Full natural exponential family

A natural exponential family is full when its parameter set is the entire natural parameter space, with no restrictions on the allowable canonical parameters beyond finiteness of its normalizing integral. Fullness must not be confused with minimality or regularity.

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.