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So I have a question here about the CMP distribution:
enter image description here

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

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    $\begingroup$ 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. $\endgroup$
    – Xi'an
    Oct 31 '15 at 15:42
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    $\begingroup$ 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? $\endgroup$
    – Xi'an
    Oct 31 '15 at 16:01
  • $\begingroup$ @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 $\endgroup$
    – Neil G
    Oct 31 '15 at 16:17
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$a$ 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.

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    $\begingroup$ Could you please write down the expected shape of the exponential family so that we all agree on the definition of those terms? $\endgroup$
    – Xi'an
    Oct 31 '15 at 16:24
  • $\begingroup$ @Xi'an what do you mean by shape? $\endgroup$
    – Neil G
    Oct 31 '15 at 17:23
  • $\begingroup$ @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. $\endgroup$
    – Neil G
    Oct 31 '15 at 18:22
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    $\begingroup$ It would be easier for the OP if you'd use the original notations. $\endgroup$
    – Xi'an
    Oct 31 '15 at 18:41
  • $\begingroup$ @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. $\endgroup$
    – Neil G
    Nov 2 '15 at 12:34

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