$B$ is a log-normal and skewed to the right. Therefore, treatment $B$ is less favorable when either 50% or 25% conditional quantile are considered. For the female group, the conditional distribution of response given treatment $A$ is a standard normal while a Cauchy distribution given treatment $B$. Therefore, if we make a comparison based on $25\%$ conditional quantile, treatment $A$ is more favorable.

![The distribution functions of the response $Y$, in a randomized clinical trial with two treatments, $A$ and $B$, for male (two panels on the left) and female (two panels on the right). The solid lines with triangle symbol, dashed line, and dotted lines are the conditional mean, $50\%$ quantile, and $25\%$ quantile functions of $Y$ given the gender and the treatment, respectively.[]{data-label="fig:plot1"}](plot1.eps){width="6in"}

In this paper, we propose a general framework for optimal individualized treatment rule estimation based on robust regression, including quantile regression and the regression based on Huber’s loss and $\epsilon$-insensitive loss. The proposed methodology has the following desired features. First, the new decision rule obtained by maximizing the conditional quantile, which is suitable for skewed, heavy-tailed errors or outliers. Second, the proposed estimator requires only modeling the contrast function between two treatments, and is therefore 