Semiparametric efficient empirical higher order influence function estimators

Robins et al. (2008, 2016b) applied the theory of higher order infuence functions (HOIFs) to derive an estimator of the mean of an outcome Y in a missing data model with Y missing at random conditional on a vector X of continuous covariates; their estimator, in contrast to previous estimators, is semiparametric efficient under minimal conditions. However the Robins et al. (2008, 2016b) estimator depends on a non-parametric estimate of the density of X. In this paper, we introduce a new HOIF estimator that has the same asymptotic properties as their estimator but does not require non-parametric estimation of a multivariate density, which is important because accurate estimation of a high dimensional density is not feasible at the moderate sample sizes often encountered in applications. We also show that our estimator can be generalized to the entire class of functionals considered by Robins et al. (2008) which include the average effect of a treatment on a response Y when a vector X suffices to control confounding and the expected conditional variance of a response Y given a vector X.