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?
[Math] Verify a distribution that is not exponential family
probability distributionsprobability theorystatistics
Best Answer
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 $h(x)=e^{-x}$.
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 $\sigma$.