Exponential family of distributions?

Question

Consider the parametric class formed by the density functions defined as follows:

$$ f(y,\theta) = \frac {2} {\Gamma (1/4)} e^{-(y-\theta)^4},\quad y\in\mathbb R,\quad\theta\in\mathbb R. $$ Does this parametric class forms an exponential family? I tried to write it in the form $$f(y, \theta)=q(y) e^{\phi(\theta)t(y)-\tau(\theta) }$$

I surely know that the right answer is yes, but how can I prove that? Writing it in logarithmic form $$f(y,\theta)=\ln \frac {2} {\Gamma (1/4)}-(y-\theta)^4$$

I assume that $$q(y)= \ln \frac {2} {\Gamma (1/4)}$$

But here I don't have two distinct functions $\phi(\theta)$ and $t(y)$, because they are with exponent $4$. And I also don't have a quantity $\tau(\theta)$.

What could I do?

Also, does this parametric class form a regular exponential family?

Thanks and sorry for my bad english.

Answer 1

Exponential family, indeed: a parametrization uses $\tau(\theta)=0$, $q(y)=2/\Gamma(1/4)$, $$t(y)=(y^4,y^3,y^2,y,1),\qquad\phi(\theta)=(-1,6\theta,-4\theta^2,6\theta^3,-\theta^4)^T. $$

Take-home message: the functions $t$ and $\phi$ used to parametrize exponential families of probability distributions are most often vector valued.