Is there a multivariate joint Amoroso distribution? The Amoroso distribution is a remarkable feat of abstraction as it exactly or asymptotically generalizes dozens of named probability distributions. Is there a published/pre-published treatment of multivariate Amoroso distributions? Either the cumulative density function (CDF) or the probability density function (PDF) would be acceptable answers.

This is so far not a full answer.  The Amoroso distribution is a generalization of the generalized gamma distribution, obtained by parametrizing its range (by replacing $x$ with $x-a$ so the lower bound of the range is no longer 0). So how do we obtain the generalized gamma?
We can do that by Weibullizing a standard gamma distribution. Start with the Gamma distribution with density in the form
$$
f(x)\; dx = \frac{1}{\Gamma(\alpha)} \left(\frac{x}{\theta}\right)^\alpha e^{\frac{x}{\theta}} \; \frac{dx}{x}
$$
Now introduce a new parameter $k$ by replacing $\frac{x}{\theta}$ with $\left(\frac{x}{\theta}\right)^k$ (and recalculate the normalizing factor). The result of this Weibullization is the generalized gamma distribution.
Now, there are multiple ways to define a multivariate gamma distribution, start with one of them and try Weibullization.
