Natural exponential family ($f_{\theta }(y)=h(y)e^{{\theta }^{T}y-\kappa (\theta )}$)

A natural exponential family uses its observation vector itself as the canonical sufficient statistic. Its cumulant function is $\kappa (\theta )=\log \int h(y)e^{{\theta }^{T}y}d\nu (y)$ and its natural parameter space consists of parameters for which this integral is finite. On the interior, differentiating gives mean parameter of an exponential family $\nabla \kappa$ and covariance matrix ${\nabla }^2\kappa$.

The convex support is the closed convex hull of the canonical sufficient-statistic support. For a minimal exponential family it has nonempty interior. Means at finite interior natural parameters lie in that interior, because a supporting hyperplane at a boundary mean would force the sufficient statistic itself onto that hyperplane.

A natural exponential family is regular when its parameter space is open. In a full regular family, differentiation under the normalizing integral is justified locally by exponential moments in a neighbourhood. Regularity does not guarantee that every possible data set has a finite maximum-likelihood estimator.

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.