I am trying to do 6.20 in Casella and Berger part d. The solutions manual says that the order statistics are minimal sufficient and not complete. I understand their logic, but why doesn't this work?
The distribution is $$f(x)=e^{-(x-\theta)}e^{-e^{-(x-\theta)}}$$ The range is $-\infty<x<\infty$ $-\infty<\theta<\infty$
I think this breaks into an exponential family with $$f(x)=e^{-x}e^{\theta}e^{-e^{-x}e^{\theta}}$$
And this would imply $\sum{e^{-x}}$ is complete sufficient. A quick computation makes me think its minimal as well. Am I correct? The sample is $X_1, X_2....X_n$ iid from this distribution.