Approximate likelihood ratios for general estimating functions

SUMMARY: The method of estimating functions (Godambe, 1991) is commonly used when one desires to conduct inference about some parameters of interest but the full distribution of the observations is unknown. However, this approach may have limited utility, due to multiple roots for the estimating function, a poorly behaved Wald test, or lack of a goodness-of-fit test. This paper presents approximate likelihood ratios that can be used along with estimating functions when any of these three problems occurs. We show that the approximate likelihood ratio provides correct large sample inference under very general circumstances, including clustered data and misspecified weights in the estimating function. Two methods of constructing the approximate likelihood ratio, one based on the quasi-likelihood approach and the other based on the linear projection approach, are compared and shown to be closely related. In particular we show that quasi-likelihood is the limit of the projection approach. We illustrate the technique with two applications.