We study the probability tail properties of Inverse Probability Weighting (IPW) estimators of the Average Treatment Effect (ATE) when there is limited overlap between the covariate distributions of the treatment and control groups. Under unconfoundedness of treatment assignment conditional on covariates, such limited overlap is manifested in the propensity score for certain units being very close (but not equal) to 0 or 1. This renders IPW estimators possibly heavy tailed, and with a slower than sqrt(n) rate of convergence. Trimming or truncation is ultimately based on the covariates, ignoring important information about the inverse probability weighted random variable Z that identifies ATE by E[Z]= ATE. We propose a tail-trimmed IPW estimator whose performance is robust to limited overlap. In terms of the propensity score, which is generally unknown, we plug-in its parametric estimator in the infeasible Z, and then negligibly trim the resulting feasible Z adaptively by its large values. Trimming leads to bias if Z has an asymmetric distribution and an infinite variance, hence we estimate and remove the bias using important improvements on existing theory and methods. Our estimator sidesteps dimensionality, bias and poor correspondence properties associated with trimming by the covariates or propensity score. Monte Carlo experiments demonstrate that trimming by the covariates or the propensity score requires the removal of a substantial portion of the sample to render a low bias and close to normal estimator, while our estimator has low bias and mean-squared error, and is close to normal, based on the removal of very few sample extremes.

翻译：本文研究在治疗组与对照组协变量分布重叠有限的情况下，逆概率加权（IPW）估计量用于平均处理效应（ATE）的概率尾部性质。在给定协变量条件下处理分配满足无混淆性时，这种有限重叠表现为某些单元的倾向得分非常接近（但不等于）0或1。这导致IPW估计量可能具有重尾特性，且收敛速度慢于√n。传统的截断或修剪方法完全基于协变量，忽略了识别ATE的关键信息——通过E[Z]=ATE定义的反概率加权随机变量Z。我们提出一种尾部修剪的IPW估计量，其性能对有限重叠具有稳健性。针对通常未知的倾向得分，我们在不可行Z中代入其参数估计量，然后通过自适应修剪大观测值对所得可行Z进行可忽略的修剪。当Z具有非对称分布和无限方差时，修剪会导致偏差，因此我们基于现有理论与方法的重要改进来估计并消除该偏差。我们的估计量规避了基于协变量或倾向得分修剪所伴随的维度问题、偏差问题以及对应关系不良问题。蒙特卡洛实验表明：基于协变量或倾向得分的修剪需要移除大量样本才能实现低偏差且接近正态的估计量，而我们的估计量仅需移除极少数极端样本，即可获得低偏差、低均方误差且接近正态的优良性质。