The Fréchet distribution, also known as inverse Weibull distribution,[2][3] is a special case of the generalized extreme value distribution. It has the cumulative distribution function
![{\displaystyle \Pr(X\leq x)=e^{-x^{-\alpha }}{\text{ if }}x>0.}](//wikimedia.org/api/rest_v1/media/math/render/svg/86f508ca017743874a40e571119ee0b642cb76af)
Quick Facts Parameters, Support ...
Fréchet
Probability density function |
Cumulative distribution function |
Parameters |
shape. (Optionally, two more parameters) scale (default: ) location of minimum (default: ) |
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Support |
![{\displaystyle x>m}](//wikimedia.org/api/rest_v1/media/math/render/svg/c70044adebdac942aae491d355d28aad0d86dc73) |
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PDF |
![{\displaystyle {\frac {\alpha }{s}}\;\left({\frac {x-m}{s}}\right)^{-1-\alpha }\;e^{-({\frac {x-m}{s}})^{-\alpha }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b971f0c59d782982ee66741ad5769631cd86d632) |
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CDF |
![{\displaystyle e^{-({\frac {x-m}{s}})^{-\alpha }}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1f7c84fadcc076ad78bf97615b2e4688fb3d485a) |
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Mean |
![{\displaystyle {\begin{cases}\ m+s\Gamma \left(1-{\frac {1}{\alpha }}\right)&{\text{for }}\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/72650606b8079267d4deb5e9baceb1b4ab305662) |
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Median |
![{\displaystyle m+{\frac {s}{\sqrt[{\alpha }]{\log _{e}(2)}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/31a72ce4ea6fe77d9c68731c0cafb36bf93dca71) |
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Mode |
![{\displaystyle m+s\left({\frac {\alpha }{1+\alpha }}\right)^{1/\alpha }}](//wikimedia.org/api/rest_v1/media/math/render/svg/afde1fa76c5e16288766a02319a0585e8933aa34) |
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Variance |
![{\displaystyle {\begin{cases}\ s^{2}\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\left(\Gamma \left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right)&{\text{for }}\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1f1d6b46d5502b2989375d81fcee28792592efb3) |
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Skewness |
![{\displaystyle {\begin{cases}\ {\frac {\Gamma \left(1-{\frac {3}{\alpha }}\right)-3\Gamma \left(1-{\frac {2}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+2\Gamma ^{3}\left(1-{\frac {1}{\alpha }}\right)}{\sqrt {\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right)^{3}}}}&{\text{for }}\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/920fdc70006de938b35dda1e8ee58a7fb0064e56) |
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Excess kurtosis |
![{\displaystyle {\begin{cases}\ -6+{\frac {\Gamma \left(1-{\frac {4}{\alpha }}\right)-4\Gamma \left(1-{\frac {3}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+3\Gamma ^{2}\left(1-{\frac {2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right]^{2}}}&{\text{for }}\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1f0e101297df7d5cbc11a6a96d305a162371856d) |
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Entropy |
, where is the Euler–Mascheroni constant. |
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MGF |
[1] Note: Moment exists if ![{\displaystyle \alpha >k}](//wikimedia.org/api/rest_v1/media/math/render/svg/690fcd2064786258688ba4d0be978ab4bb04d58c) |
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CF |
[1] |
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Close
where α > 0 is a shape parameter. It can be generalised to include a location parameter m (the minimum) and a scale parameter s > 0 with the cumulative distribution function
![{\displaystyle \Pr(X\leq x)=e^{-\left({\frac {x-m}{s}}\right)^{-\alpha }}{\text{ if }}x>m.}](//wikimedia.org/api/rest_v1/media/math/render/svg/fc8e544682a5495701d4377b0b08b71b141d885f)
Named for Maurice Fréchet who wrote a related paper in 1927,[4] further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.[5][6]