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This paper introduces the $f$-sensitivity model, a new sensitivity model thatcharacterizes the violation of unconfoundedness in causal inference. It assumesthe selection bias due to unmeasured confounding is bounded "on average";compared with the widely used point-wise sensitivity models in the literature,it is able to capture the strength of unmeasured confounding by not only itsmagnitude but also the chance of encountering such a magnitude.  We propose a framework for sensitivity analysis under our new model based ona distributional robustness perspective. We first show that the bounds oncounterfactual means under the f-sensitivity model are optimal solutions to anew class of distributionally robust optimization (DRO) programs, whose dualforms are essentially risk minimization problems. We then construct pointestimators for these bounds by applying a novel debiasing technique to theoutput of the corresponding empirical risk minimization (ERM) problems. Ourestimators are shown to converge to valid bounds on counterfactual means if anynuisance component can be estimated consistently, and to the exact bounds whenthe ERM step is additionally consistent. We further establish asymptoticnormality and Wald-type inference for these estimators under slower-than-root-nconvergence rates of the estimated nuisance components. Finally, theperformance of our method is demonstrated with numerical experiments.

Sensitivity analysis under the $f$-sensitivity models: a distributional robustness perspective