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Fréchet distribution

Continuous probability distribution

Fréchet distribution

Continuous probability distribution

name =Fréchet| type =density| pdf_image =[[File:Frechet pdf.svg|325px|PDF of the Fréchet distribution]]| cdf_image =[[File:Frechet cdf.svg|325px|CDF of the Fréchet distribution]]| parameters =\ \alpha \in (0,\infty)\ shape. (Optionally, two more parameters) \ s \in (0,\infty)\ scale (default: \ s=1\ ) \ m \in (-\infty,\infty)\ location of minimum (default: \ m=0\ ) | support =\ x m\ | pdf =\ \frac{\ \alpha\ }{s} \left( \frac{\ x - m\ }{s}\right)^{-1-\alpha} ~ e^{-(\frac{x-m}{s})^{-\alpha}}\ | cdf =\ e^{-(\frac{x-m}{s})^{-\alpha}}\ | quantile =\ m+s\left[\ -\ln p\ \right]^{-\tfrac{1}{\alpha}}\ | mean =\begin{cases} \ m\ +\ s\ \Gamma\left( 1 - \tfrac{1}{\alpha} \right) ~~ & \text{for }~ \alpha 1 \ \ \infty & \text{otherwise} \end{cases} | median =\ m\ +\ \frac{s}{\ {\ln(2)}^{\tfrac{1}{\alpha}}\ } | mode =\ m\ +\ s \left(\frac{ \alpha }{\ 1 + \alpha}\right)^{1/\alpha\ }| variance = \begin{cases} \ s^2\left[\ \Gamma\left(1 - \tfrac{2}{\alpha}\right)- \left[\Gamma\left(1 - \tfrac{1}{\alpha}\right)\right]^2\ \right] ~~ & \text{for }~ \alpha2 \ \ \infty & \text{otherwise} \end{cases} |

skewness = \begin{cases} \ \frac{\ A\ }{\ \sqrt{B^3\ }\ } ~~ & \text{for }~ \alpha 3 \ \ \infty & \text{otherwise} \end{cases} \begin{align}\text{where} ~~ A\ \equiv\ &\Gamma\left( 1 - \tfrac {3}{\alpha} \right) \ & -\ 3\ \Gamma\left( 1 - \tfrac {2}{\alpha}\right)\ \Gamma\left(1 - \tfrac {1}{\alpha}\right)\ &\quad +\ 2\Bigl[\ \Gamma\left(1 - \tfrac {1}{\alpha} \right)\ \Bigr]^3 \ \end{align} ~ \quad \text{and} ~~ B\ \equiv\ \Gamma\left( 1 - \tfrac{2}{\alpha} \right)\ -\ \Bigr[\ \Gamma\left( 1 - \tfrac{1}{\alpha} \right)\ \Bigr]^2 ~. | g_k =| kurtosis = \begin{cases} \ -6\ +\ \frac{\ C\ }{; D^2 } ~~ & \text{for } ~ \alpha 4 \ \ \infty & \text{otherwise} \end{cases} \begin{align}\text{where} ~~ C\ \equiv\ &\Gamma \left( 1 - \tfrac{4}{\alpha} \right) \ & -\ 4\ \Gamma\left( 1 - \tfrac{3}{\alpha} \right)\ \Gamma\left( 1 - \tfrac{1}{\alpha} \right) \ & \qquad +\ 3\ \Bigl[\ \Gamma\left( 1 - \tfrac{2}{\alpha} \right)\ \Bigr]^2\ \end{align} ~ \quad \text{and} ~~ D\ \equiv\ \Gamma \left( 1 - \tfrac{2}{\alpha} \right)\ -\ \Bigl[\ \Gamma\left( 1 - \tfrac{1}{\alpha} \right)\ \Bigr]^2 ~. |

entropy =\ 1 + \frac{\gamma}{\alpha} + \gamma_e +\ln \left( \frac{s}{\alpha} \right)\ , where \ \gamma_e\ is the Euler–Mascheroni constant.| mgf = Note: Moment \ k\ exists if \ \alpha k\ | char = |

The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\ \Pr(\ X \le 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 :\ \Pr(\ X \le x\ ) = \exp\left[\ -\left( \tfrac{\ x - m\ }{ s } \right)^{-\alpha}\ \right] ~~ \text{ if } ~ x m ~.

Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.

Characteristics

The single parameter Fréchet, with parameter \ \alpha\ , has standardized moment :\mu_k = \int_0^\infty x^k f(x)\ \operatorname{d} x =\int_0^\infty t^{-\frac{k}{\alpha}}e^{-t} \ \operatorname{d} t\ , (with \ t = x^{-\alpha}\ ) defined only for \ k :\ \mu_k = \Gamma\left(1-\frac{k}{\alpha}\right)\ where \ \Gamma\left(z\right)\ is the Gamma function.

In particular:

  • For \alpha1 the expectation is E[X]=\Gamma(1-\tfrac{1}{\alpha})
  • For \alpha2 the variance is \text{Var}(X)=\Gamma(1-\tfrac{2}{\alpha})-\big(\Gamma(1-\tfrac{1}{\alpha})\big)^2.

The quantile q_y of order y can be expressed through the inverse of the distribution, :q_y=F^{-1}(y)=\left(-\log_e y \right)^{-\frac{1}{\alpha}}. In particular the median is: :q_{1/2}=(\log_e 2)^{-\frac{1}{\alpha}}.

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

Especially for the 3-parameter Fréchet, the first quartile is q_1= m+\frac{s}{\sqrt[\alpha]{\log(4)}} and the third quartile q_3= m+\frac{s}{\sqrt[\alpha]{\log(\frac{4}{3})}}.

Also the quantiles for the mean and mode are: :F(mean)=\exp \left( -\Gamma^{-\alpha} \left(1- \frac{1}{\alpha} \right) \right) :F(mode)=\exp \left( -\frac{\alpha+1}{\alpha} \right).

Properties

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

Applications

Fitted cumulative Fréchet distribution to extreme one-day rainfalls
Fitted decline curve analysis. Duong model can be thought of as a generalization of the Frechet distribution.
  • In hydrology, the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. This picture illustrates an example of fitting the Fréchet distribution to ranked annually maximum one-day rainfalls in Oman showing also the 90% confidence belt based on the binomial distribution. The cumulative frequencies of the rainfall data are represented by 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).
  • 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.
  • 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 Z_i = -1/\log F_i(X_i) and then mapping from Cartesian to pseudo-polar coordinates (R, W)= (Z_1 + Z_2, Z_1/(Z_1 + Z_2)). Values of R \gg 1 correspond to the extreme data for which at least one component is large while 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 (Industrial Organization), locations (Urban Economics), or firms (Labor Economics).

References

References

  1. Fréchet, M.. (1927). "Sur la loi de probabilité de l'écart maximum". [[Annales Polonici Mathematici]].
  2. (1928). "Limiting forms of the frequency distribution of the largest and smallest member of a sample". [[Proceedings of the Cambridge Philosophical Society]].
  3. Gumbel, E.J.. (1958). "Statistics of Extremes". Columbia University Press.
  4. "Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas". Sankhya B. (2021)
  5. (2011). "Sea Level Rise, Coastal Engineering, Shorelines, and Tides". Nova Science Publishers.
  6. Coles, Stuart. (2001). "An Introduction to Statistical Modeling of Extreme Values". Springer-Verlag.
  7. (February 2008). "Theoretical analysis of inverse Weibull distribution". WSEAS Transactions on Mathematics.
  8. (2011). "The generalized inverse Weibull distribution". Springer-Verlag.
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