  In observational studies, balancing covariates in different treatment groups
is essential to estimate treatment effects. One of the most commonly used
methods for such purposes is weighting. The performance of this class of
methods usually depends on strong regularity conditions for the underlying
model, which might not hold in practice. In this paper, we investigate
weighting methods from a functional estimation perspective and argue that the
weights needed for covariate balancing could differ from those needed for
treatment effects estimation under low regularity conditions. Motivated by this
observation, we introduce a new framework of weighting that directly targets
the treatment effects estimation. Unlike existing methods, the resulting
estimator for a treatment effect under this new framework is a simple
kernel-based $U$-statistic after applying a data-driven transformation to the
observed covariates. We characterize the theoretical properties of the new
estimators of treatment effects under a nonparametric setting and show that
they are able to work robustly under low regularity conditions. The new
framework is also applied to several numerical examples to demonstrate its
practical merits.
