example, we could take $\epsilon=\sum_{j=1}^{K}\sigma_j({{\bm X}})e_j$ for any $K\geq 1$ that allows the error distribution to change with ${{\bm X}}$, used to model heterogeneous errors, where $\sigma_j({{\bm X}})$ are arbitrary positive functions and $e_{j}\perp (A,{{\bm X}})$ for all $j=1,\ldots,K$. Throughout the paper, we assume $\{(Y_i,{{\bm X}}_i,A_i,\epsilon_i),i=1,\ldots,n\}$ are i.i.d random samples of the population.

We propose to estimate $({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})$ by minimizing $$L_{3n}({{\mbox{\boldmath $\beta$}}}, {{\mbox{\boldmath $\gamma$}}})=\frac{1}{n}\sum_{i=1}^{n}M\left[Y_i-\varphi({{\bm X}}_i;{{\mbox{\boldmath $\gamma$}}})-\{A_i-\pi({{\bm X}}_i)\}C({{\bm X}}_i;{{\mbox{\boldmath $\beta$}}})\right],
\label{eq:A-general-loss}$$ where ${{\mbox{\boldmath $\gamma$}}}\in\Gamma$, ${{\mbox{\boldmath $\beta$}}}\in\mathcal{B}$ and $M:{\mathrm{I \! R} \mathit{^{\rightarrow}}} [0,\infty)$ is a convex function with minimum achieved at 0. Denote the minimizer of as $(\hat{{{\mbox{\boldmath $\beta$}}}}^R_{M},\hat{{{\mbox{\boldmath $\gamma$}}}}^{R}_{M})$, and the estimated ITR is then $\hat{g}^{R}_M({{\bm x}})\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^R_M)>0\}$. In the following, we refer the robust regression with loss function $M(x)$ as RR(M)-learning. In this article, we consider the following three types of loss functions, i.e., the pinball loss $$M(x)=\rho_\tau(x)\triangleq
\begin{cases}
(\tau-1)x, &\text{if } x<0\\
\tau x, &\text{if } x\geq0
\end{cases}$$ where $0<\tau<1$, the Huber loss $$M(x)=H_\alpha(x)\triangleq
\begin{cases}
0.5x^2, &\text{if } |x|<\alpha\\
\alpha|x|-0.5\alpha^2, &\text{if } |x|\geq\alpha
\end{cases}$$ for some $\alpha>0$, and the $\epsilon$-insensitive loss $$M(x)=J_\epsilon(x)\triangleq\max(0, |x|-\epsilon)$$ for some $\epsilon>0$. The pinball loss are frequently applied for quantile regression [@koenker2005quantile], and the Huber losses and the $\epsilon$-insensitive are robust 