**Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Proof of Equation (6)**

Define the estimated residuals based on Winsorized estimator as:

$$
\boldsymbol{\epsilon}^\*\_{m\boldsymbol{j}} = \boldsymbol{X}^\*\_{m\boldsymbol{j}} - \bar{\boldsymbol{X}}^\*\_{\boldsymbol{j}} - (\mathbf{X}^\*\_{m,-\boldsymbol{j}} - \mathbf{X}^\*\_{-\boldsymbol{j}})\boldsymbol{\mathsf{b}}\_{\boldsymbol{j}'},
$$

where *X*¯ <sup>∗</sup> *<sup>j</sup>* <sup>=</sup> 1/*<sup>n</sup>* <sup>∑</sup>*<sup>n</sup> <sup>m</sup>*=<sup>1</sup> *X*<sup>∗</sup> *mj* and *<sup>X</sup>*¯ <sup>∗</sup> <sup>−</sup>*<sup>j</sup>* <sup>=</sup> 1/*<sup>n</sup>* <sup>∑</sup>*<sup>n</sup> <sup>m</sup>*=<sup>1</sup> *X*<sup>∗</sup> *m*,−*j* . The choice of *β*ˆ *<sup>j</sup>* must satisfy the following two conditions:

$$\|\hat{\mathsf{B}}\_{j} - \mathsf{B}\_{j}\|\_{\ell\_{1}} = O\_{p}(a\_{\mathsf{u}})\_{\prime}$$

$$\min \left\{ \lambda\_{\max}^{1/2}(\Sigma) \|\hat{\mathsf{B}}\_{j} - \mathsf{B}\_{j}\|\_{\ell\_{2}}, \max\_{1 \le j \le p} \sqrt{(\hat{\mathsf{B}}\_{j} - \mathsf{B}\_{j})^{T}\hat{\mathsf{B}}\_{-j, -j}(\hat{\mathsf{B}}\_{j} - \mathsf{B}\_{j})} \right\} = O\_{p}(b\_{\mathsf{u}})\_{\prime}$$

$$\text{where } a\_{\mathsf{u}}^{(k)} = o(\sqrt{\log p / n\_{k}}), \text{ and } b\_{\mathsf{u}}^{(k)} = o(n\_{k}^{-1/4}).$$

It is noteworthy to mention that the conditions above are slightly different from the conditions in [4] due to the different convergence rates by oracle data and imputed data. The conditions above can be satisfied by the rank-based estimators introduced in [6], e.g., rank-based lasso estimator or rank-based Dantzig selector. By letting ˜*mj* = *mj* − ¯*i*, we have:

$$\left\{\epsilon\_{m}^{\*}\epsilon\_{m\bar{j}}^{\*} = \bar{\epsilon}\_{m\bar{i}}\bar{\epsilon}\_{m\bar{j}} - \bar{\epsilon}\_{m\bar{i}}\left\{(\mathbf{X}\_{m,-\bar{j}}^{\*} - \bar{\mathbf{X}}\_{-\bar{j}}^{\*})\hat{\mathfrak{k}}\_{\bar{j}} - (\mathbf{X}\_{m,-\bar{j}} - \bar{\mathbf{X}}\_{-\bar{j}})\mathfrak{k}\_{\bar{i}}\right\}\tag{A1}$$

$$-\bar{\epsilon}\_{mj}\left\{(\mathbf{X}\_{m,-i}^\* - \mathbf{X}\_{-i}^\*)\hat{\mathfrak{P}}\_i - (\mathbf{X}\_{m,-i} - \mathbf{X}\_{-i})\mathfrak{P}\_i\right\}\tag{A2}$$

$$+\left\{\mathsf{j}\_{i}^{\mathsf{T}}(\mathbf{X}\_{m,-i}^{\*}-\mathbf{X}\_{-i}^{\*})^{\mathrm{T}}(\mathbf{X}\_{m,-j}^{\*}-\mathbf{X}\_{-j}^{\*})\right\}\_{j} - \mathsf{j}\_{i}^{\mathsf{T}}(\mathbf{X}\_{m,-i}-\mathbf{X}\_{-i})^{\mathrm{T}}(\mathbf{X}\_{m,-j}-\mathbf{X}\_{-j})\mathsf{j}\_{j}\right\}.\tag{A3}$$

