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Rerandomization, a design that utilizes pretreatment covariates and improvestheir balance between different treatment groups, has received attentionrecently in both theory and practice. There are at least two types ofrerandomization that are used in practice: the first rerandomizes the treatmentassignment until covariate imbalance is below a prespecified threshold; thesecond randomizes the treatment assignment multiple times and chooses the onewith the best covariate balance. In this paper we will consider the second typeof rerandomization, namely the best-choice rerandomization, whose theory andinference are still lacking in the literature. In particular, we will focus onthe best-choice rerandomization that uses the Mahalanobis distance to measurecovariate imbalance, which is one of the most commonly used imbalance measurefor multivariate covariates and is invariant to affine transformations ofcovariates. We will study the large-sample repeatedly sampling properties ofthe best-choice rerandomization, allowing both the number of covariates and thenumber of tried complete randomizations to increase with the sample size. Weshow that the asymptotic distribution of the difference-in-means estimator ismore concentrated around the true average treatment effect underrerandomization than under the complete randomization, and propose large-sampleaccurate confidence intervals for rerandomization that are shorter than thatfor the completely randomized experiment. We further demonstrate that, withmoderate number of covariates and with the number of tried randomizationsincreasing polynomially with the sample size, the best-choice rerandomizationcan achieve the ideally optimal precision that one can expect even withperfectly balanced covariates. The developed theory and methods forrerandomization are also illustrated using real field experiments.

Asymptotic Theory of the Best-Choice Rerandomization using the Mahalanobis Distance