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Gumbel Left distribution

Portrait

Gumbel Left describes extreme minima: the smallest value in each block after normalization. It is the reflected version of maximum Gumbel and has a long tail toward the left.

Historical trail

Emil Julius Gumbel developed extreme-value statistics and applications to floods and minima. Gumbel is the light-tailed regime in the Fisher–Tippett classification.

Two useful connections

  • If X is maximum Gumbel, -X is minimum Gumbel with reflected parameters. Both are the xi=0 case of GEV; GPD models threshold excesses in the corresponding direction.
  • GEV allows the tail index to depart from zero; left Gumbel is appropriate only when that simplification is supported.

A use case

The extreme direction is the main decision: hazards may require maxima, while strength or minimum temperature requires minima. Flipping a sign without changing the question changes the analysis.

Diagnostic advice

Do not fit maximum Gumbel directly to minima without transforming the variable or changing convention; quantiles would be reversed.

Records broken in the downward direction

The lowest annual temperature, fastest response time, or weakest item in a batch all concern the left extreme. Gumbel Left is the reflection of the maxima law: changing sign turns minima into maxima and lets the same theory be reused.

Reflection also reverses the direction of the long tail. Signs for location and return levels therefore require care when moving between software packages. GEV handles minima through an equivalent transformation and can test whether the Gumbel shape, whose tail index is zero, is too restrictive. One minimum per block discards central information in exchange for focusing directly on extreme risk.

Decision guide

A good candidate when: block minima with an exponential-type tail are modelled using an explicit orientation toward small events.

Compare it with: Gumbel Right applied to negated data and GEV when tail type is uncertain. Always document sign transformations to avoid reversing extreme quantiles.

References

Gumbel Left Distribution: equations and calculator

Distribution defintion

X∼GumbelLeft(μ,σ)X\sim\mathrm{GumbelLeft}\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)=1−exp⁡(−ez(x))F_{X}\left(x\right)=1-\exp\left(-e^{z(x)}\right)

Probability density function

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

Percent point function/Sample

FX−1(u)=μ+σln⁡(−ln⁡(1−u))F^{-1}_{X}\left(u\right)=\mu+\sigma\ln\left(-\ln\left(1-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~∼GumbelLeft(0,1)\tilde{X}\sim\mathrm{GumbelLeft}\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