  We consider a longitudinal data structure consisting of baseline covariates,
time-varying treatment variables, intermediate time-dependent covariates, and a
possibly time dependent outcome. Previous studies have shown that estimating
the variance of asymptotically linear estimators using empirical influence
functions in this setting result in anti-conservative estimates with increasing
magnitudes of positivity violations, leading to poor coverage and uncontrolled
Type I errors. In this paper, we present two alternative approaches of
estimating the variance of these estimators: (i) a robust approach which
directly targets the variance of the influence function as a counterfactual
mean outcome, and (ii) a non-parametric bootstrap based approach that is
theoretically valid and lowers the computational cost, thereby increasing the
feasibility in non-parametric settings using complex machine learning
algorithms. The performance of these approaches are compared to that of the
empirical influence function in simulations across different levels of
positivity violations and treatment effect sizes.
