Regular Exponential Class on Random Vectors

Definition

Let

• $X$ : Random variable, with
  • pdf/pmf $f(x;\mathbf{\theta })$, $\mathbf{\theta }\in \Omega \subset R^m$
• $\mathcal{S}$ : Support of $X$

Suppose

$$f(x;\mathbf{\theta })=\{\begin{array}{ll}\exp \left[{\sum }_{j=1}^m{p}_j(\mathbf{\theta })K_j(x)+H(x)+q({\theta }_1,\dots ,{\theta }_m)\right]& \forall x\in \mathcal{S}\\ 0& \text{elsewhere}\end{array}$$

If

1. $\mathcal{S}$ does not depend upon $\mathbf{\theta }$
2. Space $\Omega$ contains a nonempty, $m$-dimensional open rectangle
3. $p_j(\mathbf{\theta })$ are all nontrivial, functionally independent, continuous functions of $\mathbf{\theta }$
4. If $X$ continuous random variable, then
   1. $K_j^{\prime }(x)$ are all continuous for $a<x<b$, not homogeneous linear function of the others.
   2. $H(x)$ : Continuous function of $x\in \mathcal{S}$
5. If $X$ discrete random variable, then
   1. $K_j^{\prime }(x)$ are all nontrivial functions of $x\in \mathcal{S}$, not homogeneous linear function of the others