Semiparametric efficient estimation for the auxiliary outcome problem with the conditional mean model

The authors consider semiparametric efficient estimation of parameters in the conditional mean model for a simple incomplete data structure in which the outcome of interest is observed only for a random subset of subjects but covariates and surrogate (auxiliary) outcomes are observed for all. They use optimal estimating function theory to derive the semiparametric efficient score in closed form. They show that when covariates and auxiliary outcomes are discrete, a Horvitz-Thompson type estimator with empirically estimated weights is semiparametric efficient. The authors give simulation studies validating the finite-sample behaviour of the semiparametric efficient estimator and its asymptotic variance; they demonstrate the efficiency of the estimator in realistic settings.