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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