  The inverse probability (IPW) and doubly robust (DR) estimators are often
used to estimate the average causal effect (ATE), but are vulnerable to
outliers. The IPW/DR median can be used for outlier-resistant estimation of the
ATE, but the outlier resistance of the median is limited and it is not
resistant enough for heavy contamination. We propose extensions of the IPW/DR
estimators with density power weighting, which can eliminate the influence of
outliers almost completely. The outlier resistance of the proposed estimators
is evaluated through the unbiasedness of the estimating equations. Unlike the
median-based methods, our estimators are resistant to outliers even under heavy
contamination. Interestingly, the naive extension of the DR estimator requires
bias correction to keep the double robustness even under the most tractable
form of contamination. In addition, the proposed estimators are found to be
highly resistant to outliers in more difficult settings where the contamination
ratio depends on the covariates. The outlier resistance of our estimators from
the viewpoint of the influence function is also favorable. Our theoretical
results are verified via Monte Carlo simulations and real data analysis. The
proposed methods were found to have more outlier resistance than the
median-based methods and estimated the potential mean with a smaller error than
the median-based methods.
