On natural parameter space of exponential family (Q674693)

The natural parameter space \(\Theta\) of the exponential family \[ d P_\theta (x)=C (\theta)e^{\theta'x} d\mu(x), \quad\theta =(\theta_1, \dots, \theta_k)', \tag{1} \] is a convex set in \(\mathbb{R}^k\), where \(\mu\) is a \(\sigma\)-finite measure on \(\beta^k\), the \(\sigma\)-field of all Borel sets in \(\mathbb{R}^k\). Is the converse true? For \(k=1\) it is easy to see that the answer is in the affirmative. This note considers the problem for general \(k\). Theorem. Given an arbitrary closed or open set \(\Theta\) in \(\mathbb{R}^k\), there exists an exponential family (1) having natural parameter-space \(\Theta\). If \(k>1\), then for general convex sets this assertion follows to be true.