Gamma exponential dispersion family
= Gamma exponential dispersion family
{c}
Writing $\phi=1/a$, $\theta=-r/a<0$, and $b(\theta)=-\log(-\theta)$ puts the gamma density into <exponential dispersion family> form
$$
f(y;\theta,\phi)=a(y,\phi)
\exp\left(\frac{y\theta-b(\theta)}{\phi}\right),
\qquad
a(y,\phi)=\frac{y^{1/\phi-1}}{\Gamma(1/\phi)\phi^{1/\phi}}.
$$
Its mean is $a/r$ and its variance is $a/r^2$.