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I am trying to bring together the definition of the Generalized Gamma distribution in R-package GAMLSS by Rigby and Stasinopoulos and the general definition of the Generalized Gamma distribution, which I mainly found during this research.

The definition of the GAMLSS of the pdf is: $$ f_{Y}(y \mid \mu, \sigma, \nu) = \frac{|\nu| \theta^{\theta} z^{\theta} \exp(-\theta z)}{y \cdot \Gamma(\theta)}, \quad \text{ for } y>0, \mu>0, \sigma>0, \text{-Inf}<\nu<\text{Inf} $$ with $\theta = \frac{1}{\sigma^{2}|\nu|^{2}}$ and $z = \left(\frac{y}{\mu} \right)^{\nu}$. This definition can be found in the according help file and as well in one of their books on GAMLSS: http://www.gamlss.com/wp-content/uploads/2013/01/book-2008-27-6-08.pdf

One of the mainly used definitions of the pdf of the Generalized Gamma distribution is: $$ f(y) = \frac{|a|}{\beta^{ap} \Gamma(p)} \cdot y^{ap-1} \exp(-(y/\beta)^{a}) \quad \text{for } y>0. $$

I would like to understand how the reparametrization works. I tried to rearrange the GAMLSS formula and I come to the conclusion, that $\nu = a$ and $\frac{1}{\sigma^{2}|\nu|^{2}} = p$ but with this definition there seems to be no chance to get a proper definition of $\beta$, so I must be missing something.

The only reference they give is Lopatatzidis and Green (2000) which was an unpublished paper and never seemed to be published until today.

Please, can anybody help me out here?

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You can find the right reparameterization for $\beta$ by equating the arguments of the exponential function. The conversion is $$\beta=\mu \left( \sigma^2 \nu^2\right)^{(1/\nu)}$$

By using this in the GAMLSS definition (along with the equations you already found), you can get to your desired result using algebra.

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