I am trying to better understand the importance of full rank exponential families of distributions i.e. a family of populations dominated by a $\sigma$-finite measure such that the radon-nykodym derivative can be written as
$$ f_\theta(x)=h(x)e^{\eta(\theta)^tT(x)-\zeta(\theta)} $$
I am trying to understand why the statistic $T(x)$ is minimally sufficient only when the family of populations $f_\theta$ is of full rank i.e that there exists an open set within the parameter space of our family of populations. What happens if full rank is not satisfied?
First recall that the set of all possible values of $\eta$ is convex, i.e. all values of $\eta$ on a straight line between two such possible values is another such possible value.
The set of possible values of $\eta$ is that set $H\subseteq \mathbb R^p$ for which, if $\eta\in H$ then $$ \int_D h(x) \exp(\eta \cdot T(x) - A(\eta)) \,dx < +\infty. $$ Suppose $\eta_1,\eta_2\in H$ and $w_1,w_2\ge0$ and $w_1+w_2=1,$ so that $w_1\eta_1+w_2\eta_2 \in H.$ Then \begin{align} & \exp((w_1\eta_1+w_2\eta_2)\cdot T(x) -A(\eta)) \\[8pt] \le {} & w_1\exp(\eta_1\cdot T(x)-A(\eta)) + w_2 \exp(\eta_2\cdot T(x) - A(\eta)). \end{align} Consequently the integral with this weighted average in place of $\eta$ is finite, so the weighted average is within the parameter space $H.$
Next recall that a convex subset $H$ of a Euclidean space $\mathbb R^p$ includes some open subset of the smallest affine subspace of which $H$ is a subset.
Thus if $H$ includes no open subset of $\mathbb R^p,$ then $H$ is included within some affine subspace of lower-dimension, say $q.$ So here you have a $p$-tuble of real numbers constrained to lie in a $q$-dimensional affine subspace. Consequently a point in $H$ is determined by a $q$-tuple of scalars. Thus there should be a sufficient statistic with only $q$ scalar components. Some of the scalar components of $T$ are either determined by some others or else are not necessary statistics.