Regular Exponential Class

Definition

Let $f(x;\theta )=\{\begin{array}{ll}\exp [p(\theta )K(x)+H(x)+q(\theta )]& x\in \mathcal{S}\\ 0& \text{elsewhere}\end{array}$

If

1. $\mathcal{S}$ does not depend upon $\theta$
2. $p(\theta )$ : Nontrivial continuous function of $\theta \in \Omega$
3. If $X$ : continuous random variable, then
   1. Each of $K^{\prime }(x)\not\equiv 0$
   2. $H(x)$ : Continuous function of $x\in \mathcal{S}$
4. If $X$ : discrete random variable, then
   1. $K(x)$ : Nontrivial function of $x\in \mathcal{S}$

Then we say $f(x;\theta )$ : member of the regular exponential class