is correctly specified, ${{\mbox{\boldmath $\beta$}}}$ may still not be consistent if the propensity score $\pi({{\bm X}})$ is not consistently estimated. Finally, it can be shown that lsA-learning and Q-learning are equivalent when $\pi({{\bm X}})$ is constant and both $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ take the linear form (with the space of $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ included in the space of $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$). Similar properties hold for A-learning and Q-learning [@schulte2014q].

New Proposal: Robust Regression
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Skewed, heavy-tailed, heteroscedastic errors or outliers of the response $Y$ are frequently encountered in clinical trials. It is well known that ordinary least square estimation fails to produce a reliable estimator in such situations. The immediate consequence is the efficiency loss in the estimators produced by Q-, A-, and lsA-learning. This motivates us to adopt robust regression techniques in optimal treatment regime estimation.

We consider the following additive model, $$Y_i=\varphi_0({{\bm X}}_i)+\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;\beta_0)+\epsilon_i,\; i=1,\ldots,n,
\label{eq:model_nointeraction}$$ where $\varphi_0({{\bm X}})$ is the baseline function, $C({{\bm X}};\beta_0)$ is the contrast function, $\pi({{\bm X}})$ is the propensity score, and $\epsilon$ is the error term which satisfies the conditional independence assumption $\epsilon\perp A|{{\bm X}}$. We point out that the error term defined in can be very general. For 