A smoothed Q-learning algorithm for estimating optimal dynamic treatment regimes

In this paper, we propose a smoothed Q-learning algorithm for estimating optimal dynamic treatment regimes. In contrast to the Q-learning algorithm in which nonregular inference is involved, we show that, under assumptions adopted in this paper, the proposed smoothed Q-learning estimator is asymptotically normally distributed even when the Q-learning estimator is not and its asymptotic variance can be consistently estimated. As a result, inference based on the smoothed Q-learning estimator is standard. We derive the optimal smoothing parameter and propose a data-driven method for estimating it. The finite sample properties of the smoothed Q-learning estimator are studied and compared with several existing estimators including the Q-learning estimator via an extensive simulation study. We illustrate the new method by analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness-Alzheimer's Disease (CATIE-AD) study.