Gumbel distribution: Difference between revisions
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{{Probability distribution | |
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name =Fisher-Tippett| |
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type =density| |
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pdf_image =[[Image:FT_distribution_pdf.png|center|Probability density function for Fisher-Tippett distribution: μ=0, β=1]]<br /><small>Fisher-Tippett distribution: μ=0, β=1</small>| |
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cdf_image =None uploaded yet.| |
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parameters =<math>\mu\!</math> [[location parameter|location]] ([[real numbers|real]])<br /><math>\beta>0\!</math> [[scale parameter|scale]] (real)| |
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support =<math>x \in (-\infty; +\infty)\!</math>| |
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pdf =<math>\frac{\exp(-z)\,z}{\beta}\!</math><br /> where <math>z = \exp\left[-\frac{x-\mu}{\beta}\right]\!</math>| |
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cdf =<math>\exp(-\exp[-(x-\mu)/\beta])\!</math>| |
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mean =<math>\mu + \beta\,\gamma\!</math>| |
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median =<math>\mu - \beta\,\ln(\ln(2))\!</math>| |
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mode =<math>\mu\!</math>| |
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variance =<math>\frac{\pi^2}{6}\,\beta^2\!</math>| |
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skewness =<math>\frac{12\sqrt{6}\,\zeta(3)}{\pi^3} \approx 1.14\!</math>| |
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kurtosis =<math>\frac{12}{5}</math>| |
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entropy =<math>\ln(\beta)+\gamma+1\!</math><br />for <math>\beta > \exp(-(\gamma+1))\!</math><!-- pls check-->| |
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mgf =<math>\Gamma(1-\beta\,t)\, \exp(\mu\,t)\!</math>| |
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char =<math>\Gamma(1-i\,\beta\,t)\, \exp(i\,\mu\,t)\!</math>| |
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}} |
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In [[probability theory]] and [[statistics]] the '''Gumbel distribution''' is used to find the minimum (or the maximum) of a number of samples of various distributions. |
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For example we would use it to find the maximum level of a river |
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in a particular year if we had the list of maximum values for the past ten |
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years. It is therefore useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. |
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== Test == |
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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]]. |
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Testing. |
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In particular, the Gumbel distribution is a special case of the '''Fisher-Tippett distribution''', also known as the '''log-[[Weibull distribution]]'''. |
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== Properties == |
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The [[cumulative distribution function]] is |
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:<math>F(x;\mu,\beta) = e^{-e^{(\mu-x)/\beta}}.\,</math> |
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The Gumbel distribution is the case where μ = 0 and β = 1. |
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The median is <math>\mu-\beta \ln(-\ln(0.5))</math> |
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The mean is <math>\mu+\gamma\beta</math> where <math>\gamma</math> = [[Euler-Mascheroni constant]] = 0.57721... |
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The standard deviation is |
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:<math>\beta \pi/\sqrt{6}.\,</math> |
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The mode is μ. |
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==Parameter estimation== |
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A more practical way of using the distribution could be |
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:<math>F(x;\mu,\beta)=e^{-e^{\epsilon(\mu-x)/(\mu-M)}} ;</math> |
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:<math>\epsilon=\ln(-\ln(0.5))=-0.367...\,</math> |
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where ''M'' is the [[median]]. To fit values one could get the median |
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straight away and then vary μ until it fits the list of values. |
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==Generating Fisher-Tippett variates== |
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Given a random variate ''U'' drawn from the [[uniform distribution]] in the interval <nowiki>(0, 1]</nowiki>, the variate |
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:<math>X=\mu-\beta\ln(-\ln(U))\,</math> |
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has a Fisher-Tippett distribution with parameters μ and β. This follows from the form of the cumulative distribution function given above. |
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==See also== |
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* [[order statistic]] |
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[[Category:Continuous distributions]] |
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Revision as of 16:09, 11 February 2006
Test
Testing.