Loss
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Without loss of generality, in this section we assume both the $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})$ take the linear form: $\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}})=\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\gamma$}}}$ and $C({{\bm X}};{{\mbox{\boldmath $\beta$}}})=\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}$, where $\tilde{{{\bm X}}}=(1,{{\bm X}}{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$. Denote $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)=\hat{{{\mbox{\boldmath $\beta$}}}}^{R}_{\rho(\tau)}$ and $\hat{{{\mbox{\boldmath $\gamma$}}}}(\tau)=\hat{{{\mbox{\boldmath $\gamma$}}}}^{R}_{\rho(\tau)}$. Denote ${{\bm W}}=(\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}},\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$, ${{\mbox{\boldmath $\theta$}}}(\tau)=({{\mbox{\boldmath $\beta$}}}(\tau){^{\mbox{\tiny {\sf T}}}},{{\mbox{\boldmath $\gamma$}}}(\tau){^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$, $\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)=(\hat{{{\mbox{\boldmath $\beta$}}}}(\tau){^{\mbox{\tiny {\sf T}}}},\hat{{{\mbox{\boldmath $\gamma$}}}}(\tau){^{\mbox{\tiny {\sf T}}}}){^{\mbox{\tiny {\sf T}}}}$ and $J(\tau)\triangleq{\mbox{E}}\left[f_Y({{\bm W}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\theta$}}}(\tau)|{{\bm X}},A){{\bm W}}{{\bm W}}{^{\mbox{\tiny {\sf T}}}}\right]$. Under the following regularity conditions, which is the same as the assumptions assumed in [@angrist2006quantile] and [@lee2013interpretation], we have the asymptotic normality of $\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)$, which is given in Theorem \[thm:asymptotic\_normality\_pinball\].

-   $\{(Y_i,{{\bm X}}_i,A_i,\epsilon_i),i=1,\ldots,n\}$ are i.i.d random variables;

-   the conditional density $f_Y(y|{{\bm X}}={{\bm x}},A=a))$ exists, and is bounded and uniformly continuous in y, uniformly in ${{\bm x}}$ over the support of ${{\bm X}}$;

-   $J(\tau)$ is positive definite for all $\tau\in(0,1)$, where ${{\mbox{\boldmath $\theta$}}}(\tau)$ is uniquely defined in ;

-   ${\mbox{E}}\|{{\bm X}}\|^{2+\epsilon}$ for some $\epsilon>0$.

\[thm:asymptotic\_normality\_pinball\] If regularity conditions (B1)-(B4) are hold, we have

1.  (**Uniform Consistency**) $\sup_{\tau}\|\hat{{{\mbox{\boldmath $\theta$}}}}(\tau)-{{\mbox{\boldmath $\theta$}}}(\tau)\|=o_p(1)$;

2.  