First, for term (A3), we have:


where the last term can be bounded as follows:

$$\max\_{i,j} |(\clubsuit\_i - \clubsuit\_i)^T \mathbf{E}\_{-i,-j}(\clubsuit\_j - \clubsuit\_j)| = O\_{\mathbb{P}}(\lambda\_{\max}(\mathbf{E}) \max\_{1 \le i \le p} ||\clubsuit\_i - \clubsuit\_i||\_{\ell\_2}^2) = O\_{\mathbb{P}}(b\_n^2).$$

It is not hard to show that:

$$||\dot{\Sigma} - \Sigma||\_{\infty} = O\_p\left(\sqrt{\frac{\log p}{n}}\right).$$

therefore, the second term can also be bounded

$$\max\_{i,j} |(\hat{\mathfrak{B}}\_i - \mathfrak{B}\_i)^T (\mathfrak{B}\_{-i,-j} - \mathfrak{B}\_{-i,-j})(\hat{\mathfrak{B}}\_j - \mathfrak{B}\_j)| = O\_p\left(a\_n^2 \sqrt{\frac{\log p}{n}}\right).$$

Under some mild regularity conditions (stated in [6]), we have

$$||\vec{\Sigma} - \vec{\Sigma}||\_{\infty} = O\_p\left(\sqrt{\frac{\log p \log^2 n}{n^{1/2}}}\right),$$

,

thus under the condition that max*i*,*<sup>j</sup>* |*βi*,*j*| ≤ *C*<sup>1</sup> and *λ*min(Σ) = *o*((log *p*/*n*) 3 <sup>4</sup> ), the first term can be bounded as follows:

$$\begin{split} &\max\_{i,j} |\hat{\mathsf{B}}\_{i}^{T}(\hat{\mathsf{E}}\_{-i,-j} - \hat{\mathsf{E}}\_{-i,-j})\hat{\mathsf{B}}\_{j}| \\ &\leq \max\_{i,j} |\mathsf{A}\_{i}^{T}(\hat{\mathsf{E}}\_{-i,-j} - \hat{\mathsf{E}}\_{-i,-j})\mathsf{A}\_{j}| + \max\_{i,j} |(\hat{\mathsf{B}}\_{i} - \hat{\mathsf{A}}\_{i})^{T}(\hat{\mathsf{E}}\_{-i,-j} - \hat{\mathsf{E}}\_{-i,-j})(\hat{\mathsf{B}}\_{j} - \hat{\mathsf{B}}\_{j})| \\ &= O\_{p}\left(\sqrt{\frac{\log^{2}p\log^{2}n}{n^{3/2}}} + a\_{n}^{2}\sqrt{\frac{\log p\log^{2}n}{n^{1/2}}}\right). \end{split}$$

Combining the three terms above, we have

$$\begin{split} &|\frac{1}{n}\sum\_{m=1}^{n}\left\{\hat{\mathsf{B}}\_{i}^{T}(\mathbf{X}\_{m,-i}^{\*}-\bar{\mathbf{X}}\_{-i}^{\*})^{T}(\mathbf{X}\_{m,-j}^{\*}-\bar{\mathbf{X}}\_{-j}^{\*})\hat{\mathsf{B}}\_{j}-\mathfrak{B}\_{i}^{T}(\mathbf{X}\_{m,-i}-\bar{\mathbf{X}}\_{-i})^{T}(\mathbf{X}\_{m,-j}-\bar{\mathbf{X}}\_{-j})\mathfrak{B}\_{j}\right\}| \\ &=O\_{p}\left(\sqrt{\frac{\log^{2}p\log^{2}n}{n^{3/2}}}+a\_{n}^{2}\sqrt{\frac{\log p\log^{2}n}{n^{1/2}}}+b\_{n}^{2}\right). \end{split}$$

Next, we bound term (A1), which can be rewritten as:

$$
\varepsilon\_{\rm mi} (\mathbf{X}\_{m,-j} - \bar{\mathbf{X}}\_{-j})(\hat{\mathbf{j}}\_{j} - \mathbf{j}\_{j}) + \bar{\varepsilon}\_{\rm mi} \{ (\mathbf{X}\_{m,-j} - \bar{\mathbf{X}}\_{-j}) - (\mathbf{X}\_{m,-j}^{\*} - \bar{\mathbf{X}}\_{-j}^{\*}) \} \hat{\mathbf{j}}\_{j},
$$

