Before revision:  Specifically, we show that both the IPW and trimmed IPW estimators can have different (Gaussian or non-Gaussian) limiting distributions, depending on how "close to zero" the probability weights are and on  the trimming threshold . Our method provides more robust inference for the target estimand by adapting to these different limiting distributions.
After revision:  Specifically, we show that both the IPW and trimmed IPW estimators can have different (Gaussian or non-Gaussian) limiting distributions, depending on how "close to zero" the probability weights are and on the trimming threshold is. As a remedy, we propose an inference procedure that is robust not only to small probability weights entering the IPW estimator but also to a wide range of trimming threshold choices, by adapting to these different limiting distributions.