Fréchet distribution

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Fréchet
Probability density function
Cumulative distribution function
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

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

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

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

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

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:

For $\alpha >1$ the expectation is $E[X]=\Gamma (1-{\tfrac {1}{\alpha }})$
For $\alpha >2$ 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:

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

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

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.^[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]}

Related distributions

If $X\sim U(0,1)\,$ (Uniform distribution (continuous)) then $m+s(-\log(X))^{{-1/\alpha }}\sim {\textrm {Frechet}}(\alpha ,s,m)\,$
If $X\sim {\textrm {Frechet}}(\alpha ,s,m)\,$ then $kX+b\sim {\textrm {Frechet}}(\alpha ,ks,km+b)\,$
If $X_{i}={\textrm {Frechet}}(\alpha ,s,m)\,$ and $Y=\max\{\,X_{1},\ldots ,X_{n}\,\}\,$ then $Y\sim {\textrm {Frechet}}(\alpha ,n^{{{\tfrac {1}{\alpha }}}}s,m)\,$
The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation
If $X\sim {\textrm {Weibull}}(k=\alpha ,\lambda =m)\,$ (Weibull distribution) then ${\tfrac {m^{2}}{X}}\sim {\textrm {Frechet}}(\alpha ,m)\,$

Properties

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

References

↑ 1.0 1.1 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
↑ Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values,. Springer-Verlag. ISBN 1-85233-459-2.

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

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support

beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Triangular U-quadratic uniform Wigner semicircle Xenakis

[[List of probability distributions#Supported_on_semi-infinite_intervals.2C_usually_.5B0.2C.E2.88.9E.29|Continuous univariate supported on a semi-infinite interval, usually [0,∞)]]

Benini
Benktander 1st kind
Benktander 2nd kind
Beta prime
Burr
chi-squared
chi
Coxian
Dagum
Davis
EL
Erlang
exponential
F
folded normal
Flory-Schulz
Fréchet
Gamma
Gamma/Gompertz
generalized inverse Gaussian
Gompertz
half-logistic
half-normal
Hotelling's T-squared
hyper-Erlang
hyperexponential
hypoexponential
inverse chi-squared (scaled inverse chi-squared)
inverse Gaussian
inverse gamma
Kolmogorov
Lévy
log-Cauchy
log-Laplace
log-logistic
log-normal
Maxwell–Boltzmann
Maxwell–Jüttner
Mittag–Leffler
Nakagami
noncentral chi-squared
Pareto
phase-type
Poly-Weibull
Rayleigh
relativistic Breit–Wigner
Rice
Rosin–Rammler
shifted Gompertz
truncated normal
type-2 Gumbel
Weibull
Wilks' lambda

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson SU Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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