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If $X \sim \Gamma(\alpha,\beta)$, $\alpha, \beta >0$, is a $\Gamma$-distributed r.v. with $$f^{X}(x;\alpha,\beta)= \begin{cases} \frac{\alpha^\beta}{\Gamma(\beta)}x^{\beta-1}e^{-\alpha x}, &x > 0,\\ 0, &x \le 0,\end{cases}$$ it is a known result that $X^r$, $r>0$, has the generalized $\Gamma$-distribution with density $$f^{GG}(x;\alpha,\beta, \delta)= \begin{cases} \frac{\delta \alpha^\beta}{\Gamma\left(\frac{\beta}{\delta}\right)}x^{\beta-1}e^{-(\alpha x)^{\delta}}, &x > 0,\\ 0, &x \le 0,\end{cases}$$ and $X^{-1}$ follows the inverse $\Gamma$-distribution with density $$f^{IG}(x;\alpha,\beta)= \begin{cases} \frac{\alpha^\beta}{\Gamma(\beta)}\left(\frac{1}{x}\right)^{\beta+1}e^{-\frac{\alpha}{x}}, &x > 0,\\ 0, &x \le 0.\end{cases}$$

I am able to show that $X^{-r}$, $r>0$, has a density of the form $$f(x;\alpha,\beta,\delta)= \begin{cases} \frac{\delta\alpha^\beta}{\Gamma\left(\frac{\beta}{\delta}\right)}\left(\frac{1}{x}\right)^{\beta+1}e^{-\left(\frac{\alpha}{x}\right)^{\delta}}, &x > 0,\\ 0, &x \le 0.\end{cases}$$

It seems like the latter should be a known density of something like the generalized inverse $\Gamma$-distribution but in the literature I only find links to the Stacy- or to the Amoroso-distribution which correspond to $f^{GG}$.

Question : Does anyone know the name of this distribution and a reference to it?

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  • $\begingroup$ Are $r$ and $\delta$ the same thing? $\endgroup$ – Henry Sep 11 '19 at 16:38
  • $\begingroup$ No, the reparametrization is more complicated. I just wanted to give the general form of the density. $\endgroup$ – spitzen Sep 11 '19 at 16:41
  • $\begingroup$ @Henry The actual reparametrizations for $r > 0$ would look like this:$X \sim \Gamma(\alpha, \beta) \Longrightarrow X^{r} \sim GG(\alpha^r,\frac{\beta}{r},\frac{1}{r})$, $X \sim \Gamma(\alpha, \beta) \Longrightarrow X^{-r} \sim GIG(\alpha^r,\frac{\beta}{r},\frac{1}{r})$ $\endgroup$ – spitzen Sep 12 '19 at 6:47
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I think I found an answer in the actuarial sciences. For everyone else looking for it:

https://actuarialmodelingtopics.wordpress.com/2017/06/17/transformed-gamma-distribution/

There it is called the "Inverse transformed gamma distribution".

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