An estimating function approach to finite population estimation

Godambe and Thompson (1986) define and develop simultaneous optimal estimation of superpopulation and finite population parameters based on a superpopulation model and a survey sampling design. Their theory defines the finite population parameter, \theta_N, as the solution of the optimal estimating equation for the superpopulation parameter \theta; however, some other finite population parameter, \phi, may be of interest. We propose to extend the superpopulation model in such a way that the parameter of interest, \phi, is a known function of \theta_N, say \phi = f (\theta_N). Then \phi is optimally estimated by f (\theta_s), where \theta_s is the optimal estimator of \theta_N, as given by Godambe and Thompson (1986), based on the sample s and the sampling design.