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? 
 A: There are multiple formulations of an exponential family. But whichever one chooses to follow, the basic description is that if a random variable $X\sim p_{\theta}$ where $p_{\theta}$ is a probability model (pdf or pmf), then the family of distributions $P=\{p_{\theta}:\theta\in\Omega\}$ is a one-parameter exponential family (here $\theta$ is a scalar parameter) if $p_{\theta}$ can be expressed as
$$p_{\theta}(x)=\exp\{\eta(\theta)T(x)-B(\theta)\}h(x)\quad,\,x\in\mathscr{X}\,,\tag{*}$$
where $\mathscr{X}(\subseteq \mathbb R)$ is independent of $\theta$ and $\Omega$ is some (non-degenerate) subset of $\mathbb R$. 
Here $h,T$ are functions of $x$ only and $\eta,B$ are functions of $\theta$ only.
The pdf $f(x\mid\theta)$ in the question is a Gumbel density with unit scale and (unknown) location $\theta$. 
We have $$f(x\mid\theta)=\exp\{\eta(\theta)T(x)-B(\theta)\}h(x)\quad,\,x\in\mathbb R\quad,\theta\in\mathbb R,$$
where $\eta(\theta)=-e^{\theta},\,B(\theta)=-\theta,\,T(x)=e^{-x}$ and $h(x)=e^{-x}$.

So this is definitely a member of a one-parameter exponential family. 

In fact if the scale parameter $\sigma$ (say) is known, then the general location-scale Gumbel pdf given by $$p(x)=\frac{1}{\sigma}e^{-(x-\theta)/\sigma}\exp\left(-e^{-(x-\theta)/\sigma}\right)\qquad,\,x\in\mathbb R\quad,\theta\in\mathbb R,\sigma>0$$
also belongs to the exponential family by the same logic.
If the scale $\sigma$ is unknown, then clearly $p(\cdot)$ no longer remains in the exponential family. This is because we cannot find a $T(x)$ and an $h(x)$ in the form $(*)$ which is free of $\sigma$.
