Asymptotic Theory of Rerandomization in Treatment-Control Experiments

Although complete randomization ensures covariate balance on average, the chance for observing significant differences between treatment and control covariate distributions is high especially with many covariates. Rerandomization discards randomizations that do not satisfy a pre-determined covariate balance criterion, generally resulting in better covariate balance and more precise estimates of causal effects. Previously, researchers have established finite sample theory for rerandomization under the assumptions of equal treatment group sizes, Gaussian covariate and outcome distributions, or additive causal effects. Moreover, previous theory has not derived the actual sampling distribution of the difference-in-means estimator for the average causal effect. To allow for statistical inference in more realistic settings, we develop an asymptotic theory for rerandomization without any of these assumptions. Our theory, based on the geometry of rerandomization, reveals a non-Gaussian asymptotic distribution of the difference-in-means estimator. Our work provides an approach for constructing accurate large-sample confidence intervals for the average causal effect and demonstrates the advantages of rerandomization over complete randomization.