Properties of the exponential family distributions

Given exponential family $\mathcal{P}=\{p_\theta(x) | \theta \in \Theta \}$, where
$$p_\theta(x) = h(x) \exp ( q^T(\theta) T(x) - b(\theta)  )  I_{supp}(x), \quad Z = \exp(- b(\theta))$$
Regular family (gives you completeness)
Conditions for regularity,

1. support $p_\theta(x)$ independent of $\theta$
2. finite partition function $Z(\theta) < \infty,\; \forall \theta$
3. Interior of parameter space is solid, $\mathring{\Theta} \neq \emptyset$, 
4. Interior of natural parameter space is solid $\mathring{\mathcal{Q}} \neq \emptyset$
5. Statistic vector function and the constant function are linearly independent.  i.e. $[1, T_1(x),\dotsc,T_K(x)]$ linear indep. (gives you minimal statistic)
6. twice differentiable $p_\theta(x)$ 

Curved family (only know statistic is minimal)
An exponential family where the dimension of the vector parameter $\mathbf{\theta}=(\theta_1,\dotsc,\theta_r)$ is less than the dimension of the natural statistic $\mathbf{T}(\mathbf{x})$ is called a curved family.

Identifiability of parameter vector $\mathbf{\theta}$.
When statistic is minimal, then it is a matter of ensuring $q: \Theta \mapsto \mathcal{Q}$ defines a 1-1 mapping from desired parameter space to natural parameter space.