3.6 Summary of relationships

We have previously seen the diagram for a GLM as

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That $Y$ belongs to the EF allows us to expand part of the diagram concentrating on the relationship between $Y$ and $\mu$.

With the canonical parameter $\theta$ fixed, the mean and variance of $Y$ are determined, and is represented as

Unnumbered Figure: Link

The facts that the relationship between $\mu$ and $\theta$, the mean value function, is invertible; and that the variance function determines the variance given the mean, may be portrayed by

Unnumbered Figure: Link

This diagram portrays the mean function, the inverse mean function (the link function), and the variance function.

GLMs concentrate on modelling the mean of a distribution where the variance is a function of the mean.