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Notation | |||
---|---|---|---|
Parameters |
μ ∈ ℝ — location, σ > 0 — scale, ξ ∈ ℝ — shape. | ||
Support |
x ∈ [ μ − σ / ξ , +∞ ) when ξ > 0 , | ||
| |||
CDF | for support (see above) | ||
Mean |
and is Euler’s constant. | ||
Median | |||
Mode | |||
Variance | |||
Skewness |
and is the Riemann zeta function | ||
Excess kurtosis | |||
Entropy | |||
MGF | see Muraleedharan, Soares & Lucas (2011)[1] | ||
CF | see Muraleedharan, Soares & Lucas (2011)[1] | ||
Expected shortfall |
Failed to parse (unknown function "\begin{cases}"): {\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 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]
- ^ a b Cite error: The named reference
Muraldhrn-Soares-Lucas-2011
was invoked but never defined (see the help page). - ^ Cite error: The named reference
norton
was invoked but never defined (see the help page). - ^ Weisstein, Eric W. "Extreme Value Distribution". mathworld.wolfram.com. Retrieved 2021-08-06.
- ^ Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.
- ^ 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.
- ^ Haan, Laurens; Ferreira, Ana (2007). Extreme value theory: an introduction. Springer.
- ^ von Mises, R. (1936). "La distribution de la plus grande de n valeurs". Rev. Math. Union Interbalcanique 1: 141–160.