Is there any example of canonical exponential family distribution with a natural parameter space, that is not open?
An $k$ dimensional canonical exponential family means having a p.d.f (w.r.t base measure $u(dx)$)$$\exp\{T'\eta\}c(\eta),$$ where $T\in\mathbb{R}^k$ is the sufficient statistic and $\eta$ is the natural parameter.
A natural parameter space is defined as $$\Xi=\left\{\eta\in\mathbb{R}^k: \int\exp\{t'\eta\}u(dt)<\infty\right\}$$
I wonder if there is any canonical exponential family distribution having $\Xi$ being a non-open set in $\mathbb{R}^k$?
By the way, this is NOT a duplicated question.
