Modified estimating functions

Abstract

In a parametric model the maximum likelihood estimator of a parameter of interest ψ may be viewed as the solution to the equation l' p (ψ)=0, where l P denotes the profile loglikelihood function. It is well known that the estimating function l' p (ψ) is not unbiased and that this bias can, in some cases, lead to poor estimates of ψ. An alternative approach is to use the modified profile likelihood function, or an approximation to the modified profile likelihood function, which yields an estimating function that is approximately unbiased. In many cases, the maximum likelihood estimating functions are unbiased under more general assumptions than those used to construct the likelihood function, for example under first- or second-moment conditions. Although the likelihood function itself may provide valid estimates under moment conditions alone, the modified profile likelihood requires a full parametric model. In this paper, modifications to l' p (ψ ) are presented that yield an approximately unbiased estimating function under more general conditions.