Gumbel (LogWeibull, Fisher-Tippetts, Type I Extreme Value) Distribution#

One of a class of extreme value distributions (right-skewed).

\begin{array}{rcl} f (x) & = & \exp (- (x + e^{- x})) \\ F (x) & = & \exp (- e^{- x}) \\ G (q) & = & - \log (- \log (q)) \end{array}

$\begin{eqnarray*} f\left(x\right) & = & \exp\left(-\left(x+e^{-x}\right)\right)\\ F\left(x\right) & = & \exp\left(-e^{-x}\right)\\ G\left(q\right) & = & -\log\left(-\log\left(q\right)\right)\end{eqnarray*}$

M (t) = Γ (1 - t)

$M\left(t\right)=\Gamma\left(1-t\right)$

\begin{array}{rcl} μ & = & γ = - ψ_{0} (1) \\ μ_{2} & = & \frac{π^{2}}{6} \\ γ_{1} & = & \frac{12 \sqrt{6}}{π^{3}} ζ (3) \\ γ_{2} & = & \frac{12}{5} \\ m_{d} & = & 0 \\ m_{n} & = & - \log (\log 2) \end{array}

$\begin{eqnarray*} \mu & = & \gamma=-\psi_{0}\left(1\right)\\ \mu_{2} & = & \frac{\pi^{2}}{6}\\ \gamma_{1} & = & \frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\ \gamma_{2} & = & \frac{12}{5}\\ m_{d} & = & 0\\ m_{n} & = & -\log\left(\log2\right)\end{eqnarray*}$

h [X] \approx 1.0608407169541684911

$h\left[X\right]\approx1.0608407169541684911$

Implementation: scipy.stats.gumbel_r