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Generalized Extreme Value distribution

Quick view

The Generalized Extreme Value distribution, abbreviated GEV, unifies the possible limits of block maxima or minima. Its xi parameter decides whether the tail is Gumbel-, Frechet-, or Weibull-type.

If you are coming from another distribution

xi=0 gives Gumbel as a limit; xi>0 corresponds to Frechet and xi<0 to extreme-value Weibull. The Generalized Pareto models threshold exceedances and is GEV’s natural companion.

History and terminology

Fisher and Tippett classified limiting extreme-value forms in 1928; Gnedenko completed the domain-of-attraction formulation. GEV is the modern synthesis of the three classes.

A familiar situation

GEV answers a block-maximum question: “what is the annual maximum?”. Block size changes the problem’s scale; applying it to ordinary observations can discard information.

Fitting with care

Return-period extrapolation depends on temporal dependence, stationarity, and block choice; a fit p-value does not validate those assumptions.

Three possible destinations for a sequence of maxima

Fisher and Tippett showed that normalized maxima can end in only three limiting types. GEV unifies them through xi: Gumbel at zero, Fréchet for a heavy upper tail, and maxima-type Weibull for a population with a finite upper endpoint.

Some software conventions reverse the sign of xi, so the formula must always be checked. In block analysis, block size trades bias against data quantity: small blocks may not yet be extreme, while large blocks leave few observations. Return levels extrapolate beyond the sample and usually have asymmetric uncertainty. Reporting only the fitted central curve hides the uncertainty that matters most.

Decision guide

A good candidate when: each observation is a block maximum or minimum and the data must determine whether the tail is bounded, light, or heavy.

Compare it with: Gumbel as a submodel and Generalized Pareto for exceedances. Repeat the fit with different block sizes: return levels are sensitive to that choice.

References

  • SciPy reference: scipy.stats.genextreme — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Fisher, R. A. & Tippett, L. H. C. (1928). Limiting forms of the frequency distribution of the largest and smallest member of a sample. Proceedings of the Cambridge Philosophical Society, 24, 180–190.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.

Generalized Extreme Value Distribution: equations and calculator

Distribution defintion

X∼GeneralizedExtremeValue(ξ,μ,σ)X\sim\mathrm{GeneralizedExtremeValue}\left(\xi,\mu,\sigma\right)

Distribution domain

if ξ>0: x∈(z(x),∞),if ξ=0: x∈(−∞,∞),if ξ<0: x∈(−∞,z(x))\text{if }\xi>0:\ x\in\left(z(x),\infty\right),\quad \text{if }\xi=0:\ x\in\left(-\infty,\infty\right),\quad \text{if }\xi<0:\ x\in\left(-\infty,z(x)\right)

Parameters domain and parameters constraints

ξ∈R,μ∈R,σ∈R+\xi\in\mathbb{R},\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)={exp⁡(−exp⁡(−z(x)))if  ξ=0exp⁡(−(1+ξz(x))−1/ξ)if  ξ≠0F_{X}\left(x\right)=\left\{\begin{array}{cl} \exp\Bigl(-\exp(-z(x))\Bigr) & \text{if } ~ \xi=0 \\ \exp\Bigl(-(1+\xi z(x))^{-1/\xi}\Bigr) & \text{if } ~ \xi \neq 0\end{array} \right.

Probability density function

fX(x)={1σexp⁡(−z(x))exp⁡(−exp⁡(−z(x)))if  ξ=01σ(1+ξz(x))−(1+1/ξ)exp⁡(−(1+ξz(x))−1/ξ)if  ξ≠0f_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{1}{\sigma}\exp(-z(x)) \exp\Bigl(-\exp(-z(x))\Bigr) & \text{if } ~ \xi=0 \\ \frac{1}{\sigma}\Bigl(1+\xi z(x)\Bigr)^{-(1+1/\xi)} \exp\Bigl(-(1+\xi z(x))^{-1/\xi}\Bigr) & \text{if } ~ \xi \neq 0\end{array} \right.

Percent point function/Sample

FX−1(u)={μ−σln⁡(−ln⁡(u))if  ξ=0μ+σξ((−ln⁡(u) )−ξ−1)if  ξ≠0F^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} \mu-\sigma\ln\left(-\ln\left(u\right)\right) & \text{if } ~ \xi=0 \\ \mu+\frac{}{}\frac{\sigma}{\xi}\left(\left(-\ln(u)\,\right)^{-\xi}-1\right) & \text{if } ~ \xi \neq 0\\ \end{array} \right.

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx=Γ(1−kξ)\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx=\Gamma(1-k\xi)

Parametric mean

Mean(X)={μ+σ(μ1′−1)/ξif ξ≠0,ξ<1μ+σ γif ξ=0\mathrm{Mean}(X)=\left\{\begin{array}{cl}\mu+\sigma(\mu'_{1}-1)/\xi & \text{if}\ \xi\neq 0,\xi<1\\ \mu+\sigma\,\gamma & \text{if}\ \xi=0\end{array} \right.

Parametric variance

Variance(X)={σ2 (μ2′−μ1′2)/ξ2if ξ≠0,ξ<12σ2 π26if ξ=0\mathrm{Variance}(X)=\left\{\begin{array}{cl}\sigma^2\,(\mu'_{2}-\mu'^{2}_{1})/\xi^2 & \text{if}\ \xi\neq0,\xi<\frac12\\ \sigma^2\,\frac{\pi^2}{6} & \text{if}\ \xi=0\end{array} \right.

Parametric skewness

Skewness(X)={sign(ξ)μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5if ξ≠0,ξ<13126 ζ(3)π3if ξ=0\mathrm{Skewness}(X)=\left\{\begin{array}{cl}\text{sign}(\xi)\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}} & \text{if}\ \xi\neq0,\xi<\frac{1}{3} \\ \frac{12 \sqrt{6}\,\zeta(3)}{\pi^3} & \text{if}\ \xi=0\end{array} \right.

Parametric kurtosis

Kurtosis(X)={3+μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2if ξ≠0,ξ<143+125if ξ=0\mathrm{Kurtosis}(X)=\left\{\begin{array}{cl}3+\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}} & \text{if}\ \xi\neq0,\xi<\frac{1}{4}\\ 3+\frac{12}{5} & \text{if}\ \xi=0\end{array} \right.

Parametric median

Median(X)={μ+σ(ln⁡2)−ξ−1ξif  ξ≠0μ−σln⁡ln⁡2if ξ=0\mathrm{Median}(X)=\left\{\begin{array}{cl}\mu+\sigma\frac{(\ln2)^{-\xi}-1}{\xi} & \text{if }\ \xi\neq0\\ \mu-\sigma \ln\ln2 & \text{if}\ \xi=0\end{array} \right.

Parametric mode

Mode(X)={μ+σ(1+ξ)−ξ−1ξif  ξ≠0μif  ξ=0\mathrm{Mode}(X)=\left\{\begin{array}{cl}\mu+\sigma\frac{(1+\xi)^{-\xi}-1}{\xi} & \text{if }\ \xi\neq0\\ \mu & \text{if }\ \xi=0\end{array} \right.

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}
γ:Euler-Mascheroni constant=0.5772156649\gamma:\text{Euler-Mascheroni constant}=0.5772156649