Gompertz (Truncated Gumbel) Distribution¶

For $$x\geq0$$ and $$c>0$$ . In JKB the two shape parameters $$b,a$$ are reduced to the single shape-parameter $$c=b/a$$ . As $$a$$ is just a scale parameter when $$a\neq0$$ . If $$a=0,$$ the distribution reduces to the exponential distribution scaled by $$1/b.$$ Thus, the standard form is given as

\begin{eqnarray*} f\left(x;c\right) & = & ce^{x}\exp\left[-c\left(e^{x}-1\right)\right]\\ F\left(x;c\right) & = & 1-\exp\left[-c\left(e^{x}-1\right)\right]\\ G\left(q;c\right) & = & \log\left[1-\frac{1}{c}\log\left(1-q\right)\right]\end{eqnarray*}
$h\left[X\right]=1-\log\left(c\right)-e^{c}\mathrm{Ei}\left(1,c\right),$

where

$\mathrm{Ei}\left(n,x\right)=\int_{1}^{\infty}t^{-n}\exp\left(-xt\right)dt$

Implementation: scipy.stats.gompertz

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