Existing statistical methods in causal inference often rely on the assumption that every individual has some chance of receiving any treatment level regardless of its associated covariates, which is known as the positivity condition. This assumption could be violated in observational studies with continuous treatments. In this paper, we present a novel integral estimator of the causal effects with continuous treatments (i.e., dose-response curves) without requiring the positivity condition. Our approach involves estimating the derivative function of the treatment effect on each observed data sample and integrating it to the treatment level of interest so as to address the bias resulting from the lack of positivity condition. The validity of our approach relies on an alternative weaker assumption that can be satisfied by additive confounding models. We provide a fast and reliable numerical recipe for computing our estimator in practice and derive its related asymptotic theory. To conduct valid inference on the dose-response curve and its derivative, we propose using the nonparametric bootstrap and establish its consistency. The practical performances of our proposed estimators are validated through simulation studies and an analysis of the effect of air pollution exposure (PM$_{2.5}$) on cardiovascular mortality rates.

翻译：现有因果推断统计方法常依赖于一个假设：每位个体无论其协变量如何，均有概率接受任一治疗水平，即正性条件。在连续治疗观察性研究中，该假设可能被违反。本文提出一种新颖的连续治疗因果效应（即剂量反应曲线）积分估计量，无需正性条件。该方法通过估计每个观测数据样本上治疗效应的导数函数，并将其积分至目标治疗水平，以修正因缺乏正性条件导致的偏倚。该方法的有效性依赖于一个可被加性混淆模型满足的替代性较弱假设。我们提供了实用中快速可靠的数值计算方案，并推导了其渐近理论。为对剂量反应曲线及其导数进行有效推断，我们提出使用非参数自助法并证明其一致性。通过模拟研究及空气污染暴露（PM$_{2.5}$）对心血管死亡率影响的分析，验证了所提估计量的实际效能。