Unbiased and consistent variance estimators generally do not exist for design-based treatment effect estimators because experimenters never observe more than one potential outcome for any unit. The problem is exacerbated by interference and complex experimental designs. Experimenters must accept conservative variance estimators in these settings, but they can strive to minimize conservativeness. In this paper, we show that the task of constructing a minimally conservative variance estimator can be interpreted as an optimization problem that aims to find the lowest estimable upper bound of the true variance given the experimenter's risk preference and knowledge of the potential outcomes. We characterize the set of admissible bounds in the class of quadratic forms, and we demonstrate that the optimization problem is a convex program for many natural objectives. The resulting variance estimators are guaranteed to be conservative regardless of whether the background knowledge used to construct the bound is correct, but the estimators are less conservative if the provided information is reasonably accurate. Numerical results show that the resulting variance estimators can be considerably less conservative than existing estimators, allowing experimenters to draw more informative inferences about treatment effects.

翻译：在基于设计的处理效应估计中，由于实验者永远无法观测到同一单元的多个潜在结果，通常不存在无偏且一致的方差估计量。干扰效应和复杂的实验设计进一步加剧了这一问题。在这些情况下，实验者必须接受保守的方差估计量，但可以力求最小化其保守性。本文证明，构建最小保守方差估计量的任务可被解释为一个优化问题：在给定实验者风险偏好和潜在结果先验知识的条件下，寻找真实方差的最小可估计上界。我们在二次型类别中刻画了可采纳界的集合，并证明该优化问题对于许多自然目标函数是凸规划。由此得到的方差估计量无论构建界时所依据的背景知识是否正确，都能保证保守性；但若所提供信息具有合理准确性，估计量的保守程度将降低。数值结果表明，所得方差估计量相比现有估计量可显著降低保守性，使实验者能够对处理效应作出信息量更大的统计推断。