This is documentation for an old release of SciPy (version 1.3.2). Read this page in the documentation of the latest stable release (version 1.15.1).
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