7.5 The Exponential Class of Distributions

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Definition 7.5.1: Regular exponential class

Definition

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

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

Theorem 7.5.1

Let

• $X_1,\dots ,X_n$ : Random sample, with
  • Distribution that represents a regular case of the exponential class
  • pdf/pmf $f(x;\theta )=\{\begin{array}{ll}\exp [p(\theta )K(x)+H(x)+q(\theta )]& x\in \mathcal{S}\\ 0& \text{elsewhere}\end{array}$
• $Y_1={\sum }_{i=1}^{n}K(X_i)$

Then

1. pdf/pmf of $Y_1$ has the form $f_{Y_1}(y_1;\theta )=R(y_1)\exp [p(\theta )y_1+nq(\theta )]$ for $y_1\in {\mathcal{S}}_{Y_1}$ and some function $R(y_1)$. Neither ${\mathcal{S}}_{Y_1}$ nor $R(y_1)$ depends on $\theta$
2. $E(Y_1)=-1\frac{q^{\prime }(\theta )}{p^{\prime }(\theta )}$
3. $\mathrm{Var}(Y_1)=n\frac{1}{p^{\prime }(\theta {)}^3}\{p^{\prime \prime }(\theta )q^{\prime }(\theta )-q^{\prime \prime }(\theta )p^{\prime }(\theta )\}$

Remark

Theorem 7.5.2 fits into the 4th case of this theorem.

Theorem 7.5.2

Let

• $X$ : Random variable, with
  • $f(x;\theta ),\gamma <\theta <\delta$ : pdf/pmf
  • Distribution is a regular case of exponential class
• $X_1,X_2,\dots ,X_n$ : Random sample from the distribution of $X$
• $Y_1={\sum }_{i=1}^{n}K\left(X_i\right)$

Then $Y_1$ is a complete sufficient statistic for $\theta$