AIDS Clinical Trials Group Protocol 175 (ACTG175) data [@hammer1996trial], HIV-infected subjects were randomized to four regimes with equal probabilities, and our objective is to find the optimal ITR for each patient based on their age, weight, race, gender and some other baseline measurements. The response CD4 count of the data follows a skewed, heteroscedastic errors, which weakens the efficiency of classical Q- and A-learning. A method to estimate optimal ITR which is robust against skewed, heavy-tailed, heteroscedastic errors or outliers is highly valuable. One possible solution is to construct the optimal decision rule based on the conditional median or quantiles of response given covariates than based on average effects.

In the following, we present a simple example where a quantile-based decision rule is more preferable than a mean-based decision rules. We use higher value of response $Y$ to indicate more favorable outcomes. Figure \[fig:plot1\] plots the conditional density of $Y$ under two treatments, $A$ and $B$, given a binary covariate $X$ which takes the value of male and female. Under the comparison based on conditional means, $A$ and $B$ are exactly equivalent. However, conditional quantiles provide us more insight. For the male group, the conditional distribution of response given treatment 