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. As the basis of $L^\infty$-sensitivity analysis, it assumes the logit difference between the observed and full data propensity scores is uniformly bounded. In this article, we introduce a new $L^2$-sensitivity analysis framework which is flexible, sharp and efficient. We allow the strength of unmeasured confounding to vary across units and only require it to be bounded marginally for partial identification. We derive analytical solutions to the optimization problems under our $L^2$-models, which can be used to obtain sharp bounds for the average treatment effect (ATE). We derive efficient influence functions and use them to develop efficient one-step estimators in both analyses. We show that multiplier bootstrap can be applied to construct simultaneous confidence bands for our ATE bounds. In a real-data study, we demonstrate that $L^2$-analysis relaxes the interpretation of $L^\infty$-analysis and provides a much more reliable calibration process using observed covariates. Finally, we provide an extension of our theoretical results to the conditional average treatment effect (CATE).

翻译：针对无混杂假设的敏感性分析是观察性研究的重要组成部分。边际敏感性模型因其可解释性和数学性质在此领域日益流行。作为$L^\infty$敏感性分析的基础，该模型假设观测数据与完整数据倾向得分之间的对数几率差一致有界。本文提出一种灵活、尖锐且高效的$L^2$敏感性分析新框架。我们允许未测量混杂强度随个体变化，仅要求其在部分识别时边际有界。我们推导了$L^2$模型下优化问题的解析解，可用于获得平均处理效应（ATE）的尖锐边界。通过推导有效影响函数，我们在两种分析中均构建了高效的一步估计量。研究表明，乘子自助法可用于构造ATE边界的联合置信带。在实际数据研究中，我们证明$L^2$分析放松了$L^\infty$分析的解释，并利用观测协变量提供了更可靠的校准过程。最后，我们将理论结果扩展至条件平均处理效应（CATE）。