$ f(x | \theta) = e^{-(x-\theta)} \exp(-e^{-(x-\theta)}) , -\infty < x < \infty, -\infty < \theta <\infty $ , where I can identify $h(x)=e^{-x}, c(\theta)=e^\theta, w(\theta)=e^\theta, t(x)= -e^{-x}$ not an exponential family?
The definition of exponential family is $ f(x | \theta) = h(x)c(\theta) \exp\left( \sum_{i=1}^{k} w_i(\theta)t_i(x) \right) $.
Then by using theorem complete statistics in the exponential family , since here $w_1(\theta)=e^\theta$ contains an open set in $\Bbb{R}^1$. Thus, my statistic $\sum_i (-e^{-x_i})$ is complete.
However, the answer showed that it is not complete by using ancillary method. Can anyone show me where my method got wrong?