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Completely randomized experiments have been the gold standard for drawingcausal inference because they can balance all potential confounding on average.However, they may suffer from unbalanced covariates for realized treatmentassignments. Rerandomization, a design that rerandomizes the treatmentassignment until a prespecified covariate balance criterion is met, hasrecently got attention due to its easy implementation, improved covariatebalance and more efficient inference. Researchers have then suggested to usethe treatment assignments that minimize the covariate imbalance, namely theoptimally balanced design. This has caused again the long-time controversybetween two philosophies for designing experiments: randomization versusoptimal and thus almost deterministic designs. Existing literature argued thatrerandomization with overly balanced observed covariates can lead to highlyimbalanced unobserved covariates, making it vulnerable to modelmisspecification. On the contrary, rerandomization with properly balancedcovariates can provide robust inference for treatment effects while sacrificingsome efficiency compared to the ideally optimal design. In this paper, we showit is possible that, by making the covariate imbalance diminishing at a properrate as the sample size increases, rerandomization can achieve its ideallyoptimal precision that one can expect with perfectly balanced covariates, whilestill maintaining its robustness. We further investigate conditions on thenumber of covariates for achieving the desired optimality. Our results rely ona more delicate asymptotic analysis for rerandomization. The derived theory forrerandomization provides a deeper understanding of its large-sample propertyand can better guide its practical implementation. Furthermore, it also helpsreconcile the controversy between randomized and optimal designs in anasymptotic sense.

Rerandomization with Diminishing Covariate Imbalance and Diverging Number of Covariates