Verify a distribution that is not exponential family

I understand that if the support of a distribution depends on the parameter $\theta$, it is not exponential family even if its pdf can be written in the form $f(x | \theta) = h(x)c(\theta) \exp\left( \sum_{i=1}^{k} w_i(\theta)t_i(x) \right)$. For example, Verifying Exponential Family. But why the density $f(x | \theta) = e^{-(x-\theta)} \exp(-e^{-(x-\theta)}) , -\infty < x < \infty, -\infty < \theta <\infty$ , where I can identify $h(x)=e^{-x}, c(\theta)=e^\theta, w(\theta)=e^\theta, t(x)= -e^{-x}$ not an exponential family?

• I thought it was contingent on all of your real valued functions being greater then 0? If so, $t(x)\not\gt{0}$. I might be wrong on this, but I thought that was the case. – Eleven-Eleven Apr 25 '13 at 15:30
• I would tag probability here as well. There is a statistics stack exchange and most users who see statistics tags i think ignore these questions. I'm studying to be an actuary and have had questions such as these and they can go unanswers AND unseen. For example, you asked your question 36 minutes ago and have only 7 view, yet a question on operator theory was asked 4 minutes ago and has 43 views... – Eleven-Eleven Apr 25 '13 at 15:54
• @ChristopherErnst Sorry but ALL your questions have answers. Where is the problem? – Did Apr 25 '13 at 19:52
• @Did, sorry I meant to write "seen", not "had". Had implies they were mine, and that wasn't the case. I've just noticed that many of the questions related to statistics go unnoticed much more than other fields. I didn't know why that was until I noticed a comment on another question and there is a whole stack exchange devoted to statistics. And I really didn't mean unseen, I meant viewed with much less frequency. – Eleven-Eleven Apr 25 '13 at 20:02
• – StubbornAtom Aug 13 '18 at 18:17