Is this an exponential distribution? (Variant of Gumbel distribution)

Question

The distribution function is

$$F_x(x\mid\lambda) = e^{-e^{-\lambda x}}, \qquad \lambda > 0$$

Is this an exponential family?

The pdf that I obtained was

$$f(x\mid\lambda) = \lambda e^{-\lambda x} e^{-e^{-\lambda x}}$$

and the joint pdf of an iid sample is

$$f(x^n\mid\lambda) = \lambda^n e^{-\lambda \sum_{i=1}^n x_i} e^{-\sum_{i=1}^n (e^{-\lambda x_i})}$$

I don't see how I can bring this to the form that is required for it to be an exponential family. Am I right in concluding that this isn't an exponential family?

Also, is it possible to use the Karlin Rubin theorem to obtain a UMP test and thus a confidence interval for $\lambda$?

Answer 1

I don't see how this is a variant of the Gumbel distribution. It is the Gumbel distribution with scale parameter $\beta=1/\lambda$ and mean 0.

Writing the pdf:

$$f(x|\lambda)=\lambda \exp(-\lambda x-e^{-\lambda x})\neq h(x)\exp(\eta(\lambda)T(x)-A(\lambda)),$$

specifically because $e^{-\lambda x}$ cannot be factored into $\eta(\lambda)T(x)$. So it's not of exponential family.