I want to show if this generalized gamma (GG) distribution is a member of the exponential family. I don't know how to start since the exponential family has only 2 paramters and the GG has 3 parameters to be estimated. Below are definitions of the GG distr. and exponential family distr. for your perusal.

The density of the generalized gamma distribution with $y>0$ , shape parameter $\kappa$, scale parameter $\omega>0$ and location parameter $\eta>0$. \begin{equation} f_{Y}(y;\kappa,\eta,\omega)=\frac{\lambda^\lambda}{\omega\,y\,\Gamma(\lambda)\, \sqrt\lambda}\,\exp({z\,\sqrt{\lambda}-u}). \end{equation} with $\lambda=|\kappa|^{-2}$ , $z=sign(\kappa)\frac{\ln(y)-\eta}{\omega}$, and $u=\lambda\,\exp(|\kappa|\,z)$.


A random variable $Y$ is a member of the exponential family if its probability density function can be written in the form \begin{equation} f(y;\theta,\phi)=\exp\Bigg[\frac{y\,\theta-b(\theta)}{a(\phi)} + c(y,\theta)\Bigg] \end{equation}

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    $\begingroup$ I'm honestly not sure where you picked up that definition of "exponential family", but it doesn't look like anything I would call exponential family. AFAIK, people talk about specific exponential families, with different specific statistics, and different base measures. I don't think there is THE exponential family. Here is the wiki page: en.wikipedia.org/wiki/Exponential_family $\endgroup$ – Guillaume Dehaene Aug 11 '15 at 16:43
  • $\begingroup$ I don't see why you have written $sign(\kappa)$ and absolute values of $\kappa$ since $\kappa>0$? However, it looks like your formula could be simplified with some work so that it matches the definition in en.wikipedia.org/wiki/Generalized_gamma_distribution which is easily seen to be exponential familty (as defined in the wikipedia page!) $\endgroup$ – MarkoJ Aug 12 '15 at 16:51
  • $\begingroup$ @GuillaumeDehaene Thank you so much for the hint. I already looked through what wiki page. However i meant the exponential family for GLM. Looking at this lecture notes <stat.duke.edu/courses/Fall05/sta216/lecture2.pdf>, the density of the exponential family is given as above. $\endgroup$ – Zico Aug 14 '15 at 7:43
  • $\begingroup$ @MarkoJ thank you for correction. I edited it. However my parameterization of the generalized gamma is from the paper <citeseerx.ist.psu.edu/viewdoc/…> . I tried reconciling the two distributions but the definition of parameters are different in both distributions. $\endgroup$ – Zico Aug 14 '15 at 7:51
  • $\begingroup$ @Zico Realized that the generalized gamma actually seems to not be from exp. family since the exponent of x depends on parameter p. Also realised that there is a definition for overdispersed exponential family that matches your definition for exp. family, see en.wikipedia.org/wiki/Generalized_linear_model, and it is a generalization used in the context of GLMs. Page 24 in the paper you linked shows that your GG density is indeed quite similar to GG density in wikipedia but it seems to me, though this is not my area of expertise, that neither is from exp. family (either definition). $\endgroup$ – MarkoJ Aug 14 '15 at 18:16

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