Notation |
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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 |
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CDF |
for support (see above) |
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Mean |
;\\{}\end{cases}}}
where gk ≡ Γ( 1 − k ξ ) , (see Gamma function) and is Euler’s constant. |
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Median |
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Mode |
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Variance |
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Skewness |
;\\{}\end{cases}}}
where is the sign function and is the Riemann zeta function |
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Excess kurtosis |
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Entropy |
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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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