# Conway–Maxwell–Poisson (CMP) distribution and exponential family

So I have a question here about the CMP distribution:

My understanding is that $b(\theta)$ is only a function of $\theta$ but why is $v$ able to be included in that function, would $v$ not be a dispersion parameter instead; as that is the purpose of $v$.

I'm just having a hard time grasping the concept of exponential families as the parameterisation of $a(\phi), b(\theta)$ and $c(\phi, y)$ seem so arbitrary for me.

Update: the link at where I got this from: http://www.stat.purdue.edu/~ovitek/STAT526-Spring11_files/pdfs/hw8-sol.pdf

• Your question is unclear: the notations $\theta$, $\phi$, $a(\cdot)$, $b(\cdot)$, $c(\cdot,\cdot)$ are not standard so please rephrase how you define an exponential family using such notations. Oct 31 '15 at 15:42
• Before getting your reply, I must add that the answer sounds incorrect since the natural parameter should be $\theta=(\ln \lambda,\nu)$. Could you provide the reference from which you extracted this excerpt? Oct 31 '15 at 16:01
• @Xi'an: Oh I see, I looked at the equation, where you can see that the natural parameters are what you wrote. It's just that below in his "Therefore,…" he is not consistent Oct 31 '15 at 16:17

$a$ is your exponentiated carrier measure, and $b$ is your log-normalizer. The log-normalizer is a function of the parameter vector, which is $\theta = [\lambda, \nu]$. $\phi$ should to be the vector $y$. $c$ makes no sense at all — remove it.
• @Xi'an: The density is \begin{align} \newcommand{\mbx}{\mathbf x} \newcommand{\btheta}{\boldsymbol{\theta}} f(\mbx \mid \btheta) &= \exp\bigl(\eta(\btheta) \cdot T(\mbx) - g(\btheta) + h(\mbx)\bigr) \end{align} where $g$ is the log-normalizer, $h$ is the carrier measure, $\eta$ is the mapping to natural parameters and T is the mapping to sufficient statistics. Oct 31 '15 at 18:22