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?

  • $\begingroup$ At stats.stackexchange.com/a/519715/919 I describe a general method to solve this problem -- maybe it will help you with this one. $\endgroup$ – whuber May 12 at 19:44
  • $\begingroup$ Thank you @Whuber I have two parameters. For using this method, should I differentiate first w.r.t x and then w.r.t alpha and finally w.r.t eta? $\endgroup$ – user321525 May 13 at 5:36

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