Let $X_1\dots X_n$ be iid observations with pdf $f(x|\theta)=e^{-(x-\theta)}e^{-e^{-(x-\theta)}}$ for $-\infty<x<\infty$ and $-\infty<\theta<\infty$. I need to find a complete sufficient statistic, or show that one does not exist.
I tried the following. We can write the pdf as $f(x|\theta)= e^{-x} e^{\theta} e^{-e^{-x}e^{\theta}}$. Then, I believe this means that this belongs to the exponential family with $t(x)=e^{-x}$. Hence, I would claim that a complete and sufficient statistic is $\sum_{i=1}^n e^{-x_i}$. The textbook solutions, however, claims that the sufficient statistic is the order statistics, which are not complete. Where's the mistake in my logic?