  We study variants of the average treatment effect on the treated with
population parameters replaced by their sample counterparts. For each estimand,
we derive the limiting distribution with respect to a semiparametric efficient
estimator of the population effect and provide guidance on variance estimation.
Included in our analysis is the well-known sample average treatment effect on
the treated, for which we obtain some unexpected results. Unlike the ordinary
sample average treatment effect, we find that the asymptotic variance for the
sample average treatment effect on the treated is point-identified and
consistently estimable, but it potentially exceeds that of the population
estimand. To address this shortcoming, we propose a modification that yields a
new estimand, the mixed average treatment effect on the treated, which is
always estimated more precisely than both the population and sample effects. We
also introduce a second new estimand that arises from an alternative
interpretation of the treatment effect on the treated with which all
individuals are weighted by the propensity score.
