  Propensity score trimming, which discards subjects with propensity scores
below a threshold, is a common way to address positivity violations that
complicate causal effect estimation. However, most works on trimming assume
treatment is discrete and models for the outcome regression and propensity
score are parametric. This work proposes nonparametric estimators for trimmed
average causal effects in the case of continuous treatments based on efficient
influence functions. For continuous treatments, an efficient influence function
for a trimmed causal effect does not exist, due to a lack of pathwise
differentiability induced by trimming and a continuous treatment. Thus, we
target a smoothed version of the trimmed causal effect for which an efficient
influence function exists. Our resulting estimators exhibit doubly-robust style
guarantees, with error involving products or squares of errors for the outcome
regression and propensity score, which allows for valid inference even when
nonparametric models are used. Our results allow the trimming threshold to be
fixed or defined as a quantile of the propensity score, such that confidence
intervals incorporate uncertainty involved in threshold estimation. These
findings are validated via simulation and an application, thereby showing how
to efficiently-but-flexibly estimate trimmed causal effects with continuous
treatments.
