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