  This study designs an adaptive experiment for efficiently estimating average
treatment effects (ATEs). In each round of our adaptive experiment, an
experimenter sequentially samples an experimental unit, assigns a treatment,
and observes the corresponding outcome immediately. At the end of the
experiment, the experimenter estimates an ATE using the gathered samples. The
objective is to estimate the ATE with a smaller asymptotic variance. Existing
studies have designed experiments that adaptively optimize the propensity score
(treatment-assignment probability). As a generalization of such an approach, we
propose optimizing the covariate density as well as the propensity score.
First, we derive the efficient covariate density and propensity score that
minimize the semiparametric efficiency bound and find that optimizing both
covariate density and propensity score minimizes the semiparametric efficiency
bound more effectively than optimizing only the propensity score. Next, we
design an adaptive experiment using the efficient covariate density and
propensity score sequentially estimated during the experiment. Lastly, we
propose an ATE estimator whose asymptotic variance aligns with the minimized
semiparametric efficiency bound.
