- 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}!
Inprobability theory andstatistics the Gumbel distribution (named afterEmil 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) andLeonard 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
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