What is the name of the functions in exponential dispersion family?

Question

If an exponential family is given by:

$g(y|\theta) = exp\{\theta^TT(y)-A(\theta)\}h(y)$

then the functions $h(y)$, $A(\theta)$ and $T(y)$ are defined by names:
$T(y)$ is a sufficient statistic
$A(\theta)$ is a cumulant function
$h(y)$ is an underlying measure

For an exponential dispersion family:

$f_Y(y;\theta, \phi) = exp\left\{\frac{(y\theta - b(\theta)}{a(\phi)}) + c(y, \phi) \right\}$

does $a(\phi)$, $b(\theta)$ and $c(y, \phi)$ similarly have names? This notation is from Nelder and McCullaghs Generalized Linear Models

Answer 1

The exponential dispersion family is most easily compared to the natural exponential family (which is like the exponential family with $T(y) = y$).

The natural exponential family

$$f(y|\theta) = \exp\left(\theta^Ty-A(\theta)\right)\cdot h(y)$$

The exponential dispersion family

$$f(y|\theta,\lambda) = \exp\left(\theta^Ty-\lambda A(\theta)\right) \cdot h(y,\lambda)$$

You get the same functions $A(\theta)$ and $h(x)$, but now there is an additional dependency on a parameter $\lambda$ which scales the precision (inverse of variance) of the distribution (and also adds an additional dependency of the precision on $y$).

Your expression with $a$, $b$ and $c$ rearranges these terms, but that does not make the difference between the exponential family and the exponential dispersion family.