= Maximum likelihood existence in a full regular natural exponential family
{title2=$\nabla\kappa(\widehat\theta)=\bar Y$}
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 $\|\theta\|\to\infty$. 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 https://www.stat.umn.edu/geyer/5421/notes/expfam.html[Geyer's exponential-family lecture notes].
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