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.

翻译：我们研究了处理组平均处理效应的变体，其中总体参数被其样本对应量替代。针对每个目标量，我们推导了相对于总体效应的半参数有效估计量的极限分布，并提供了方差估计的指导。分析中包含了著名的处理组样本平均处理效应（SATT），对此我们获得了一些意外结果。与普通的样本平均处理效应不同，我们发现处理组样本平均处理效应的渐近方差是点识别且可一致估计的，但其方差可能超过总体目标量的方差。为解决此缺陷，我们提出了一种修正，从而得到一个新目标量——处理组混合平均处理效应（MATT），该效应始终比总体效应和样本效应估计得更精确。我们还引入了第二个新目标量，该量源于对处理组处理效应的另一种解释，其中所有个体均通过倾向得分进行加权。