where the first term can be further decomposed into two parts,

$$\mathbb{E}\_{\text{mi}}(\mathbf{X}\_{m,-j} - \mathbf{X}\_{-j})(\hat{\mathfrak{P}}\_{j} - \mathfrak{P}\_{\mathfrak{I}}) = \mathbb{E}\_{\text{mi}}(X\_{\text{mi}} - \mathbf{X}\_{\text{i}})(\hat{\mathfrak{P}}\_{\text{i},j} - \mathfrak{P}\_{\mathfrak{I},j})I\{i \neq j\} + \sum\_{\mathbf{l} \neq \mathbf{i}, \mathbf{j}} \mathbb{E}\_{\text{mi}}(X\_{\text{ml}} - \mathbf{X}\_{\text{l}})(\hat{\mathfrak{P}}\_{\text{l},\mathbf{j}} - \mathfrak{P}\_{\mathfrak{I},\mathbf{j}}) .$$

To bound ∑ *l*=*i*,*j* ˜*mi*(*Xml* <sup>−</sup> *<sup>X</sup>*¯*l*)(*β*<sup>ˆ</sup> *<sup>l</sup>*,*<sup>j</sup>* − *βl*,*j*), we use the independence between *mi* and *Xm*,−*i*. It is easy to show that

$$\max\_{1 \le i \le l} |\frac{1}{n} \sum\_{m=1}^n \tilde{\epsilon}\_{mi} (X\_{ml} - \bar{X}\_l)| = O\_p \left( \sqrt{\frac{\log p}{n}} \right)^j$$

,

which indicates that

$$\max\_{i,j} \left| \frac{1}{n} \sum\_{m=1}^{n} \left( \sum\_{l \neq i,j} \vec{\epsilon}\_{mi} (X\_{ml} - \vec{X}\_{l}) (\hat{\beta}\_{l,j} - \beta\_{l,j}) \right) \right| = O\_p \left( a\_n \sqrt{\frac{\log p}{n}} \right).$$

By the independence between *mi* and *X*<sup>∗</sup> *<sup>m</sup>* − *Xm*, it is not hard to show

$$\begin{split} &|\frac{1}{n}\sum\_{m=1}^{n}\bar{\varepsilon}\_{m\boldsymbol{l}}\{ (\mathbf{X}\_{m,-j}-\hat{\mathbf{X}}\_{-j})-(\mathbf{X}\_{m,-j}^{\*}-\mathbf{X}\_{-j}^{\*})\}\hat{\mathfrak{b}}\_{j}| \\ &\leq |\frac{1}{n}\sum\_{m=1}^{n}\bar{\varepsilon}\_{m\boldsymbol{l}}\{ (\mathbf{X}\_{m,-j}-\hat{\mathbf{X}}\_{-j})-(\mathbf{X}\_{m,-j}^{\*}-\mathbf{X}\_{-j}^{\*})\}\boldsymbol{\mathfrak{b}}\_{j}| + |\frac{1}{n}\sum\_{m=1}^{n}\bar{\varepsilon}\_{m\boldsymbol{l}}\{ (\mathbf{X}\_{m,-j}-\hat{\mathbf{X}}\_{-j})-(\mathbf{X}\_{m,-j}^{\*}-\hat{\mathbf{X}}\_{-j}^{\*})\}\boldsymbol{\mathfrak{b}}\_{j}| - \hat{\mathbf{b}}\_{j}| \\ &=O\_{p}\left(\frac{\log p}{n}+a\_{\mathrm{il}}\sqrt{\frac{\log p}{n}}\right). \end{split}$$

Combing term (A1) and term (A2), we have

$$\begin{split} \frac{1}{n} \sum\_{m=1}^{n} \mathbb{E}\_{mi} \{ (\mathbf{X}\_{m,-j}^{\*} - \mathbf{X}\_{-j}^{\*}) \hat{\mathfrak{B}}\_{j} - (\mathbf{X}\_{m,-j} - \mathbf{X}\_{-j}) \} \mathbf{\hat{g}}\_{j} &= \frac{1}{n} \sum\_{m=1}^{n} \mathbb{E}\_{mi} (X\_{mi} - X\_{i}) (\mathbb{\hat{\beta}\_{i,j}} - \boldsymbol{\beta}\_{i,j}) I \{ i \neq j \}, \\ &+ O\_{p} \Big( \frac{\log p}{n} + a\_{n} \sqrt{\frac{\log p}{n}} \Big). \end{split}$$

