How to check if an exponential family is regular?

Question

A strictly $k$-parameter exponential family $$f=\exp\left(\sum \eta_i(\theta)T_i(x)-B(\theta)\right)h(x)$$ is regular if the natural parameter space $\eta(\theta)$ contains a $k$-dimensional open set.

I was given this counterexample but couldn't figure out why it is not regular: $$\exp\left[\sum_{j=1}^{k-1} \left(\log \theta_j−\log\left(1−\sum_{l=1}^{k-1}\theta_l\right)\right)N_j+n\log\left(1−\sum_{l=1}^{k-1}\theta_l\right)\right].$$ For reference this is the likelihood of $n$ i.i.d. random variables $x_i$, taking value $j\in \{1,...,k\}$ with probability $\theta_j$, and $N_j=\sum_i1_{x_i=j}$.

And in general, how do I check if an exponential family is regular?

Answer 1

The easy way to see this one isn't regular is that the $\theta_j$ need to add up to 1, so (given sufficient smoothness) they can only fill up at most a $k-1$-dimensional set.

Somewhat more generally, you could work out the information matrix and see that it is of rank less than $k$ everywhere in the interior of the parameter space. I don't know if that's a necessary condition to be non-regular, though.

(To make the calculations easier, you might start with $j=2$)