  With many pretreatment covariates and treatment factors, the classical
factorial experiment often fails to balance covariates across multiple
factorial effects simultaneously. Therefore, it is intuitive to restrict the
randomization of the treatment factors to satisfy certain covariate balance
criteria, possibly conforming to the tiers of factorial effects and covariates
based on their relative importances. This is rerandomization in factorial
experiments. We study the asymptotic properties of this experimental design
under the randomization inference framework without imposing any distributional
or modeling assumptions of the covariates and outcomes. We derive the joint
asymptotic sampling distribution of the usual estimators of the factorial
effects, and show that it is symmetric, unimodal, and more "concentrated" at
the true factorial effects under rerandomization than under the classical
factorial experiment. We quantify this advantage of rerandomization using the
notions of "central convex unimodality" and "peakedness" of the joint
asymptotic sampling distribution. We also construct conservative large-sample
confidence sets for the factorial effects.
