  Consider sensitivity analysis for estimating average treatment effects under
unmeasured confounding, assumed to satisfy a marginal sensitivity model. At the
population level, we provide new representations for the sharp population
bounds and doubly robust estimating functions, recently derived by Dorn, Guo,
and Kallus. We also derive new, relaxed population bounds, depending on
weighted linear outcome quantile regression. At the sample level, we develop
new methods and theory for obtaining not only doubly robust point estimators
for the relaxed population bounds with respect to misspecification of a
propensity score model or an outcome mean regression model, but also
model-assisted confidence intervals which are valid if the propensity score
model is correctly specified, but the outcome quantile and mean regression
models may be misspecified. The relaxed population bounds reduce to the sharp
bounds if outcome quantile regression is correctly specified. For a linear
outcome mean regression model, the confidence intervals are also doubly robust.
Our methods involve regularized calibrated estimation, with Lasso penalties but
carefully chosen loss functions, for fitting propensity score and outcome mean
and quantile regression models. We present a simulation study and an empirical
application to an observational study on the effects of right heart
catheterization.
