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Gumbel Right distribution

What it describes

Gumbel Right is the classical distribution of extreme maxima. It models the maximum in each block when the underlying variable belongs to the type-I domain of attraction.

It spans the real line and has an exponentially decaying right tail. mu controls position and sigma scale; it imposes no finite upper endpoint.

History and names

Gumbel popularized extreme-value statistics and applications in hydrology, engineering, and meteorology. The law is also called type-I extreme value.

How it connects to other distributions

It is the xi=0 case of GEV and the reflection of Gumbel Left. Power-law tails favour Frechet; maxima with a finite upper endpoint favour the Weibull GEV regime.

Where it appears

  • annual maxima of flood, rainfall, wind, or load
  • structural design and return-level analysis

Fitting cautions

Choosing Gumbel by habit can hide Frechet tails or a finite-endpoint Weibull regime; comparing a free-shape GEV is informative but not definitive.

The record across many light-tailed seasons

When the original distribution has a relatively light tail, such as Normal or Exponential, normalized block maxima may converge to Gumbel. The shape became known as type I extreme value, and Gumbel developed it into a practical tool for floods and other records.

Its right tail decays exponentially and has no finite upper endpoint. The standard mean is displaced from the location parameter by Euler’s constant. In design work, the important object is usually a return level and its uncertainty rather than the curve’s mode. Climate trends, measurement changes, or dependence between seasons break the idea of identically distributed blocks and call for a nonstationary model.

Decision guide

A good candidate when: block maxima are modelled and the tail reasonably belongs to the zero-shape limiting case of GEV.

Compare it with: GEV to allow bounded or heavy tails. A reasonable central plot is not enough: compare return levels and their uncertainty.

References

  • SciPy reference: scipy.stats.gumbel_r — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Gumbel, E. J. (1958). Statistics of Extremes. Columbia University Press.
  • 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.

Gumbel Right Distribution: equations and calculator

Distribution defintion

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

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

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

Cumulative distribution function

FX(x)=exp⁡(−e−z(x))F_{X}\left(x\right)=\exp\left(-e^{-z(x)}\right)

Probability density function

fX(x)=1σexp⁡(−(z(x)+e−z(x)))f_{X}\left(x\right)=\frac{1}{\sigma}\exp\left(-\left(z(x)+e^{-z(x)}\right)\right)

Percent point function/Sample

FX−1(u)=μ~−σln⁡(−ln⁡(u))F^{-1}_{X}\left(u\right)=\tilde{\mu}-\sigma\ln\left(-\ln\left(u\right)\right)

Non-central parametric moments

μ~k′=E[X~k]=∫−∞∞xkfX~(x)dx\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{-\infty}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx

Parametric mean

Mean(X)=μ+σμ~1′=μ+γσ\mathrm{Mean}(X)=\mu+\sigma\tilde{\mu}'_{1}=\mu+\gamma\sigma

Parametric variance

Variance(X)=σ2(μ~2′−μ~1′2)=σ2π26\mathrm{Variance}(X)=\sigma^{2}(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\sigma^{2}\frac{\pi^{2}}{6}

Parametric skewness

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5=126ζ(3)π3\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}=\frac{12\sqrt{6}\zeta(3)}{\pi^{3}}

Parametric kurtosis

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2=3+125\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}=3+\frac{12}{5}

Parametric median

Median(X)=μ−σln⁡(−ln⁡(12))\mathrm{Median}(X)=\mu-\sigma\ln\left(-\ln\left(\frac{1}{2}\right)\right)

Parametric mode

Mode(X)=μ\mathrm{Mode}(X)=\mu

Additional information and definitions

X~∼GumbelRight(0,1)\tilde{X}\sim\mathrm{GumbelRight}\left(0,1\right)
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:Euler-Mascheroni constant=0.5772156649\gamma:\text{Euler-Mascheroni constant}=0.5772156649
ζ(3):Apeˊry’s constant=1.2020569031\zeta(3):\text{Apéry's constant}=1.2020569031