One of the few points on which theoretical statisticians of all persuasions are agreed is the role played by the likelihood function in statistical modelling and inference. We have devoted the ﬁrst two chapters to illustrating this point. Given a statistical model we prefer to use likelihood inferences. However, there are many practical problems for which a complete probability mechanism (statistical model) is too complicated to specify fully or is not available, except perhaps for assumptions about the ﬁrst two moments, hence precluding a classical likelihood approach.
Typical examples are structured dispersions of non-Gaussian data for modelling the mean and dispersion. These are actually quite common in practice: for example, in the analysis of count data the standard Poisson regression assumes the variance is equal to the mean: V (μ) = μ. However, often there is extra-Poisson variation so we would like to ﬁt V (μ) = φμ with φ > 1, but it is now no longer obvious what distribution to use. In fact, Jørgensen (1986) showed that there is no GLM family on the integers that satisﬁes this mean-variance relationship, so there is no simple adjustment to the standard Poisson density to allow overdispersion. Wedderburn’s (1974) quasi-likelihood approach deals with this problem, and the analyst needs to specify only the mean-variance relationship rather than a full distribution for the data.