Theory and Methods of Statistics

2.2.12 Exponential Family of Distributions

A family of pdf’s or pmf’s is said to be an exponential family if it is of the form

$\begin{array}{l}\hfill f\left(x,\mathit{\theta }\right)=c\left(\mathit{\theta }\right)exp\left[\sum _{j=1}^k{Q}_j\left(\mathit{\theta }\right)T_j(x)\right]r(x),\mathit{\theta }\in \Theta ,\end{array}$

where $r(x)>0,c\left(\mathit{\theta }\right)>0$, and each T_j(x) is a real-valued function of x, while each $Q_j\left(\mathit{\theta }\right)$ is a real-valued function of θ.

Note

The set $\left\{x:f\left(x,\mathit{\theta }\right)>0\right\}=\left\{x:r(x)>0\right\}$ does not depend on θ.

Many distributions ...