the other hand, only requires modeling the contrast function of the treatments at each time point, is therefore more flexible and robust to a model misspecification. See [@schulte2014q] for a complete review and comparison of these two methods under various scenarios, in terms of the parameter estimation accuracy and the estimation of expected outcomes. Q- and A-learning have good performance when model is correctly specified but are sensitive to model misspecification. To overcome this shortcoming, several “direct” methods have been proposed, which maximize value functions directly instead of modeling the conditional mean. See [@ZhaoYingQi2012OWL; @Zhang2013Robust] for example.

All existing methods for optimal individualized treatment rule estimation, including Q-learning and A-learning, belong to mean regression as they estimate the optimal estimator by maximizing expected outcomes. In the case of single decision point, Q-learning is equivalent to the least-squares regression. Least-squares estimates are optimal if the errors are i.i.d. normal random variables. However, skewed, heavy-tailed, heteroscedastic errors or outliers of the response are frequently encountered. In such situations, the efficiency of the least square estimates is impaired. One extreme example is that when the response takes i.i.d. Cauchy errors, neither Q-learning nor A-learning can consistently estimate the optimal ITR. For example, in 