Godambe estimating functions and asymptotic optimal inference

Godambe (1985) introduced a class of optimum estimating functions which can be regarded as a generalization of quasilikelihood score functions. The "optimality" established by Godambe (1985) within a certain class is for estimating functions and it is based on finite samples. The question that arises naturally is what (if any) asymptotic optimality properties do the estimators and tests based on optimum estimating functions possess. In this paper, we establish, via presenting a convolution theorem, asymptotic optimality of estimators and tests obtained from Godambe optimum estimating functions. It is noted that we do not require the knowledge of the likelihood function.