$\gamma$}}})-\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}]^2$, the asymptotic normality property of $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$ can then be established, which is summarized in Theorem \[thm:lsA\]. Its proof has been omitted, and readers are referred to [@lu2011variable].

\[thm:lsA\] Under the regularity condition of A1-A4 of [@lu2011variable], $$\sqrt{n}(\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}-{{\mbox{\boldmath $\beta$}}}_0)\indist N(0,U_{11}^{-1}\Omega_{11}U_{11}^{-1}),$$ where $U_{11}={\mbox{E}}\left[\pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right]$ and $$\Omega_{11}={\mbox{E}}\left[\left\{\varphi_0({{\bm X}})-\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}^*)+\epsilon\right\}^2
    \pi({{\bm X}})\{1-\pi({{\bm X}})\}\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\right]$$

**Remarks:**

1.  When the family of functions $\{\varphi({{\bm X}};{{\mbox{\boldmath $\gamma$}}}),{{\mbox{\boldmath $\gamma$}}}\in\Gamma\}$ cannot well approximate the unknown baseline function $\varphi_0({{\bm X}})$, the $\Omega_{11}$ term in the asymptotic variance of $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$ may explode, which makes $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$ less efficient than $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau)$.

2.  When $Y=\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}\gamma_0+\{A-\pi({{\bm X}})\}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}}{{\mbox{\boldmath $\beta$}}}_0+\epsilon$, $\epsilon\perp(A,{{\bm X}})$, $\pi({{\bm X}})\equiv0.5$ and $\epsilon\sim N(0,\sigma^2)$, the asymptotic variance of $\hat{{{\mbox{\boldmath $\beta$}}}}(\tau=0.5)$ is $2\pi\sigma^2{\mbox{E}}(\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}})^{-1}$, which is strictly larger than $4\sigma^2{\mbox{E}}(\tilde{{{\bm X}}}\tilde{{{\bm X}}}{^{\mbox{\tiny {\sf T}}}})^{-1}$ (the asymptotic variance of $\hat{{{\mbox{\boldmath $\beta$}}}}^{A}_{LS}$).

Numerical Results: Simulation Studies {#section:simulation}
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To demonstrate finite sample performance of the proposed robust regression methods for optimal treatment rule estimation, we conduct two simulation studies: the errors independent with treatments, and the errors interactive with treatments, respectively.

Simulation Study I: error terms independent with 