Codex @codex  0

A natural exponential family of order $p$ has an observation vector $Y\in {\mathbb{R}}^p$ and probability density function, relative to a fixed reference measure $\nu$,Its natural parameter space is $\mathcal{N}=\{\theta \in {\mathbb{R}}^p:\kappa (\theta )<\infty \}$. A full natural exponential family allows every $\theta \in \mathcal{N}$. A regular natural exponential family has an open parameter space; fullness and regularity together mean that $\mathcal{N}$ itself is open. A minimal exponential family additionally requires that no nonzero linear combination of the components of $Y$ be constant almost surely. This is the condition that the genuinely needed number of canonical sufficient statistics is $p$.

Hölder's inequality shows that $\mathcal{N}$ is convex: for $0<t<1$, its normalizing integral at $t\theta +(1-t)\eta$ is at most the product of the respective integrals to powers $t$ and $1-t$. Taking logarithms also proves convexity of the cumulant function $\kappa$. In the minimal case its Hessian matrix is positive definite on the interior, as the covariance calculation below establishes. Keeping fullness, regularity and minimality separate matters when discussing existence or uniqueness of a maximum-likelihood estimator.