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We consider a longitudinal data structure consisting of baseline covariates,time-varying treatment variables, intermediate time-dependent covariates, and apossibly time dependent outcome. Previous studies have shown that estimatingthe variance of asymptotically linear estimators using empirical influencefunctions in this setting result in anti-conservative estimates with increasingmagnitudes of positivity violations, leading to poor coverage and uncontrolledType I errors. In this paper, we present two alternative approaches ofestimating the variance of these estimators: (i) a robust approach whichdirectly targets the variance of the influence function as a counterfactualmean outcome, and (ii) a non-parametric bootstrap based approach that istheoretically valid and lowers the computational cost, thereby increasing thefeasibility in non-parametric settings using complex machine learningalgorithms. The performance of these approaches are compared to that of theempirical influence function in simulations across different levels ofpositivity violations and treatment effect sizes.

Robust variance estimation and inference for causal effect estimation