against heavy tailed errors or outliers. A dramatic difference of pinball loss, Huber loss and $\epsilon$-insensitive loss, compared with the square loss, is that they penalize large deviances linearly instead of quadratically. This property makes them more robust when dealing with responses with non-normal type of errors.

Asymptotic Properties {#section:asymptotic}
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Consistency of Robust Regression: Pinball Loss
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Under the conditional independence assumption $\epsilon\perp A|{{\bm X}}$, we have $$\begin{aligned}
Q({{\bm X}},A)=& \varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};\beta_0)+\mu_\epsilon({{\bm X}});\\
Q_\tau({{\bm X}},A)=& \varphi_0({{\bm X}})+\{A-\pi({{\bm X}})\}C({{\bm X}};\beta_0)+F^{-1}_{\epsilon}({{\bm X}};\tau).\end{aligned}$$ where $\mu_{\epsilon}({{\bm X}})$ and $F^{-1}_{\epsilon}({{\bm X}};\tau)$ denote the mean and the $\tau$-th quantile of $\epsilon$ conditional on ${{\bm X}}$ respectively. Therefore, in this situation, we have $g^{\mathrm{opt}}_\mu=g^{\mathrm{opt}}_{\tau}={\mathrm{1}}\{C({{\bm X}};{{\mbox{\boldmath $\beta$}}}_0)>0\}$. In other words, the underlying ITR which maximize the population mean and $\tau$-th quantile are equivalent. For a good ITR $\hat{g}={\mathrm{1}}\{C({{\bm X}};\hat{{{\mbox{\boldmath $\beta$}}}})>0\}$, it is reasonable to require $\hat{{{\mbox{\boldmath $\beta$}}}}$ to be a consistent estimator of ${{\mbox{\boldmath $\beta$}}}_0$. This consistency result is first shown for the robust regression with pinball loss, which is given in Theorem 1. We allocate all the proofs into the Appendix A.

Under regularity conditions (A1)-(A8) in the Appendix A, if the contrast function in is correctly specified and $\pi({{\bm x}})$ is known, then 