Fréchet distribution

Fréchet

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha \in (0,\infty )}$ shape. (Optionally, two more parameters) ${\displaystyle s\in (0,\infty )}$ scale (default: ${\displaystyle s=1\,}$) ${\displaystyle m\in (-\infty ,\infty )}$ location of minimum (default: ${\displaystyle m=0\,}$)
Support	${\displaystyle x>m}$
PDF	${\displaystyle {\frac {\alpha }{s}}\;\left({\frac {x-m}{s}}\right)^{-1-\alpha }\;e^{-({\frac {x-m}{s}})^{-\alpha }}}$
CDF	${\displaystyle e^{-({\frac {x-m}{s}})^{-\alpha }}}$
Mean	${\displaystyle {\begin{cases}\ m+s\Gamma \left(1-{\frac {1}{\alpha }}\right)&{\text{for }}\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}}$
Median	${\displaystyle m+{\frac {s}{\sqrt[{\alpha }]{\log _{e}(2)}}}}$
Mode	${\displaystyle m+s\left({\frac {\alpha }{1+\alpha }}\right)^{1/\alpha }}$
Variance	${\displaystyle {\begin{cases}\ s^{2}\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\left(\Gamma \left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right)&{\text{for }}\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}}$
Skewness	${\displaystyle {\begin{cases}\ {\frac {\Gamma \left(1-{\frac {3}{\alpha }}\right)-3\Gamma \left(1-{\frac {2}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+2\Gamma ^{3}\left(1-{\frac {1}{\alpha }}\right)}{\sqrt {\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right)^{3}}}}&{\text{for }}\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}}$
Excess kurtosis	${\displaystyle {\begin{cases}\ -6+{\frac {\Gamma \left(1-{\frac {4}{\alpha }}\right)-4\Gamma \left(1-{\frac {3}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+3\Gamma ^{2}\left(1-{\frac {2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right]^{2}}}&{\text{for }}\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}}$
Entropy	${\displaystyle 1+{\frac {\gamma }{\alpha }}+\gamma +\ln \left({\frac {s}{\alpha }}\right)}$, where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.
MGF	[1] Note: Moment ${\displaystyle k}$ exists if ${\displaystyle \alpha >k}$
CF	[1]

The Fréchet distribution, also known as inverse Weibull distribution,^[2][3] is a special case of the generalized extreme value distribution. It has the cumulative distribution function

${\displaystyle \Pr(X\leq x)=e^{-x^{-\alpha }}{\text{ if }}x>0.}$

where α > 0 is a shape parameter. It can be generalised to include a location parameter m (the minimum) and a scale parameter s > 0 with the cumulative distribution function

${\displaystyle \Pr(X\leq x)=e^{-\left({\frac {x-m}{s}}\right)^{-\alpha }}{\text{ if }}x>m.}$

Named for

The single parameter Fréchet with parameter ${\displaystyle \alpha }$ has standardized moment

${\displaystyle \mu _{k}=\int _{0}^{\infty }x^{k}f(x)dx=\int _{0}^{\infty }t^{-{\frac {k}{\alpha }}}e^{-t}\,dt,}$

(with ${\displaystyle t=x^{-\alpha }}$) defined only for ${\displaystyle k<\alpha }$:

${\displaystyle \mu _{k}=\Gamma \left(1-{\frac {k}{\alpha }}\right)}$

where ${\displaystyle \Gamma \left(z\right)}$ is the Gamma function.

In particular:

• For ${\displaystyle \alpha >1}$ the expectation is ${\displaystyle E[X]=\Gamma (1-{\tfrac {1}{\alpha }})}$
• For ${\displaystyle \alpha >2}$ the variance is ${\displaystyle {\text{Var}}(X)=\Gamma (1-{\tfrac {2}{\alpha }})-{\big (}\Gamma (1-{\tfrac {1}{\alpha }}){\big )}^{2}.}$

The quantile ${\displaystyle q_{y}}$ of order ${\displaystyle y}$ can be expressed through the inverse of the distribution,

${\displaystyle q_{y}=F^{-1}(y)=\left(-\log _{e}y\right)^{-{\frac {1}{\alpha }}}}$.

In particular the median is:

${\displaystyle q_{1/2}=(\log _{e}2)^{-{\frac {1}{\alpha }}}.}$

The mode of the distribution is ${\displaystyle \left({\frac {\alpha }{\alpha +1}}\right)^{\frac {1}{\alpha }}.}$

Especially for the 3-parameter Fréchet, the first quartile is ${\displaystyle q_{1}=m+{\frac {s}{\sqrt[{\alpha }]{\log(4)}}}}$ and the third quartile ${\displaystyle q_{3}=m+{\frac {s}{\sqrt[{\alpha }]{\log({\frac {4}{3}})}}}.}$

Also the quantiles for the mean and mode are:

${\displaystyle F(mean)=\exp \left(-\Gamma ^{-\alpha }\left(1-{\frac {1}{\alpha }}\right)\right)}$

${\displaystyle F(mode)=\exp \left(-{\frac {\alpha +1}{\alpha }}\right).}$

Applications

• In
  plotting positions as part of the cumulative frequency analysis
  .

However, in most hydrological applications, the distribution fitting is via the generalized extreme value distribution as this avoids imposing the assumption that the distribution does not have a lower bound (as required by the Frechet distribution). ^{[citation needed]}

• In decline curve analysis, a declining pattern the time series data of oil or gas production rate over time for a well can be described by the Fréchet distribution.^[8]
• One test to assess whether a multivariate distribution is asymptotically dependent or independent consists of transforming the data into standard Fréchet margins using the transformation ${\displaystyle Z_{i}=-1/\log F_{i}(X_{i})}$ and then mapping from Cartesian to pseudo-polar coordinates ${\displaystyle (R,W)=(Z_{1}+Z_{2},Z_{1}/(Z_{1}+Z_{2}))}$. Values of ${\displaystyle R\gg 1}$ correspond to the extreme data for which at least one component is large while ${\displaystyle W}$ approximately 1 or 0 corresponds to only one component being extreme.
• In Economics it is used to model the idiosyncratic component of preferences of individuals for different products (
  Labor Economics
  ).

Related distributions

• If ${\displaystyle X\sim U(0,1)\,}$ (
  Uniform distribution (continuous)
  ) then ${\displaystyle m+s(-\log(X))^{-1/\alpha }\sim {\textrm {Frechet}}(\alpha ,s,m)\,}$
• If ${\displaystyle X\sim {\textrm {Frechet}}(\alpha ,s,m)\,}$ then ${\displaystyle kX+b\sim {\textrm {Frechet}}(\alpha ,ks,km+b)\,}$
• If ${\displaystyle X_{i}\sim {\textrm {Frechet}}(\alpha ,s,m)\,}$ and ${\displaystyle Y=\max\{\,X_{1},\ldots ,X_{n}\,\}\,}$ then ${\displaystyle Y\sim {\textrm {Frechet}}(\alpha ,n^{\tfrac {1}{\alpha }}s,m)\,}$
• The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation
• If ${\displaystyle X\sim {\textrm {Frechet}}(\alpha ,s,m=0)\,}$ then its reciprocal is Weibull-distributed: ${\displaystyle X^{-1}\sim {\textrm {Weibull}}(k=\alpha ,\lambda =s^{-1})\,}$

Properties

• The Frechet distribution is a max stable distribution
• The negative of a random variable having a Frechet distribution is a min stable distribution

See also

• Type-2 Gumbel distribution
• Fisher–Tippett–Gnedenko theorem

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References