Fisher-Tippett distribution

Fisher-Tippett distribution

Probability distribution
name =Fisher-Tippett
type =density
pdf_

cdf_

parameters =mu! location (real)
eta>0! scale (real)
support =x in (-infty; +infty)!
pdf =frac{z,e^{-z
{eta}!
where z = e^{-frac{x-mu}{eta!
cdf =exp(-e^{-(x-mu)/eta})!
mean =mu + eta,gamma!
median =mu - eta,ln(ln(2))!
mode =mu!
variance =frac{pi^2}{6},eta^2!
skewness =frac{12sqrt{6},zeta(3)}{pi^3} approx 1.14!
kurtosis =frac{12}{5}
entropy =ln(eta)+gamma+1!
for eta > e^{-(gamma+1)}!
mgf =Gamma(1-eta,t), e^{mu,t}!
char =Gamma(1-i,eta,t), e^{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.

Properties

The cumulative distribution function of the Fisher-Tippett distribution is

:F(x;mu,eta) = e^{-e^{(mu-x)/eta.,

The median is mu-eta ln(-ln(0.5))

The mean is mu+gammaeta where gamma = Euler-Mascheroni constant = 0.57721...

The standard deviation is

:eta pi/sqrt{6}.,

The mode is μ.

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)) = 0.3665dots

The mean is gamma, the Euler-Mascheroni constant 0.57721...

The standard deviation is

: pi/sqrt{6} = 1.2825dots,.

The mode is 0.

Parameter estimation

A more practical way of using the distribution could be

:F(x;mu,eta)=e^{-e^{varepsilon(mu-x)/(mu-M) ;

:varepsilon=ln(-ln(0.5))=-0.367dots,

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

Generating Fisher-Tippett variates

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

:X=mu-etaln(-ln(U)),

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

ee also

* order statistic


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Fisher-Tippett-Verteilung — Die Fisher Tippett Verteilung (nach Ronald Aylmer Fisher) ist eine stetige Wahrscheinlichkeitsverteilung. Wie die Rossi Verteilung und die Frechet Verteilung gehört sie zu den Extremwertverteilungen. Sie berechnet den in einem Zeitraum T zu… …   Deutsch Wikipedia

  • Ronald Fisher — R. A. Fisher Born 17 February 1890(1890 02 17) East Finchley, London …   Wikipedia

  • Leonard Henry Caleb Tippett — (1902 1985), physicist and statistician, born in London. Tippett graduated in physics in the early 1920s at Imperial College. He studied for his MSc in statistics under Professor Karl Pearson at the Galton Laboratory, University College London… …   Wikipedia

  • Ronald Fisher — Ronald Aylmer Fisher Sir Ronald Aylmer Fisher (* 17. Februar 1890 in London, England; † 29. Juli 1962 in Adelaide, Australien) war einer der bedeutendsten Theoretischen Biologen, Genetiker, Evolutionstheoretiker und …   Deutsch Wikipedia

  • Fischer-Tippett-Verteilung — Die Fisher Tippett Verteilung (nach Ronald Aylmer Fisher) ist eine stetige Wahrscheinlichkeitsverteilung. Wie die Rossi Verteilung und die Frechet Verteilung gehört sie zu den Extremwertverteilungen. Sie berechnet den in einem Zeitraum T zu… …   Deutsch Wikipedia

  • Ronald Aylmer Fisher — Sir Ronald Aylmer Fisher (* 17. Februar 1890 in London, England; † 29. Juli 1962 in Adelaide, Australien) war einer der bedeutendsten Theoretischen Biologen, Genetiker, Evolutionstheoretiker und …   Deutsch Wikipedia

  • Fréchet distribution — Probability distribution name =Fréchet type =density pdf cdf parameters =alpha in (0,infty] shape support =x>0 pdf =alpha ; x^{ 1 alpha} ; e^{ x^{ alpha cdf =e^{ x^{ alpha mean =Gammaleft(1 frac{1}{alpha} ight) ext{ if } alpha>1 median… …   Wikipedia

  • Generalized extreme value distribution — Probability distribution name =Generalized extreme value type =density pdf cdf parameters =mu in [ infty,infty] , location (real) sigma in (0,infty] , scale (real) xiin [ infty,infty] , shape (real) support =x>mu sigma/xi,;(xi > 0) x …   Wikipedia

  • List of probability distributions — Many probability distributions are so important in theory or applications that they have been given specific names.Discrete distributionsWith finite support* The Bernoulli distribution, which takes value 1 with probability p and value 0 with… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”