Approximate likelihood ratios for general estimating functions

The method of estimating functions (Godambe, 1991) is commonlyused when one desires to conduct inference about some parametersof interest but the full distribution of the observations isunknown. However, this approach may have limited utility, dueto multiple roots for the estimating function, a poorly behavedWald test, or lack of a goodness-of-fit test. This paper presentsapproximate likelihood ratios that can be used along with estimatingfunctions when any of these three problems occurs. We show thatthe approximate likelihood ratio provides correct large sampleinference under very general circumstances, including clustereddata and misspecified weights in the estimating function. Twomethods of constructing the approximate likelihood ratio, onebased on the quasi-likelihood approach and the other based onthe linear projection approach, are compared and shown to beclosely related. In particular we show that quasi-likelihoodis the limit of the projection approach. We illustrate the techniquewith two applications.