en>AnomieBOT: Dating maintenance tags: {{Unreferenced}}

2014-11-18T06:27:26Z

Dating maintenance tags: {{Unreferenced}}

en>Syed Ahsan Kamal at 13:21, 21 November 2011

2011-11-21T13:21:44Z

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{{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 =<math>\alpha \in (0,\infty) </math> [[shape parameter|shape]]. <br> (Optionally, two more parameters) <br> <math> s \in (0,\infty) </math> [[scale parameter|scale]] (default: <math> s=1 \, </math>) <br> <math> m \in (-\infty,\infty) </math> [[location parameter|location]] of minimum (default: <math> m=0 \, </math>) |
support =<math>x>m</math>|
pdf =<math>\frac{\alpha}{s} \; \left(\frac{x-m}{s}\right)^{-1-\alpha} \; e^{-(\frac{x-m}{s})^{-\alpha}}</math>|
cdf =<math>e^{-(\frac{x-m}{s})^{-\alpha}}</math> |
mean =<math>\begin{cases}
\ m+s\Gamma\left(1-\frac{1}{\alpha}\right) & \text{for } \alpha>1 \\
\ \infty & \text{otherwise}
\end{cases}</math> |
median =<math>m+\frac{s}{\sqrt[\alpha]{\log_e(2)}}</math> |
mode =<math>m+s\left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}</math>|
variance = <math>\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}</math> |

skewness = <math>\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}</math> |
g_k =|
kurtosis = <math>\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}</math> |
</math> |
entropy =<math> 1 + \frac{\gamma}{\alpha} + \gamma +\ln \left( \frac{s}{\alpha} \right) </math>, where <math>\gamma</math> is the [[Euler–Mascheroni constant]].|
mgf = <ref name=R1/> Note: Moment <math>k</math> exists if <math>\alpha>k</math> |
char = <ref name=R1/> |
}}

The '''Fréchet distribution''' is a special case of the [[generalized extreme value distribution]]. It has the cumulative distribution function
:<math>\Pr(X \le x)=e^{-x^{-\alpha}} \text{ if } x>0. </math>
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
:<math>\Pr(X \le x)=e^{-\left(\frac{x-m}{s}\right)^{-\alpha}} \text{ if } x>m. </math>

Named for [[Maurice Fréchet]] who wrote a related paper in 1927, further work was done by [[Fisher–Tippett distribution|Fisher and Tippett]] in 1928 and by [[Emil Julius Gumbel|Gumbel]] in 1958.

==Characteristics==
The single parameter Fréchet with parameter <math>\alpha</math> has [[standardized moment]]
:<math>\mu_k=\int_0^\infty x^k f(x)dx=\int_0^\infty t^{-\frac{k}{\alpha}}e^{-t} \, dt,</math>
(with <math>t=x^{-\alpha}</math>) defined only for <math>k<\alpha</math>:
:<math>\mu_k=\Gamma\left(1-\frac{k}{\alpha}\right)</math>
where <math>\Gamma\left(z\right)</math> is the [[Gamma function]].

In particular:
* For <math>\alpha>1</math> the [[Expected value|expectation]] is <math>E[X]=\Gamma(1-\tfrac{1}{\alpha})</math>
* For <math>\alpha>2</math> the [[variance]] is <math>\text{Var}(X)=\Gamma(1-\tfrac{2}{\alpha})-\big(\Gamma(1-\tfrac{1}{\alpha})\big)^2.</math>

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

The [[mode (statistics)|mode]] of the distribution is <math>\left(\frac{\alpha}{\alpha+1}\right)^\frac{1}{\alpha}.</math>

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

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

[[File:FitFrechetDistr.tif|thumb|220px|Fitted cumulative Fréchet distribution to extreme one-day rainfalls]]

==Applications==

* In [[hydrology]], the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.<ref name=Coles1/> 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 position]]s 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|date=May 2011}}

==Related distributions==

*If <math> X \sim U(0,1) \,</math> ([[Uniform distribution (continuous)]]) then <math> m + s(-\log(X))^{-1/\alpha} \sim \textrm{Frechet}(\alpha,s,m)\,</math>
*If <math> X \sim \textrm{Frechet}(\alpha,s,m)\,</math> then <math> k X + b \sim \textrm{Frechet}(\alpha,k s,k m + b)\,</math>
*If <math> X_i=\textrm{Frechet}(\alpha,s,m) \, </math> and <math> Y=\max\{\,X_1,\ldots,X_n\,\} \, </math> then <math> Y \sim \textrm{Frechet}(\alpha,n^{\tfrac{1}{\alpha}} s,m) \,</math>
*The [[cumulative distribution function]] of the Frechet distribution solves the maximum [[stability postulate]] equation
*If <math>X \sim \textrm{Weibull}(k=\alpha, \lambda=m)\,</math> ([[Weibull distribution]]) then <math> \tfrac{m^2}{X} \sim \textrm{Frechet}(\alpha,m)\,</math>

==Properties==
*The Frechet distribution is a [[Stability postulate|max stable distribution]]
*The negative of a random variable having a Frechet distribution is a [[Stability postulate|min stable distribution]]

==See also==
* [[Type-2 Gumbel distribution]]
* [[Fisher–Tippett–Gnedenko theorem]]
* [[CumFreq]] (application software for probability distributions including Fréchet)

{{More footnotes|date=May 2011}}
==References==
{{reflist|refs=

<ref name=R1>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</ref>

<ref name=Coles1>{{cite book | author=Coles, Stuart | title=An Introduction to Statistical Modeling of Extreme Values, | url = http://books.google.com/books?id=2nugUEaKqFEC&lpg=PP1&pg=PP1#v=onepage&q=&f=false | publisher=Springer-Verlag | year=2001 | isbn = 1-85233-459-2 }}</ref>
}}

==Publications==
* 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==
*[http://www.bankofengland.co.uk/publications/workingpapers/wp287.pdf Bank of England working paper]
*[http://www.emeraldinsight.com/Insight/ViewContentServlet?Filename=Published/EmeraldFullTextArticle/Articles/0830160102.html#0830160102006.png An application of a new extreme value distribution to air pollution data]
*[http://www.maths.lth.se/matstat/wafo/documentation/wafodoc/wafo/wstats/wfrechstat.html Wave Analysis for Fatigue and Oceanography]

{{ProbDistributions|continuous-semi-infinite}}

{{DEFAULTSORT:Frechet distribution}}
[[Category:Continuous distributions]]
[[Category:Extreme value data]]
[[Category:Probability distributions]]

Shift theorem - Revision history

en>AnomieBOT: Dating maintenance tags: {{Unreferenced}}

en>Syed Ahsan Kamal at 13:21, 21 November 2011