Micro-randomized trials (MRTs) are increasingly utilized for optimizing mobile health interventions, with the causal excursion effect (CEE) as a central quantity for evaluating interventions under policies that deviate from the experimental policy. However, MRT often contains missing data due to reasons such as missed self-reports or participants not wearing sensors, which can bias CEE estimation. In this paper, we propose a two-stage, doubly robust estimator for CEE in MRTs when longitudinal outcomes are missing at random, accommodating continuous, binary, and count outcomes. Our two-stage approach allows for both parametric and nonparametric modeling options for two nuisance parameters: the missingness model and the outcome regression. We demonstrate that our estimator is doubly robust, achieving consistency and asymptotic normality if either the missingness or the outcome regression model is correctly specified. Simulation studies further validate the estimator's desirable finite-sample performance. We apply the method to HeartSteps, an MRT for developing mobile health interventions that promote physical activity.

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