A unifying approach for doubly-robust ₁ regularized estimation of causal contrasts

We consider inference about a scalar parameter under a non-parametric model on a one-step estimator computed as a plug in estimator plus the mean of an estimator of the parameter's influence function. We focus a class of parameters that have influence function which depends on two dimensional nuisance functions and such that the bias of the one-step of the parameter of interest is the expectation of the product of the errors of the two nuisance functions. Our class includes many treatment effect contrasts of interest in causal inference and, such as ATE, ATT, an integrated causal contrast with a continuous, and the mean of an outcome missing not at random. We propose of the target parameter that entertain approximately sparse models for the nuisance functions allowing for the number of confounders to be even larger than the sample size. By employing splitting, cross-fitting and ₁-regularized regression estimators the nuisance functions based on objective functions whose directional agree with those of the parameter's influence function, we obtain of the target parameter with two desirable robustness properties:(1) they are rate doubly-robust in that they are root-n consistent and normal when both nuisance functions follow approximately sparse, even if one function has a very non-sparse regression coefficient, so as the other has a sufficiently sparse regression coefficient, and (2) are model doubly-robust in that they are root-n consistent and normal even if one of the nuisance functions does not follow an sparse model so long as the other nuisance function follows an sparse model with a sufficiently sparse regression coefficient.