Balancing Covariates in Randomized Experiments with the Gram–Schmidt Walk Design
本文形式化了实验设计中协变量平衡与稳健性之间的权衡，提出一种由稳健性参数控制的实验设计，能在有限样本中平衡所有线性协变量函数，并给出渐近正态性和方差估计的条件。
The design of experiments involves a compromise between covariate balance and robustness.This article provides a formalization of this tradeoff and describes an experimental design that allows experimenters to navigate it.The design is specified by a robustness parameter that bounds the worst-case mean squared error of an estimator of the average treatment effect.Subject to the experimenter's desired level of robustness, the design aims to simultaneously balance all linear functions of potentially many covariates.Less robustness allows for more balance.We show that the mean squared error of the estimator is bounded in finite samples by the minimum of the loss function of an implicit ridge regression of the potential outcomes on the covariates.Asymptotically, the design perfectly balances all linear functions of a growing number of covariates with a diminishing reduction in robustness, effectively allowing experimenters to escape the compromise between balance and robustness in large samples.Finally, we describe conditions that ensure asymptotic normality and provide a conservative variance estimator, which facilitate the construction of asymptotically valid confidence intervals.