  The seminal work of Morgan and Rubin (2012) considers rerandomization for all
the units at one time. In practice, however, experimenters may have to
rerandomize units sequentially. For example, a clinician studying a rare
disease may be unable to wait to perform an experiment until all the
experimental units are recruited. Our work offers a mathematical framework for
sequential rerandomization designs, where the experimental units are enrolled
in groups. We formulate an adaptive rerandomization procedure for balancing
treatment/control assignments over some continuous or binary covariates, using
Mahalanobis distance as the imbalance measure. We prove in our key result,
Theorem 3, that given the same number of rerandomizations (in expected value),
under certain mild assumptions, sequential rerandomization achieves better
covariate balance than rerandomization at one time.
