# Regression model with non-constant variance

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?

• If the $y_{i}$'s are count data then why have you written there that they follow a normal distribution? Commented Dec 12, 2011 at 4:23
• 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. Commented Dec 12, 2011 at 15:36
• Just checking - are you assuming the variance is proportional to the mean? Commented Dec 12, 2011 at 17:48
• Constant CV means that the standard deviation is proportional to the mean, not the variance. Commented Dec 13, 2011 at 5:00
• 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.
– whuber
Commented Dec 19, 2011 at 15:18

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.