name for the distribution of a gamma RV raise to 1/p?

Question

This should be a very easy answer for those who know the distribution. Lately, I am dealing a lot with the following distribution:

$\rho\left(x|u,s,p\right)=\frac{x^{pu-1}p}{s^{u}\Gamma\left(u\right)}\exp\left(-\frac{x^{p}}{s}\right)$

It is obtained by raising a gamma distributed random variable with shape $u$ and scale $s$ to the power $\frac{1}{p}$ ($p>0$). The resulting distribution is a generalization of the $\chi$-distribution (for $p=2$ and $u=\frac{n}{p}$) or, for arbitrary $p>0$, the generalization of the radial distribution of a multivariate $p$-generalized Normal (for $u=\frac{n}{p}$).

My question is: Is there an "official" name for that distribution?

Answer 1

Unless I am mistaken, this distribution could be called a $p$-Gamma distribution because the Gamma distribution is Infinitely Divisible

Answer 2

I found the answer. It is embarrassingly simple: The distribution is called Generalized Gamma Distribution. Who would have thought of that? The corresponding publication is:

Stacy EW. A Generalization of the Gamma Distribution. The Annals of Mathematical Statistics. 1962;33(3):pp. 1187-1192. Available at: http://www.jstor.org/stable/2237889.

Answer 3

I don't know of a name for a general power of a general gamma distribution. But here are two special cases.

If the power is -1 and the gamma shape arbitrary, it's an inverse gamma distribution. And if the power is arbitrary and the gamma shape is 1 (i.e. exponential) then it's a Weibull distribution.