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.