Maximum likelihood existence in a full regular natural exponential family ($\nabla \kappa (\hat{\theta })=\bar{Y}$)

In a full regular minimal exponential family, a finite maximum-likelihood estimator exists precisely when the observed sufficient-statistic mean is in the interior of the convex support of an exponential family. It is unique because the log likelihood is strictly concave. For an interior mean, positive base-measure mass near finitely many support points surrounding the mean makes $\kappa (\theta )-{\theta }^T\bar{Y}$ grow at least linearly as $\Vert \theta \Vert \to \infty$. At a finite boundary of the full open natural domain, the normalizing integral diverges. Thus the likelihood attains its maximum in the domain. On a boundary face, no finite parameter can have the required mean. A regular restriction should instead use the exact mean-image criterion. These full-family conventions are also discussed in Geyer's exponential-family lecture notes.