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

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

• Could you please explain what you mean by "defined by names" and "have names"?
– whuber
May 16, 2022 at 12:40
• Hi whuber, i have edited my question.
– DHJ
May 16, 2022 at 13:20
• 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. May 16, 2022 at 13:53
• 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.
– DHJ
May 16, 2022 at 14:26

The exponential dispersion family is most easily compared to the natural exponential family (which is like the exponential family with $$T(y) = y$$).
$$f(y|\theta) = \exp\left(\theta^Ty-A(\theta)\right)\cdot h(y)$$
$$f(y|\theta,\lambda) = \exp\left(\theta^Ty-\lambda A(\theta)\right) \cdot h(y,\lambda)$$
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.