Notation |
![{\displaystyle {\textrm {GEV}}(\ \mu ,\ \sigma ,\ \xi \ )}](//wikimedia.org/api/rest_v1/media/math/render/svg/1fc546ddaefe8b73993c258d0efb44b5a48114c1) |
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Parameters |
μ ∈ ℝ — location, σ > 0 — scale, ξ ∈ ℝ — shape. |
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Support |
x ∈ [ μ - σ / ξ , +∞ ) when ξ > 0 ,
x ∈ ( −∞, +∞ ) when ξ = 0 ,
x ∈ ( −∞, μ - σ / ξ ] when ξ < 0 . |
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PDF |
where ![{\displaystyle \ t(x)\equiv {\begin{cases}\left[1+\xi \ \left(\ {\tfrac {\ x-\mu \ }{\sigma }}\right)\right]^{-{\tfrac {\ 1\ }{\xi }}}&~{\mathsf {if}}~\xi \neq 0\ ,\\{}\\\exp \left(\ -{\tfrac {\ x-\mu \ }{\sigma }}\ \right)&~{\mathsf {if}}~\xi =0\ ~.\\{}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/53f59ee014117d03940cf8761c95dda722fb1aa5) |
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CDF |
for support (see above) |
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Mean |
;\\{}\end{cases}}}
![{\displaystyle ={\begin{cases}{}\\\mu +{\tfrac {\ \sigma \ (g_{1}-1)\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\xi <1\ ,\\{}\\\mu +\sigma \ \gamma &~{\mathsf {if}}~\xi =0\ ,\\{}\\\infty &~{\mathsf {if}}~\xi \geq 1\ ;\\{}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/992a12cd7dfdbf4296874e1d0654861943432e73)
where gk ≡ Γ( 1 − k ξ ) , (see Gamma function) and is Euler’s constant. |
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Median |
![{\displaystyle ={\begin{cases}\mu +\sigma \cdot {\frac {\ (\ln 2)^{-\xi }\ -\ 1\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\\{}\\\mu -\sigma \cdot \ln \ln 2&~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/b16b5056bccb9038d1668672c4cbbf34746df093) |
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Mode |
![{\displaystyle ={\begin{cases}\mu +\sigma \cdot {\frac {\ (1+\xi )^{-\xi }\ -\ 1\ }{\xi }}&~{\mathsf {if}}~\xi \neq 0\ ,\\\mu &~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8eff0a252be449a96d550c68b4b9ba028f568ac8) |
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Variance |
![{\displaystyle ={\begin{cases}\sigma ^{2}\cdot {\tfrac {\ g_{2}-g_{1}^{2}\ }{\xi ^{2}}}&~{\mathsf {if}}~\xi \neq 0~{\mathsf {and}}~\xi <{\frac {\ 1\ }{2}}\ ,\\{}\\\sigma ^{2}\cdot {\frac {\ \pi ^{2}\ }{6}}&~{\mathsf {if}}~\xi =0\ ,\\{}\\\infty &~{\mathsf {if}}~\xi \geq {\tfrac {\ 1\ }{2}}~.\\{}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/71f85115653ccd10c1f2fc6b0d1098d1b96e6f83) |
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Skewness |
;\\{}\end{cases}}}
where is the sign function and is the Riemann zeta function |
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Excess kurtosis |
![{\displaystyle ={\begin{cases}{}\\{\frac {\ g_{4}\ -\ 4\ g_{3}\ g_{1}\ +\ 6\ g_{1}^{2}g_{2}\ -\ 3\ g_{1}^{4}\ }{\left(g_{2}\ -\ g_{1}^{2}\right)^{2}}}&~{\mathsf {if}}~\xi \neq 0~{\mathsf {and}}~\xi <{\tfrac {\ 1\ }{4}}\ ,\\{}\\{\tfrac {12}{\ 5\ }}&~{\mathsf {if}}~\xi =0~.\\{}\end{cases}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e81de46663fe516e38054cf6b59d916ed4df11c4) |
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Entropy |
![{\displaystyle =\ \ln(\sigma )\ +\ \gamma \ \xi \ +\ \gamma \ +\ 1\ }](//wikimedia.org/api/rest_v1/media/math/render/svg/203921fb35ba418d3a5b3f1c3f8415563b2831dc) |
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MGF |
see Muraleedharan, Soares & Lucas (2011)[1] |
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CF |
see Muraleedharan, Soares & Lucas (2011)[1] |
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Expected shortfall |
;\\{\ }\end{cases}}}
where is the lower incomplete gamma function and is the logarithmic integral function.[2] |
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