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I have a dataset to analysis and the following information is known: $y_i \sim N(\mu_i, \theta(\mu_i)^2)$ The link function is $\ln(\mu_i) = (\beta)^\top X$

$y_i$s are count data. The model parameter is beta and theta. I need to find an estimation method and fit a model.

I have looked at the over-dispersion model and negative binomial models. But they don't seem quite right. Can anybody point me to the right direction?

Update: After some more research - I am looking for any R-package that will model data with the above properties?

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    $\begingroup$ If the $y_{i}$'s are count data then why have you written there that they follow a normal distribution? $\endgroup$
    – Macro
    Commented Dec 12, 2011 at 4:23
  • $\begingroup$ Yeah, it's a bit strange to me as well. The goal is to come up with a model to fit the data with the above property and compare that model with a negative binomial model. $\endgroup$
    – shelly
    Commented Dec 12, 2011 at 15:36
  • $\begingroup$ Just checking - are you assuming the variance is proportional to the mean? $\endgroup$
    – jbowman
    Commented Dec 12, 2011 at 17:48
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    $\begingroup$ Constant CV means that the standard deviation is proportional to the mean, not the variance. $\endgroup$
    – Hong Ooi
    Commented Dec 13, 2011 at 5:00
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    $\begingroup$ Look into a Poisson generalized linear model with log link, Shelly: this is appropriate for count data and for largish $\mu_i$ the distribution will be approximately Normal with variance proportional to the mean. $\endgroup$
    – whuber ♦
    Commented Dec 19, 2011 at 15:18

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Essentially answered in comments:

Look into a Poisson generalized linear model with log link, Shelly: this is appropriate for count data and for largish $𝜇_𝑖$ the distribution will be approximately Normal with variance proportional to the mean.

– whuber

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