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Sensitivity analysis of unconfounded assumptions is an important component of observational studies. Marginal sensitivity models are becoming increasingly popular for this purpose due to their interpretability and mathematical properties. After reviewing the original marginal sensitivity model, which imposes $L^\infty$ constraints on the maximum logit difference between the observed data propensity score and the full data propensity score, we propose a more flexible $L^2$ analysis framework. introduce. Sensitivity values ​​are interpreted as the ‘average’ amount of unmeasured confounding in the analysis. Derive analytical solutions to stochastic optimization problems under the $L^2$ model. This can be used to limit the average therapeutic effect (ATE). We obtain the best-valued efficient influence functions and use them to develop an efficient one-step estimator. We show that we can apply multiplier bootstrapping to construct simultaneous confidence bands for ATE. The proposed method is illustrated by simulations and real data studies.

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