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| {{Probability distribution | | |||
| name =Fisher-Tippett| | |||
| type =density| | |||
| pdf_image =]<br /><small>Fisher-Tippett distribution: μ=0, β=1</small>| | |||
| cdf_image =None uploaded yet.| | |||
| parameters =<math>\mu\!</math> ] (])<br /><math>\beta>0\!</math> ] (real)| | |||
| support =<math>x \in (-\infty; +\infty)\!</math>| | |||
| pdf =<math>\frac{\exp(-z)\,z}{\beta}\!</math><br /> where <math>z = \exp\left\!</math>| | |||
| cdf =<math>\exp(-\exp)\!</math>| | |||
| mean =<math>\mu + \beta\,\gamma\!</math>| | |||
| median =<math>\mu - \beta\,\ln(\ln(2))\!</math>| | |||
| mode =<math>\mu\!</math>| | |||
| variance =<math>\frac{\pi^2}{6}\,\beta^2\!</math>| | |||
| skewness =<math>\frac{12\sqrt{6}\,\zeta(3)}{\pi^3} \approx 1.14\!</math>| | |||
| kurtosis =<math>\frac{12}{5}</math>| | |||
| entropy =<math>\ln(\beta)+\gamma+1\!</math><br />for <math>\beta > \exp(-(\gamma+1))\!</math><!-- pls check-->| | |||
| mgf =<math>\Gamma(1-\beta\,t)\, \exp(\mu\,t)\!</math>| | |||
| char =<math>\Gamma(1-i\,\beta\,t)\, \exp(i\,\mu\,t)\!</math>| | |||
| }} | |||
| In ] and ] the '''Gumbel distribution''' 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. | |||
| == Test == | |||
| The distribution of the samples could be of the normal or exponential type. The Gumbel distribution, and similar distributions, are used in ]. | |||
| Testing. | |||
| In particular, the Gumbel distribution is a special case of the '''Fisher-Tippett distribution''', also known as the '''log-]'''. | |||
| == Properties == | |||
| The ] is | |||
| :<math>F(x;\mu,\beta) = e^{-e^{(\mu-x)/\beta}}.\,</math> | |||
| The Gumbel distribution is the case where μ = 0 and β = 1. | |||
| The median is <math>\mu-\beta \ln(-\ln(0.5))</math> | |||
| The mean is <math>\mu+\gamma\beta</math> where <math>\gamma</math> = ] = 0.57721... | |||
| The standard deviation is | |||
| :<math>\beta \pi/\sqrt{6}.\,</math> | |||
| The mode is μ. | |||
| ==Parameter estimation== | |||
| A more practical way of using the distribution could be | |||
| :<math>F(x;\mu,\beta)=e^{-e^{\epsilon(\mu-x)/(\mu-M)}} ;</math> | |||
| :<math>\epsilon=\ln(-\ln(0.5))=-0.367...\,</math> | |||
| where ''M'' is the ]. To fit values one could get the median | |||
| straight away and then vary μ until it fits the list of values. | |||
| ==Generating Fisher-Tippett variates== | |||
| Given a random variate ''U'' drawn from the ] in the interval <nowiki>(0, 1]</nowiki>, the variate | |||
| :<math>X=\mu-\beta\ln(-\ln(U))\,</math> | |||
| has a Fisher-Tippett distribution with parameters μ and β. This follows from the form of the cumulative distribution function given above. | |||
| ==See also== | |||
| * ] | |||
| ] | |||
Revision as of 16:09, 11 February 2006
Test
Testing.