How can write the probability density function of generalized exponential distribution as exponential family?

Question

I want to use GAM method and generalized exponential distribution for response variable. I know GAM method is a generalized GLM method and the distribution of response variable must be in exponential family. The probability density (pdf) of generalized exponential distribution is as following :

$$ f(x ; \alpha, \eta)=\alpha \eta \exp\left\{ -\eta x \right\}\cdot \left( 1-\exp(-\eta x) \right)^{\alpha - 1}, \quad x>0 $$ CDF of this distribution is as following : $$ F(x; \alpha, \eta) = \left(1-\exp(-\eta x)\right)^\alpha, \quad x>0 $$ The $\alpha$ is shape parameter and the $\eta# is scale parameter. How can I write this pdf as exponential family? That is, is the generalized exponential distribution a member of the exponential family? Also known as the *exponentiated exponential distribution, a special case of the Exponentiated Weibull distribution

Answer 1

No, you cannot. Just take the logarithm of the density, and you will see that the expression do not take the form of the argument to the exponential function needed for an exponential family. You need that the interaction of the argument $x$ and the parameters $\theta$ can be separated as $ \eta(\theta) \cdot T(x)$, where the dot represents a dot product. You don´t get that form.

But you want this for a gam, maybe look into generalized non-linear models, see how to fit a generalized nonlinear model in R? or https://stackoverflow.com/questions/59493784/writing-a-custom-model-for-gnm-package-but-need-a-function-not-in-derivative-ta