Full Rank Exponential Families 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?  
 A: 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.
