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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

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    $\begingroup$ Could you please explain what you mean by "defined by names" and "have names"? $\endgroup$
    – whuber
    May 16 at 12:40
  • $\begingroup$ Hi whuber, i have edited my question. $\endgroup$
    – DHJ
    May 16 at 13:20
  • $\begingroup$ To the best of my knowledge, the ED model can be represented as $f_Y(y;\theta,\lambda)=\exp(\lambda(\theta^TT(y)-A(\theta)))h(y,\lambda)$ where $A,T,h$ have the same names, $\sigma^2$ is the dispersion parameter and $\lambda=1/\sigma^2$. I believe that replacing $\sigma^2$ with $\phi$ would fit your notation. $\endgroup$
    – Spätzle
    May 16 at 13:53
  • $\begingroup$ Thank you for your answer. This notation is from Nelder and McCullaghs Generalized Linear Models, and i have seen it in a couple of GLM related articles. $\endgroup$
    – DHJ
    May 16 at 14:26

1 Answer 1

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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.

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