Generalized Linear Models with Varying Dispersion

SUMMARY Generalized linear models are further generalized to include a linear predictor for the dispersion as well as for the mean. It is shown how the convenient structure of generalized linear models can be carried over to this more general setting by considering the mean and dispersion structure separately. Mean and dispersion submodels are formulated for this, the dependent variable for the dispersion submodel being the deviance components of the mean submodel. The fact that both submodels are essentially generalized linear models themselves is used to derive simple expressions for the likelihood equations and for asymptotic tests. Estimation algorithms are proposed which have good convergence properties. The results apply mainly to the normal, inverse Gaussian and gamma distributions but can be extended to discrete distributions by using quasi-likelihoods. The methods developed are applied to a well-known data set.