*2.1. Winsorized Estimator of the Latent Gaussian Variables*

In practice, the transformation functions *f* (*k*) = (*f* (*k*) <sup>1</sup> , ..., *f* (*k*) *<sup>p</sup>* ) in the nonparanormal distribution are unknown. However, one can use a Winsorized estimator to approximate *f* (*k*), i.e., to impute the latent Gaussian variables (oracle data) (*X*(*k*) *<sup>m</sup>*<sup>1</sup> , ..., *<sup>X</sup>*(*k*) *mp*)1≤*m*≤*nk* . To illustrate the Winsorized estimator, we define the following quantile function:

$$\hat{h}\_j^{(k)}(t) = \Phi^{-1}(\mathcal{F}\_j^{(k)}(t)), \ 1 \le j \le p\_{\prime}$$

where *F*˜(*k*) *<sup>j</sup>* is some estimator of the cumulative distribution function of *<sup>Y</sup>*(*k*) *<sup>j</sup>* , and a natural choice for *F*˜(*k*) *<sup>j</sup>* would be the empirical cumulative distribution function (eCDF)

$$\mathcal{F}\_j^{(k)}(t) = \frac{1}{n\_k} \sum\_{m=1}^{n\_k} I\{\mathcal{Y}\_{mj}^{(k)} \le t\}.$$

One major drawback of the eCDF above is that under high dimensionality, the variance of *F*ˆ(*k*) *<sup>j</sup>* (*t*) could be too large. To overcome the problem, Liu et al. (2009) considered a truncated (Winsorized) estimator as follows:

$$\mathcal{F}\_{j}^{(k)} = \begin{cases} \delta\_{n} & \mathcal{F}\_{j}^{(k)}(t) < \delta\_{n} \\ \mathcal{F}\_{j}^{(k)}(t) & \delta\_{n} \le \hat{\mathcal{F}}\_{j}^{(k)}(t) \le 1 - \delta\_{n} \\ 1 - \delta\_{n} & \mathcal{F}\_{j}^{(k)}(t) > 1 - \delta\_{n} \end{cases}$$

where *δ<sup>n</sup>* serves as the truncation parameter that should be carefully chosen. Liu et al. (2009) [5] suggested *δ<sup>n</sup>* = 1/(4*n*1/4*π* log *n*) to balance the bias and variance of eCDF, and so we will use this value in our calculations. To estimate the transformation functions and impute the latent Gaussian variable *X*, we define

$$X\_{m\mathbf{j}}^{(k)\*} = f\_{\mathbf{j}}^{(k)}(Y\_{m\mathbf{j}}^{(k)}) = \boldsymbol{\mu}\_{\mathbf{j}}^{(k)} + \boldsymbol{\sigma}\_{\mathbf{j}}^{(k)}\boldsymbol{\hbar}\_{\mathbf{j}}^{(k)}(Y\_{m\mathbf{j}}^{(k)}),$$

where ˜ *h* (*k*) *<sup>j</sup>* (*t*), *μ*ˆ (*k*) *<sup>j</sup>* and *σ*ˆ (*k*) *<sup>j</sup>* are given below:

$$
\tilde{h}\_j^{(k)}(t) = \Phi^{-1}(\tilde{F}\_j^{(k)}(t)),
$$

$$
\hat{\mu}\_j^{(k)} = \frac{1}{n\_k} \sum\_{m=1}^{n\_k} Y\_{mj}^{(k)}.
$$

$$
\phi\_j^{(k)} = \sqrt{\frac{1}{n\_k} \sum\_{m=1}^{n\_k} (Y\_{mj}^{(k)} - \hat{\mu}\_j^{(k)})^2}.
$$

The Winsorized estimator *X*(*k*)<sup>∗</sup> *mj* generally works well in approximating the unknown *<sup>X</sup>*(*k*) *mj* , and it could be used to estimate the oracle sample covariance. Let <sup>Σ</sup><sup>ˆ</sup> (*k*) be the sample covariance matrix by the oracle data, and <sup>Σ</sup>˜ (*k*) be the sample covariance matrix by (*X*(*k*)<sup>∗</sup> <sup>1</sup> , ..., *<sup>X</sup>*(*k*)<sup>∗</sup> *<sup>p</sup>* ), that is

$$\mathbf{E}^{(k)} = \frac{1}{n\_k} \sum\_{m=1}^{n\_k} (X\_m^{(k)\*} - \tilde{\mu}^{(k)}) (X\_m^{(k)\*} - \tilde{\mu}^{(k)})^T, \mathbf{0}$$

where *<sup>μ</sup>*˜(*k*) = (1/*nk*) <sup>∑</sup>*nk <sup>m</sup>*=<sup>1</sup> *<sup>X</sup>*(*k*)<sup>∗</sup> *<sup>m</sup>* . Liu et al. (2009) established the following consistency results under mild regularity conditions:

$$||\mathfrak{L}^{(k)} - \mathfrak{L}^{(k)}||\_{\infty} = O\_p\left(\sqrt{\frac{\log p \log^2 n\_k}{n\_k^{1/2}}}\right).$$

When estimating the precision matrix Ω(*k*), one can consider a modified graphical lasso based on imputed data, i.e.,

$$\boldsymbol{\hat{\mathbf{D}}}\_{\text{glasso}}^{(k)} = \arg\min\_{\mathbf{D}} \left\{ \text{tr}(\mathbf{D}\boldsymbol{\Sigma}^{(k)}) - \log|\mathbf{D}| + \lambda ||\mathbf{D}||\_1 \right\}.\tag{2}$$

.

Liu et al. (2009) showed the following convergence, which elucidated the asymptotic equivalence between the oracle data and imputed data in the structural estimation of NPNGM

$$\|\mathbf{\hat{\mathbf{n}}}\_{\text{glasso}}^{(k)} - \mathbf{\hat{\mathbf{n}}}^{(k)}\|\_{F} = O\_p\left(\sqrt{\frac{(\|\mathbf{\hat{\mathbf{n}}}^{(k)}\|\_{0} + p)\log p \log^2 n\_k}{n\_k^{1/2}}}\right)$$
