  In this paper, we consider recent progress in estimating the average
treatment effect when extreme inverse probability weights are present and focus
on methods that account for a possible violation of the positivity assumption.
These methods aim at estimating the treatment effect on the subpopulation of
patients for whom there is a clinical equipoise. We propose a systematic
approach to determine their related causal estimands and develop new insights
into the properties of the weights targeting such a subpopulation. Then, we
examine the roles of overlap weights, matching weights, Shannon's entropy
weights, and beta weights. This helps us characterize and compare their
underlying estimators, analytically and via simulations, in terms of the
accuracy, precision, and root mean squared error. Moreover, we study the
asymptotic behaviors of their augmented estimators (that mimic doubly robust
estimators), which lead to improved estimations when either the propensity or
the regression models are correctly specified. Based on the analytical and
simulation results, we conclude that overall overlap weights are preferable to
matching weights, especially when there is moderate or extreme violations of
the positivity assumption. Finally, we illustrate the methods using a real data
example marked by extreme inverse probability weights.
