# Deviance in a distribution that has a shape parameter

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.

• 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. – JDL Aug 31 '16 at 13:43
• 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. – Scortchi - Reinstate Monica Aug 31 '16 at 13:52
• @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. – Ernest A Aug 31 '16 at 13:53
• So it's like a correlation parameter? (Except that normally zero correlation implies independence) – JDL Aug 31 '16 at 13:56
• @Scortchi Thanks for the tip. I'll try to find some examples with the negative binomial. – Ernest A Aug 31 '16 at 13:56