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Frechet distribution

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The Frechet distribution models positive maxima with a heavy tail. It is one of the three limiting laws of extreme-value theory and represents the regime where extreme observations decay slowly.

If you are coming from another distribution

It is the type-II extreme-value form and is linked to Pareto through power transformations. The GEV unifies the three classes; Frechet corresponds to positive extreme-value shape under a common convention.

History and terminology

Maurice Fréchet studied limiting laws for sample maxima in 1927. Together with Gumbel and Weibull, it forms the classical Fisher–Tippett–Gnedenko classification.

A familiar situation

It should be used for block maxima or mechanisms supporting a regular power tail. In extremes, a small change in alpha can greatly change extrapolated quantiles.

Fitting with care

Do not extrapolate long return periods from a few maxima without uncertainty intervals, threshold diagnostics, and stability checks.

The maximum when the population is already heavy-tailed

In 1927 Fréchet identified a limiting law for maxima drawn from populations with power tails. After recording the largest annual flow, claim, or load in each block, suitable normalization may lead to this shape.

Its heavy tail has operational consequences. Only moments below the relevant tail index exist, so a theoretical mean or variance may be absent. Fréchet is one of the three regimes unified by GEV and corresponds to an unbounded upper endpoint. A sample containing one spectacular maximum does not establish its domain of attraction. Diagnosis needs multiple blocks, a stable tail index, and block definitions that do not destroy independence.

Decision guide

A good candidate when: block maxima are being modelled and evidence supports a heavy tail with no finite upper endpoint.

Compare it with: GEV to estimate tail type and Generalized Pareto for threshold exceedances. Do not mix block maxima with ordinary observations.

References

  • SciPy reference: scipy.stats.invweibull — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.
  • Frechet, M. (1927). Sur la loi de probabilité de l’écart maximum. Annales de la Société Polonaise de Mathématique, 6, 93–116.
  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.

Frechet Distribution: equations and calculator

Distribution defintion

X∼Frechet(α,Loc,Sc)X\sim\mathrm{Frechet}\left(\alpha,\text{Loc},\text{Sc}\right)

Distribution domain

x∈[Loc,∞)x\in\left[\text{Loc},\infty\right)

Parameters domain and parameters constraints

α∈R+,Loc∈R,Sc∈R+\alpha\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=e(−z(x))−αF_{X}\left(x\right)=e^{(-z(x))^{-\alpha}}

Probability density function

fX(x)=αSc (z(x))−1−α e−(z(x))−αf_{X}\left(x\right)=\frac{\alpha}{\text{Sc}} \,\left(z(x)\right)^{-1-\alpha} \,e^{-(z(x))^{-\alpha}}

Percent point function/Sample

FX−1(u)=Loc+Sc(−ln⁡(u))−1αF^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\left(-\ln(u)\right)^{-\frac{1}{\alpha}}

Non-central parametric moments

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

Parametric mean

Mean(X)=Loc+Sc⋅μ~1′if α>1\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot\tilde{\mu}'_{1} \quad \text{if } \alpha>1

Parametric variance

Variance(X)=Sc2⋅(μ~2′−μ~1′2)if α>2\mathrm{Variance}(X)=\text{Sc}^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}) \quad \text{if } \alpha>2

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5if α>3\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}} \quad \text{if } \alpha>3

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2if α>4\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}} \quad \text{if } \alpha>4

Parametric median

Median(X)=Loc+Scln⁡(2)α\mathrm{Median}(X)=\text{Loc}+\frac{\text{Sc}}{\sqrt[\alpha]{\ln(2)}}

Parametric mode

Mode(X)=Loc+Sc(α1+α)1/α\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}

Additional information and definitions

Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}