Estimating Individualized Decision Rules with Tail Controls
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With the emergence of precision medicine, estimating optimal individualized decision rules (IDRs) has attracted tremendous attentions in many scientific areas. Most existing literature has focused on finding optimal IDRs that can maximize the expected outcome for each individual. Motivated by complex individualized decision making procedures and popular conditional value at risk (CVaR) measures, we propose two new robust criteria to estimate optimal IDRs: one is to control the average lower tail of the subjects' outcomes and the other is to control the individualized lower tail of each subject's outcome. In addition to optimizing the individualized expected outcome, our proposed criteria take risks into consideration, and thus the resulting IDRs can prevent adverse events caused by the heavy lower tail of the outcome distribution. Interestingly, from the perspective of duality theory, the optimal IDR under our criteria can be interpreted as the decision rule that maximizes the "worst-case" scenario of the individualized outcome within a probability constrained set. The corresponding estimating procedures are implemented using two proposed efficient non-convex optimization algorithms, which are based on the recent developments of difference-of-convex (DC) and majorization-minimization (MM) algorithms that can be shown to converge to the sharpest stationary points of the criteria. We provide a comprehensive statistical analysis for our estimated optimal IDRs under the proposed criteria such as consistency and finite sample error bounds. Simulation studies and a real data application are used to further demonstrate the robust performance of our methods.
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