  Modern longitudinal studies collect feature data at many timepoints, often of
the same order of sample size. Such studies are typically affected by {dropout}
and positivity violations. We tackle these problems by generalizing effects of
recent incremental interventions (which shift propensity scores rather than set
treatment values deterministically) to accommodate multiple outcomes and
subject dropout. We give an identifying expression for incremental intervention
effects when dropout is conditionally ignorable (without requiring treatment
positivity), and derive the nonparametric efficiency bound for estimating such
effects. Then we present efficient nonparametric estimators, showing that they
converge at fast parametric rates and yield uniform inferential guarantees,
even when nuisance functions are estimated flexibly at slower rates. We also
study the variance ratio of incremental intervention effects relative to more
conventional deterministic effects in a novel infinite time horizon setting,
where the number of timepoints can grow with sample size, and show that
incremental intervention effects yield near-exponential gains in statistical
precision in this setup. Finally we conclude with simulations and apply our
methods in a study of the effect of low-dose aspirin on pregnancy outcomes.
