Theorem 2 shows that $\hat{Q}\left({{\mbox{\boldmath $\beta$}}}(\tau),{{\mbox{\boldmath $\gamma$}}}(\tau)\right)$ is a weighted least square approximation to $Q_{\tau}({{\bm X}},A)$. In other word, $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}(\tau))+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}}(\tau))$ is close to $\varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};{{\mbox{\boldmath $\beta$}}}_0(\tau))$. So even though it is not true that ${{\mbox{\boldmath $\beta$}}}(\tau)={{\mbox{\boldmath $\beta$}}}_0(\tau)$ holds exactly, the difference between them is small in general . This coupled with the fact that $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}\inprob{{\mbox{\boldmath $\beta$}}}(\tau)$ (proved in Theorem \[thm:asymptotic\_normality\_pinball\]), leads to the conclusion that approximately ITR $\hat{g}^{R}_{\rho(\tau)}({{\bm x}})$ $(\triangleq{\mathrm{1}}\{C({{\bm x}};\hat{{{\mbox{\boldmath $\beta$}}}}^R_{\rho(\tau)})>0\})$ maximizes the $\tau$-th conditional quantile. This observation is justified numerically in Section 4.2.

2.  When there exists ${{\mbox{\boldmath $\gamma$}}}_0\in\Gamma$ such that $\varphi_0({{\bm X}})\equiv\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}_0)$, then we have ${{\mbox{\boldmath $\beta$}}}(\tau)={{\mbox{\boldmath $\beta$}}}_0(\tau)$.

Consistency of Robust Regression: Other Losses
----------------------------------------------

Under model and the assumption $\epsilon\perp A|{{\bm X}}$, similar consistency results can be established for Huber loss and the $\epsilon$-insensitive loss, as stated in Theorem \[thm:consistency\_huber\].

\[thm:consistency\_huber\] Under regularity conditions (A1)-(A8), if the contrast function in is correctly specified and $\pi({{\bm x}})$ is known, then we have

(a) $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{H(\alpha)}\inprob{{\mbox{\boldmath $\beta$}}}_0$ for all $\alpha>0$, where $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{H(\alpha)}$ is the solution of when $M(x)=H_{\alpha}(x)$;

(b) $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{J(\epsilon)}\inprob{{\mbox{\boldmath $\beta$}}}_0$ for all $\epsilon>0$, where $\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{J(\epsilon)}$ is the solution of when $M(x)=J_{\epsilon}(x)$.

Asymptotic Normality: Pinball 