  Causal excursion effect (CEE) characterizes the effect of an intervention
under policies that deviate from the experimental policy. It is widely used to
study the effect of time-varying interventions that have the potential to be
frequently adaptive, such as those delivered through smartphones. We study the
semiparametric efficient estimation of CEE and we derive a semiparametric
efficiency bound for CEE with identity or log link functions under working
assumptions, in the context of micro-randomized trials. We propose a class of
two-stage estimators that achieve the efficiency bound and are robust to
misspecified nuisance models. In deriving the asymptotic property of the
estimators, we establish a general theory for globally robust Z-estimators with
either cross-fitted or non-cross-fitted nuisance parameters. We demonstrate
substantial efficiency gain of the proposed estimator compared to existing ones
through simulations and a real data application using the Drink Less
micro-randomized trial.
