Regular Exponential Class

Definition

A PDF of form $f(x|\theta )=\exp [p(\theta )K(x)+H(x)+q(\theta )]I(x\in \mathcal{S})$ is said to be a member of the regular exponential class if

• $\mathcal{S}$, support of $x$, does not depend on $\theta$
• $p(\theta )$ is a non-trivial (constant) continuous function of $\theta \in \Omega$
• If $X$ is a continuous random variable, then $K^{\prime }(x)\ne 0$ and $H(x)$ is a continuous function of $x\in \mathcal{S}$
• If $X$ is a discrete random variable, then $K(x)$ is non-trivial function of $x\in \mathcal{S}$

Properties

Joint PDF

Let $X_1,X_2,\dots ,X_n$ be a Random Sample from a regular exponential class, then the joint pdf of $X_1,X_2,\dots ,X_n$ is ${\prod }_{i=1}^{n}f(x_i|\theta )=\exp \left[p(\theta ){\sum }_{i=1}^{n}K(x_i)+{\sum }_{i=1}^{n}H(x_i)+nq(\theta )\right]{\prod }_{i=1}^{n}I(x_i\in \mathcal{S})$

Monotone Likelihood Ratio

Let $X_1,X_2,\dots ,X_n$ be a Random Sample from a regular exponential class $f(x|\theta )=\exp [p(\theta )K(x)+H(x)+q(\theta )]I(x\in \mathcal{S})$ If $p(\theta )$ is monotone function of $\theta$, then likelihood ratio has Monotone Likelihood Ratio property in $y=\sum _{i=1}^{n}K(x_i)$

Facts

Let $X_1,X_2,\dots ,X_n$ be a Random Sample from a regular exponential class, and $Y_1=\sum _{i=1}^{n}K(x_i)$ be a Statistic, then

• The pdf of $Y_1$ is $f_{Y_1}(y_1|\theta )=R(y_1)\exp [p(\theta )y_1+nq(\theta )]$ where $R(y_1)$ is a function of $Y_1$ only.
• $E(Y_1)=-n\frac{q^{\prime }(\theta )}{p^{\prime }(\theta )}$
• $\mathrm{Var}(Y_1)=n\frac{1}{p^{\prime }(\theta {)}^3}\{p^{\prime \prime }(\theta )q^{\prime }(\theta )-q^{\prime \prime }(\theta )p^{\prime }(\theta )\}$
• $Y_1$ is a complete sufficient Statistic