medicine; Quantile regression; Robust regression.

Introduction
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Given the same drug or treatment, different patients may respond quite differently. Factors causing individual variability in drug response are multi-fold and complex. This has raised increasing interests of individualized medicine, where customized medicine or treatment is recommended to each individual according to his/her characteristics, including genetic, physiological, demographic, environmental, and other clinical information. The rule that applied in personalized medicine to match each patient with a target treatment is called individualized treatment rule (ITR), and our goal is to find the “optimal” one, which if followed by the whole patient population would lead to the “best” outcome. For many complex diseases such as cancer and AIDS, the optimal individualized treatment rule or regime is a dynamical treatment process, involving a sequence of treatment decisions made at different time points throughout the disease evolving course.

Q-learning [@watkins1992q; @murphy2005generalization] and A-learning [@murphy2003optimal; @robins2004optimal] are two main approaches for finding optimal dynamic individualized treatment rules based on clinical trials or observational data. Q-learning is based on posing a regression model to estimate the conditional expectation of the outcome at each time point, and then applying a backward recursive procedure to fit the model. A-learning, on 