Estimation with missing not at random binary outcomes via exponential tilts

We study the problem of missing not at random (MNAR) datasets with binary outcomes. We propose an exponential tilt based approach that bypasses any knowledge on 'nonresponse instruments' or 'shadow variables' that are usually required for statistical estimation. We establish a sufficient condition for identifiability of tilt parameters and propose an algorithm to estimate them. Based on these tilt parameter estimates, we propose importance weighted and doubly robust estimators for any mean functions of interest, and validate their performances in a synthetic dataset. In an experiment with the Waterbirds dataset, we utilize our tilt framework to perform unsupervised transfer learning, when the responses are missing from a target domain of interest, and achieve a prediction performance that is comparable to a gold standard.