  We present estimators for smooth Hilbert-valued parameters, where smoothness
is characterized by a pathwise differentiability condition. When the parameter
space is a reproducing kernel Hilbert space, we provide a means to obtain
efficient, root-n rate estimators and corresponding confidence sets. These
estimators correspond to generalizations of cross-fitted one-step estimators
based on Hilbert-valued efficient influence functions. We give theoretical
guarantees even when arbitrary estimators of nuisance functions are used,
including those based on machine learning techniques. We show that these
results naturally extend to Hilbert spaces that lack a reproducing kernel, as
long as the parameter has an efficient influence function. However, we also
uncover the unfortunate fact that, when there is no reproducing kernel, many
interesting parameters fail to have an efficient influence function, even
though they are pathwise differentiable. To handle these cases, we propose a
regularized one-step estimator and associated confidence sets. We also show
that pathwise differentiability, which is a central requirement of our
approach, holds in many cases. Specifically, we provide multiple examples of
pathwise differentiable parameters and develop corresponding estimators and
confidence sets. Among these examples, four are particularly relevant to
ongoing research by the causal inference community: the counterfactual density
function, dose-response function, conditional average treatment effect
function, and counterfactual kernel mean embedding.
