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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?

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    $\begingroup$ 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. $\endgroup$ Apr 25, 2013 at 15:30
  • $\begingroup$ 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... $\endgroup$ Apr 25, 2013 at 15:54
  • $\begingroup$ @ChristopherErnst Sorry but ALL your questions have answers. Where is the problem? $\endgroup$
    – Did
    Apr 25, 2013 at 19:52
  • $\begingroup$ @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. $\endgroup$ Apr 25, 2013 at 20:02
  • $\begingroup$ Related:math.stackexchange.com/questions/1883900/…. $\endgroup$ Aug 13, 2018 at 18:17

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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 $\sigma$.

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