On the efficiency of quasi-likelihood estimation

Abstract

A quasi-likelihood method has been proposed by Wedderburn (1974) for the estimation of parameters in regression models when there is some assumed relationship between the mean and variance of each observation but not necessarily a fully specified likelihood. If the underlying distribution comes from a natural exponential family the quasi-likelihood estimates maximize the likelihood and so have full asymptotic efficiency; under more general distributions there is some loss of efficiency, which is investigated here. Three types of model are discussed in detail: models with constant variance, models with constant coefficient of variation and models with overdispersion relative to some exponential family. The asymptotic efficiency of quasi-likelihood estimation is calculated under some particular distributions, and then more generally via an approximation for ‘small departures’ from the corresponding natural exponential family. The possibility of refinement of the quasi-likelihood approach, to incorporate additional information about the underlying distribution, is considered.