ABSTRACT
This chapter utilizes the word ‘non-linear’ in a specialized sense because all generalized linear models are in a strict sense non-linear. It describes a number of models in which unknown parameters enter either the variance or the link function or both. Intrinsically non-linear parameters complicate the fitting algorithm either by introducing an extra level of iteration or by introducing covariates that change at each iteration. The discrete distributions in their standard forms do not contain such parameters, although quasi-likelihood arguments extended the analysis to include an unknown dispersion parameter also. While link functions in generalized linear models are usually assumed known, it may be useful on occasion to assume that the link comes from a class of functions, members of the class being indexed by one or more unknown parameters. Exploration of the class of functions, used as transformations of the data rather than of the fitted values, was considered by G. E. P. Box and D. R. Cox.
