  Causal inference with observational studies often relies on the assumptions
of unconfoundedness and overlap of covariate distributions in different
treatment groups. The overlap assumption is violated when some units have
propensity scores close to 0 or 1, and therefore both practical and theoretical
researchers suggest dropping units with extreme estimated propensity scores.
However, existing trimming methods ignore the uncertainty in this design stage
and restrict inference only to the trimmed sample, due to the non-smoothness of
the trimming. We propose a smooth weighting, which approximates the existing
sample trimming but has better asymptotic properties. An advantage of the new
smoothly weighted estimator is its asymptotic linearity, which ensures that the
bootstrap can be used to make inference for the target population,
incorporating uncertainty arising from both the design and analysis stages. We
also extend the theory to the average treatment effect on the treated,
suggesting trimming samples with estimated propensity scores close to 1.

Methodology