4. Let  , then the cumulative distribution of   is:


which is the cdf for  .

5. Let  , then the cumulative distribution of   is:


which is the cumulative distribution of  .

See alsoEdit


  1. ^ a b 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
  2. ^ Weisstein, Eric W. "Extreme Value Distribution". Retrieved 2021-08-06.
  3. ^ Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.
  4. ^ Jenkinson, Arthur F (1955). "The frequency distribution of the annual maximum (or minimum) values of meteorological elements". Quarterly Journal of the Royal Meteorological Society. 81 (348): 158–171. doi:10.1002/qj.49708134804.
  5. ^ Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.
  6. ^ Mises, R. von. (1936). "La distribution de la plus grande de n valeurs". Rev. Math. Union Interbalcanique 1: 141–160.
  7. ^ Moscadelli, Marco. "The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee." Available at SSRN 557214 (2004).
  8. ^ 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, SSRN 2558747
  9. ^ CumFreq for probability distribution fitting [1]
  10. ^ Kjersti Aas, lecture, NTNU, Trondheim, 23 Jan 2008

Further readingEdit