  Robins et al. (2008, 2017) applied the theory of higher order influence
functions (HOIFs) to derive an estimator of the mean $\psi$ 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 the minimal conditions of Robins et al. (2009b),
together with an additional (non-minimal) smoothness condition on the density g
of X, because the Robins et al. (2008, 2017) estimator depends on a
nonparametric estimate of g. In this paper, we introduce a new HOIF estimator
that has the same asymptotic properties as the original one, but does not
impose any smoothness requirement on g. This is important for two reasons.
First, one rarely has the knowledge about the properties of g. Second, even
when g is smooth, if the dimension of X is even moderate, accurate
nonparametric estimation of its density is not feasible at the sample sizes
often encountered in applications. In fact, to the best of our knowledge, this
new HOIF estimator remains the only semiparametric efficient estimator of
$\psi$ under minimal conditions, despite the rapidly growing literature on
causal effect estimation. 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. Simulation experiments are also conducted, which
demonstrate that our new estimator outperforms those of Robins et al. (2008,
2017) in finite samples, when g is not very smooth.
