We consider the gamma pdf of a random variable $Y$ that is given by:

$$f(y) = (s^a \Gamma (a))^{-1} y^{a-1}e^{-\frac{y}{s}},$$

where $y \geq 0$, $s$ is the scale parameter and $a$ the shape parameter. If we then reparameterise by setting $a=\frac{1}{\phi}$ and $s=\mu \phi$, we can show that $V[Y]=\phi\mu^{2}$. My question is, how can one then use this to assess whether data are best modelled as a normal, overdispersed Poisson or gamma distribution? Hypothesis test of some sort?

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    $\begingroup$ Your question is a little strange because (a) how one decides to model data shouldn't depend on how the possible distributions are parameterized and (b) there are such enormous differences between normal, Poisson, and Gamma distributions that it's unlikely anyone would be choosing among them; and if they were, the choice would be obvious. Do you have a specific data set you are trying to model? $\endgroup$ – whuber Dec 28 '18 at 14:55
  • $\begingroup$ First of all, thank you for your comment. Yeah, it did seem very odd to me too and so I thought I'd ask here. No, no data involved. It is the part of one of my assignment questions and it is all theoretical. I've done the other parts of the question, but I can't figure this bit out. $\endgroup$ – amator2357 Dec 28 '18 at 15:03
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    $\begingroup$ I prefer to write $f(y)\, dy = (\Gamma(a))^{-1} (y/s)^{a-1} e^{-y/s} \, (dy/s).$ It makes it clear why the exponent of $s$ is what it is, so that that's not just a brute fact to be memorized. $\qquad$ $\endgroup$ – Michael Hardy Dec 28 '18 at 19:49
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    $\begingroup$ can you clarify whether you're considering a single sample with a fixed mean or whether the mean (and hence implicitly the variance) may vary, e.g. as a function of covariates? $\endgroup$ – Ben Bolker Dec 28 '18 at 21:11

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