Optimal estimating functions, quasi-likelihood and statistical modelling

The value of studying quasi-likelihood and recent extensions within the framework of estimating function theory is explored. The notion of a quasi-score function as an optimal surrogate for the true score function in semi-parametric models is outlined. It is shown that such quasi-score functions may be identified with optimal estimating functions within certain classes of unbiased estimating functions, defined in terms of an underlying semi-parametric model. Wedderburn's quasi-score function is seen to be a special case, for a particular semi-parametric model and a particular class of estimating functions. Applications of the estimating function approach to problems with overdispersed data and longitudinal data are discussed. A recent formulation of the analysis of case-control data via estimating functions is also discussed as well as the role of orthogonality and Bayesian inference in estimating function approaches.