This paper establishes the functional average as an important estimand for causal inference. The significance of the estimand lies in its robustness against traditional issues of confounding. We prove that this robustness holds even when the probability distribution of the outcome, conditional on treatment or some other vector of adjusting variables, differs almost arbitrarily from its counterfactual analogue. This paper also examines possible estimators of the functional average, including the sample mid-range, and proposes a new type of bootstrap for robust statistical inference: the Hoeffding bootstrap. After this, the paper explores a new class of variables, the $\mathcal{U}$ class of variables, that simplifies the estimation of functional averages. This class of variables is also used to establish mean exchangeability in some cases and to provide the results of elementary statistical procedures, such as linear regression and the analysis of variance, with causal interpretations. Simulation evidence is provided. The methods of this paper are also applied to a National Health and Nutrition Survey data set to investigate the causal effect of exercise on the blood pressure of adult smokers.

翻译：本文确立了功能性平均估计量作为因果推断中的重要估计量。该估计量的意义在于其对传统混杂问题的稳健性。我们证明，当结果变量的概率分布（在给定处理或某些调整变量向量的条件下）与其反事实模拟几乎任意不同时，这种稳健性依然成立。本文还考察了功能性平均的可能估计量（包括样本中程数），并提出了一种用于稳健统计推断的新型自助法：霍夫丁自助法。随后，本文探索了一类新变量——$\mathcal{U}$类变量，该变量可简化功能性平均的估计。此类变量在某些情况下还可用于建立均值可交换性，并为基本统计程序（如线性回归和方差分析）的结果赋予因果解释。提供了模拟证据。本文方法还应用于美国国家健康与营养调查数据集，以研究运动对成年吸烟者血压的因果效应。