# 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

## 1 Answer

$a$ 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.

• Could you please write down the expected shape of the exponential family so that we all agree on the definition of those terms? Oct 31 '15 at 16:24
• @Xi'an what do you mean by shape? Oct 31 '15 at 17:23
• It would be easier for the OP if you'd use the original notations. Oct 31 '15 at 18:41
• @Xi'an: As you pointed out in your comment, his notation doesn't make any sense. It would be better for him to cite a reference and adapt to it. Nov 2 '15 at 12:34