Bounds and semiparametric inference in $L^\infty$- and $L^2$-sensitivity analysis for observational studies

Sensitivity analysis for the unconfoundedness assumption is a crucial component of observational studies. The marginal sensitivity model has become increasingly popular for this purpose due to its interpretability and mathematical properties. After reviewing the original marginal sensitivity model that imposes a $L^\infty$-constraint on the maximum logit difference between the observed and full data propensity scores, we introduce a more flexible $L^2$-analysis framework; sensitivity value is interpreted as the "average" amount of unmeasured confounding in the analysis. We derive analytic solutions to the stochastic optimization problems under the $L^2$-model, which can be used to bound the average treatment effect (ATE). We obtain the efficient influence functions for the optimal values and use them to develop efficient one-step estimators. We show that multiplier bootstrap can be applied to construct a simultaneous confidence band of the ATE. Our proposed methods are illustrated by simulation and real-data studies.