Rejoinder: Learning Optimal Distributionally Robust Individualized Treatment Rules.

We thank the opportunity offered by editors for this discussion and the discussants for their insightful comments and thoughtful contributions. We also want to congratulate Kallus (2020) for his inspiring work in improving the effciency of policy learning by retargeting. Motivated from the discussion in Dukes and Vansteelandt (2020), we first point out interesting connections and distinctions between our work and Kallus (2020) in Section 1. In particular, the assumptions and sources of variation for consideration in these two papers lead to different research problems with different scopes and focuses. In Section 2, following the discussions in Li et al. (2020); Liang and Zhao (2020), we also consider the efficient policy evaluation problem when we have some data from the testing distribution available at the training stage. We show that under the assumption that the sample sizes from training and testing are growing in the same order, efficient value function estimates can deliver competitive performance. We further show some connections of these estimates with existing literature. However, when the growth of testing sample size available for training is in a slower order, efficient value function estimates may not perform well anymore. In contrast, the requirement of the testing sample size for DRITR is not as strong as that of efficient policy evaluation using the combined data. Finally, we highlight the general applicability and usefulness of DRITR in Section 3.