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Generalized extreme value distribution

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Parameters   μ   ∈   ℝ   —   location,
  σ   >   0   —   scale,
  ξ   ∈   ℝ   —   shape.

  x   ∈   [ μ - σ/ ξ , +∞ )   when   ξ > 0 ,
  x   ∈   ( −∞, +∞ )        when   ξ = 0 ,

  x   ∈   ( −∞, μ - σ/ ξ ]   when   ξ < 0  .

CDF   for   support (see above)

where   gkΓ( 1 − k ξ ) , (see Gamma function)
and     is Euler’s constant.

where is the sign function
and is the Riemann zeta function
Excess kurtosis
MGF see Muraleedharan, Soares & Lucas (2011)[1]
CF see Muraleedharan, Soares & Lucas (2011)[1]
Expected shortfall
where is the lower incomplete gamma function and 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]


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 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

Example of probability density functions for distributions of the GEV family. [8]

Summary statistics[edit]

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 and is the gamma function.

Link to Fréchet, Weibull, and Gumbel families[edit]

The shape parameter governs the tail behavior of the distribution. The sub-families defined by three cases: and 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 for all
  • Type II or Fréchet extreme value distribution, case for all
Let and
  • Type III or reversed Weibull extreme value distribution, case for all
Let and

The subsections below remark on properties of these distributions.

Modification for minima rather than maxima[edit]

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 for in the distribution function, and subtracting the cumulative distribution from one: That is, replace with . Doing so yields yet another family of distributions.

Alternative convention for the Weibull distribution[edit]

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 formulation of extreme value theory here. This arises because the ordinary Weibull distribution is used for 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, whereas when applying the ordinary Weibull distribution in reliability applications the lower bound is usually known to be zero.

Ranges of the distributions[edit]

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 variables[edit]

One can link the type I to types II and III in 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.[9][10]
Fitted GEV probability distribution to monthly maximum one-day rainfalls in October, Surinam[11]
  • 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.[12]
  • In hydrology the GEV distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges.[13] 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.

Example for Normally distributed variables[edit]

Let be i.i.d. normally distributed random variables with mean 0 and variance 1. The Fisher–Tippett–Gnedenko theorem[14] 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 distributions[edit]

  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 also[edit]


  1. ^ a b Muraleedharan, G; Guedes Soares, C.; Lucas, Cláudia (2011). "Characteristic and moment generating functions of generalised extreme value distribution (GEV)". In Wright, Linda L. (ed.). Sea Level Rise, Coastal Engineering, Shorelines and Tides. Nova Science Publishers. Chapter-14, pp. 269–276. ISBN 978-1-61728-655-1.
  2. ^ Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation" (PDF). Annals of Operations Research. 299 (1–2). Springer: 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1. S2CID 254231768. Archived from the original (PDF) on 2023-03-31. Retrieved 2023-02-27.
  3. ^ Weisstein, Eric W. "Extreme Value Distribution". mathworld.wolfram.com. Retrieved 2021-08-06.
  4. ^ Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.
  5. ^ 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. Bibcode:1955QJRMS..81..158J. doi:10.1002/qj.49708134804.
  6. ^ Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.
  7. ^ von Mises, R. (1936). "La distribution de la plus grande de n valeurs". Rev. Math. Union Interbalcanique 1: 141–160.
  8. ^ Norton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2021). "Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation". Annals of Operations Research. 299 (1–2): 1281–1315. arXiv:1811.11301. doi:10.1007/s10479-019-03373-1.
  9. ^ Moscadelli, Marco. "The modelling of operational risk: experience with the analysis of the data collected by the Basel Committee." Available at SSRN 557214 (2004).
  10. ^ 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
  11. ^ CumFreq for probability distribution fitting [1]
  12. ^ Kjersti Aas, lecture, NTNU, Trondheim, 23 Jan 2008
  13. ^ Liu, Xin; Wang, Yu (2022). "Quantifying annual occurrence probability of rainfall-induced landslide at a specific slope". Computers and Geotechnics. 149: 104877. Bibcode:2022CGeot.14904877L. doi:10.1016/j.compgeo.2022.104877. S2CID 250232752.
  14. ^ David, Herbert A.; Nagaraja, Haikady N. (2004). Order statistics. John Wiley & Sons. p. 299.

Further reading[edit]