Fisher-Tippett distribution

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Fisher-Tippett
Probability density function
Cumulative distribution function
Parameters	$\mu\!$ location (real) $\beta>0\!$ scale (real)
Support	$x \in (-\infty; +\infty)\!$
Probability density function (pdf)	$\frac{z\,\exp(-z)}{\beta}\!$ where $z = \exp\left[-\frac{x-\mu}{\beta}\right]\!$
Cumulative distribution function (cdf)	$\exp(-\exp[-(x-\mu)/\beta])\!$
Mean	$\mu + \beta\,\gamma\!$
Median	$\mu - \beta\,\ln(\ln(2))\!$
Mode	$\mu\!$
Variance	$\frac{\pi^2}{6}\,\beta^2\!$
Skewness	$\frac{12\sqrt{6}\,\zeta(3)}{\pi^3} \approx 1.14\!$
Excess kurtosis	$\frac{12}{5}$
Entropy	$\ln(\beta)+\gamma+1\!$ for $\beta > \exp(-(\gamma+1))\!$
Moment-generating function (mgf)	$\Gamma(1-\beta\,t)\, \exp(\mu\,t)\!$
Characteristic function	$\Gamma(1-i\,\beta\,t)\, \exp(i\,\mu\,t)\!$

In probability theory and statistics the Gumbel distribution (named after Emil Julius Gumbel (1891–1966)) is used to find the minimum (or the maximum) of a number of samples of various distributions. For example we would use it to find the maximum level of a river in a particular year if we had the list of maximum values for the past ten years. It is therefore useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur.

The distribution of the samples could be of the normal or exponential type. The Gumbel distribution, and similar distributions, are used in extreme value theory.

In particular, the Gumbel distribution is a special case of the Fisher-Tippett distribution (named after Sir Ronald Aylmer Fisher (1890–1962) and Leonard Henry Caleb Tippett (1902–1985)), also known as the log-Weibull distribution.

1 Properties
- 1.1 Properties of the Gumbel distribution
2 Parameter estimation
3 Generating Fisher-Tippett variates
4 See also

[edit] Properties

The cumulative distribution function of the Fisher-Tippett distribution is

$F(x;\mu,\beta) = e^{-e^{(\mu-x)/\beta}}.\,$

The median is $μ - βln( - ln(0.5))$

The mean is $μ + γβ$ where $γ$ = Euler-Mascheroni constant = 0.57721...

The standard deviation is

$\beta \pi/\sqrt{6}.\,$

The mode is μ.

[edit] Properties of the Gumbel distribution

The standard Gumbel distribution is the case where μ = 0 and β = 1 with cumulative distribution function

$F(x) = e^{-e^{(-x)}}.\,$

and probability density function

$f(x) = e^{-x} e^{-e^{-x}}.$

The median is $-\ln(\ln(2)) = 0.3665\dots$

The mean is $γ$ , the Euler-Mascheroni constant 0.57721...

The standard deviation is

$\pi/\sqrt{6} = 1.2825\dots\,.$

The mode is 0.

[edit] Parameter estimation

A more practical way of using the distribution could be

$F(x;\mu,\beta)=e^{-e^{\varepsilon(\mu-x)/(\mu-M)}} ;$

$\varepsilon=\ln(-\ln(0.5))=-0.367\dots\,$

where M is the median. To fit values one could get the median straight away and then vary μ until it fits the list of values.

[edit] Generating Fisher-Tippett variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1], the variate

$X=\mu-\beta\ln(-\ln(U))\,$

has a Fisher-Tippett distribution with parameters μ and β. This follows from the form of the cumulative distribution function given above.

A piece of graph paper that incorporates the Gumbel distribution.

[edit] See also

order statistic

v • d • e

Probability distributions

Discrete univariate with finite support

Benford · Bernoulli · binomial · categorical · hypergeometric · Rademacher · discrete uniform · Zipf · Zipf-Mandelbrot

Discrete univariate with infinite support

Boltzmann · Conway-Maxwell-Poisson · compound Poisson · discrete phase-type · extended negative binomial · Gauss-Kuzmin · geometric · logarithmic · negative binomial · parabolic fractal · Poisson · Skellam · Yule-Simon · zeta

Continuous univariate supported on a bounded interval

Beta · Kumaraswamy · raised cosine · triangular · U-quadratic · uniform · Wigner semicircle

Continuous univariate supported on a semi-infinite interval

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · 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 · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

Categories: Continuous distributions | Extreme value data

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Fisher-Tippett distribution

From Wikipedia, the free encyclopedia

Contents

[edit] Properties

[edit] Properties of the Gumbel distribution

[edit] Parameter estimation

[edit] Generating Fisher-Tippett variates

[edit] See also

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