Summarizing all the results above, by Equations (22) and (23) of Liu (2013), we have

$$\begin{split} \frac{1}{n} \sum\_{m=1}^{n} \varepsilon\_{m}^{\*} \varepsilon\_{mj}^{\*} &= \frac{1}{n} \sum\_{m=1}^{n} \tilde{\varepsilon}\_{m} \varepsilon\_{mj} - \frac{1}{n} \sum\_{m=1}^{n} \tilde{\varepsilon}\_{mi} (X\_{mi} - \bar{X}\_{i}) \left(\hat{\beta}\_{i,j} - \beta\_{i,j}\right) I\{i \neq j\} \\ &- \frac{1}{n} \sum\_{m=1}^{n} \tilde{\varepsilon}\_{mj} (X\_{mj} - \bar{X}\_{j}) (\hat{\beta}\_{j,i} - \beta\_{j,i}) I\{i \neq j\} \\ &+ O\_{p} \left(\sqrt{\frac{\log^{2} p \log^{2} n}{n^{3/2}}} + a\_{n}^{2} \sqrt{\frac{\log p \log^{2} n}{n^{1/2}}} + a\_{n} \sqrt{\frac{\log p}{n}} + b\_{n}^{2}\right). \end{split}$$

As <sup>1</sup> *n n* ∑ *m*=1 ˜*mi*(*Xmi* <sup>−</sup> *<sup>X</sup>*¯*i*) = <sup>1</sup> *n n* ∑ *m*=1 ˜2 *mi* <sup>+</sup> <sup>1</sup> *<sup>n</sup>* ˜*mi*(*Xm*,−*<sup>i</sup>* <sup>−</sup> *<sup>X</sup>*¯ <sup>−</sup>*i*)*βi*, and Var(*Xm*,−*iβi*)=(*σiiωii* <sup>−</sup> <sup>1</sup>)/*ωii* <sup>≤</sup> *<sup>C</sup>*, we have 1 *n* log *p* 

$$\max\_{1 \le i \le p} |\frac{1}{n} \sum\_{m=1}^n \tilde{\varepsilon}\_{mi} (\mathbf{X}\_{m,-i} - \mathbf{X}\_{-i}) \mathfrak{F}| = O\_p \left( \frac{\log p}{n} \right),$$

therefore

$$\begin{split} \frac{1}{n} \sum\_{m=1}^{n} \bar{\epsilon}\_{mi} (X\_{mi} - \bar{X}\_{i}) &= \frac{1}{n} \sum\_{m=1}^{n} \bar{\epsilon}\_{mi}^{2} + O\_{p}(\log p/n) \\ &= \frac{1}{n} \sum\_{m=1}^{n} \epsilon\_{mi}^{\*2} + O\_{p} \left( \sqrt{\frac{\log^{2} p \log^{2} n}{n^{3/2}}} + a\_{n}^{2} \sqrt{\frac{\log p \log^{2} n}{n^{1/2}}} + a\_{n} \sqrt{\frac{\log p}{n}} + b\_{n}^{2} \right), \end{split}$$

$$\begin{split} \frac{1}{n} \sum\_{m=1}^{n} \mathfrak{c}\_{mi}^{\*} \mathfrak{c}\_{mj}^{\*} &= \frac{1}{n} \sum\_{m=1}^{n} \mathfrak{c}\_{mi} \mathfrak{c}\_{mj} - \frac{1}{n} \sum\_{m=1}^{n} \mathfrak{c}\_{mi}^{\*2} (\hat{\beta}\_{i,j} - \beta\_{i,j}) I \{i \neq j\} - \frac{1}{n} \sum\_{m=1}^{n} \mathfrak{c}\_{mj}^{\*2} (\hat{\beta}\_{j,i} - \beta\_{j,i}) I \{i \neq j\} \\ &+ O\_p \left( \sqrt{\frac{\log^2 p \log^2 n}{n^{3/2}}} + a\_n^2 \sqrt{\frac{\log p \log^2 n}{n^{1/2}}} + a\_n \sqrt{\frac{\log p}{n}} \right) . \end{split}$$

In addition

$$\frac{1}{n}\sum\_{m=1}^n \mathfrak{e}\_{mi}^2 = \frac{1}{n}\sum\_{m=1}^n \mathfrak{e}\_{mi}^2 + O\_p\left(\sqrt{\frac{\log^2 p \log^2 n}{n^{3/2}}} + a\_n^2 \sqrt{\frac{\log p \log^2 n}{n^{1/2}}} + a\_n \sqrt{\frac{\log p}{n}} + b\_n^2\right).$$

Equation (6) follows immediately by central limit theorem.
