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Consider the following log-likelihood function

$$ \ell (\vec{\mu}, \rho \mid \vec{y}) = \sum_{i=1}^I \log f(\mu_{i1}, \dotsc, \mu_{iJ}, \rho \mid y_{i1}, \dotsc, y_{iJ}) $$

where $\vec{y}$ ($I \times J$) is the response, $\vec{\mu}$ ($I \times J$) are the corresponding expected values and $\rho$ is some shape parameter.

How do we define the full model? In the context of GLMs, the full model is defined as a model with $N = I \times J$ parameters, one for each observation. But here if we set $\hat\mu_{ij} = y_{ij}$ for all $i,j$ there is still one parameter left $\rho$ which has to be assigned some value.

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  • $\begingroup$ I think it depends on the exact distribution and how it's affected by the shape parameter. If you meant to say a scale parameter, then things get much easier --- It's the ratio of the predicted deviance to the actual deviance. $\endgroup$ – JDL Aug 31 '16 at 13:43
  • $\begingroup$ Good question. Such models are usually shoe-horned into the generalized linear model framework by stipulating a relation between variance and mean (the latter being a function of the parameters), & estimating a dispersion parameter (not necessarily by maximum-likelihood). The gamma & negative binomial "GLM"s are examples. $\endgroup$ – Scortchi - Reinstate Monica Aug 31 '16 at 13:52
  • $\begingroup$ @JDL it's not a scale parameter, it's a parameter related to the joint probability distribution of the $Y_{i1}, \dotsc, Y_{iJ}$. If $\rho = 1$ it means the responses are independently distributed. $\endgroup$ – Ernest A Aug 31 '16 at 13:53
  • $\begingroup$ So it's like a correlation parameter? (Except that normally zero correlation implies independence) $\endgroup$ – JDL Aug 31 '16 at 13:56
  • $\begingroup$ @Scortchi Thanks for the tip. I'll try to find some examples with the negative binomial. $\endgroup$ – Ernest A Aug 31 '16 at 13:56

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