Fréchet distribution

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Fréchet
Probability density function
PDF of the Fréchet distribution
Cumulative distribution function
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}}
Mean \begin{cases}
                  \ m+s\Gamma\left(1-\frac{1}{\alpha}\right)  & \text{for } \alpha>1  \\
                  \ \infty              & \text{otherwise}
                \end{cases}
Median m+\frac{s}{\sqrt[\alpha]{\log_e(2)}}
Mode m+s\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}
Variance \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 \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}
Ex. kurtosis \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  1 + \frac{\gamma}{\alpha} + \gamma +\ln \left( \frac{s}{\alpha} \right) , where \gamma is the Euler–Mascheroni constant.
MGF [1] Note: Moment k exists if \alpha>k
CF [1]

The Fréchet 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)=e^{-\left(\frac{x-m}{s}\right)^{-\alpha}} \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[edit]

The single parameter Fréchet with parameter \alpha has standardized moment

\mu_k=\int_0^\infty x^k f(x)dx=\int_0^\infty t^{-\frac{k}{\alpha}}e^{-t} \, dt,

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

\mu_k=\Gamma\left(1-\frac{k}{\alpha}\right)

where \Gamma\left(z\right) is the Gamma function.

In particular:

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).
Fitted cumulative Fréchet distribution to extreme one-day rainfalls

Applications[edit]

  • In hydrology, the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.[2] The blue 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).[citation needed]
  • One useful test to assess whether a multivariate distribution is asymptotically dependent or independent consists transforming the data into standard Frechet margins using transformation  Z_i = -1/log F_i(X_i) and then mapping from the cartesian to pseudo-polar coordinates  (R,W)= (Z_1 +Z_2, Z_1/(Z_1 + Z_2)) .  R >> 1 corresponds to the extreme data for which at least only one component is large while  W approximately 1 or 0 corresponds to only one component being extreme.

Related distributions[edit]

Properties[edit]

See also[edit]

References[edit]

  1. ^ a b Muraleedharan. G, C. Guedes Soares and Cláudia Lucas (2011). "Characteristic and Moment Generating Functions of Generalised Extreme Value Distribution (GEV)". In Linda. L. Wright (Ed.), Sea Level Rise, Coastal Engineering, Shorelines and Tides, Chapter 14, pp. 269–276. Nova Science Publishers. ISBN 978-1-61728-655-1
  2. ^ Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values,. Springer-Verlag. ISBN 1-85233-459-2. 

Publications[edit]

  • Fréchet, M., (1927). "Sur la loi de probabilité de l'écart maximum." Ann. Soc. Polon. Math. 6, 93.
  • Fisher, R.A., Tippett, L.H.C., (1928). "Limiting forms of the frequency distribution of the largest and smallest member of a sample." Proc. Cambridge Philosophical Society 24:180–190.
  • Gumbel, E.J. (1958). "Statistics of Extremes." Columbia University Press, New York.
  • Kotz, S.; Nadarajah, S. (2000) Extreme value distributions: theory and applications, World Scientific. ISBN 1-86094-224-5

External links[edit]