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(-eta*x) * ((1-exp(-eta*x))^(alpha-1))
CDF of this distribution is as following :
F(x; alpha, eta) = (1-exp(-eta*x))^alpha
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?