>0, \\
& ~~~~~~~~~~~~\sigma_{min}\{(\hat{\Sigma}_S^{-1}+\gamma ~t_2~I)-\frac{\lambda}{2}*sign(\hat{\Sigma}_S^{-1}+\gamma t_2 I)\}>\epsilon \Big \},
$$

and $t_2$ is average of the diagonal elements of $\hat{\Sigma}_S $. The minimization in (3.8) over $\Omega$ is for fixed $ (\lambda, \gamma) \in \hat{\mathscr{S}}^{S}_2$. The estimator in (3.8) is positive definite and well-conditioned. The consistency result of the estimator $\hat{\Omega}_S$ is given in following theorem.

Let $(\lambda, \gamma) \in \hat{\mathscr{S}}^{S}_2$ and let $\hat{\Omega}_S$ be as defined in (3.8). Under Assumptions B0, B1, and B2, $$\|\hat{\Omega}_S-\Omega_0\|_F=O_P \Big ( \sqrt{\frac{(s+p) log~p}{n}} \Big ).$$

### Weighted JPEN Estimator for The Inverse Covariance Matrix

Similar to weighted JPEN covariance matrix estimator $\hat{\Sigma}_{K,A}$, a weighted JPEN estimator of the inverse covariance matrix is obtained by adding positive weights $a_i $ to the term $(\sigma_i(Z)-1)^2$ in (3.8). The weighted JPEN estimator is $\hat{\Omega}_{{K},A}:=({S^{+}})^{-1/2}\hat{Z}_A ({S^{+}})^{-1/2}$, where $$\hat{Z}_A=\operatorname*{arg\,min}_{Z=Z^T,tr(Z)=tr(\hat{R}_K^{-1})|(\lambda, \gamma) \in \hat{\mathscr{S}}^{K,A}_2}~~\Big[  ||Z-\hat{R}_K^{-1}||^2_F +  \lambda \|Z^-\|_1  + \gamma\sum_{i=1}^{p} a_i\{\sigma_i(Z)-1\}^2\Big],$$ with $$\begin{aligned}
\hat{\mathscr{S}}^{{K},A}_2& = \Big \{(\lambda,\gamma): \lambda \asymp \gamma \asymp \sqrt{\frac{log~p}{n}}, \lambda \le \frac{(2~\sigma_{min}({R}_K^{-1}))(1+\gamma t_1 max(A_{ii})^{-1})}{(1+\gamma~ min(A_{ii})^{-1}p}+ \frac{\gamma min(A_{ii})}{p} \Big \},\end{aligned}$$ and $A=\text{diag}(A_{11},A_{22},\cdots A_{pp})$ with $A_{ii}=a_i$. The optimization problem in (3.10) is convex and yields a positive definite estimator for $(\lambda,\gamma) \in \hat{\mathscr{S}}^{K,A}_2$. A simple excercise shows that the estimator $\hat{Z}_{A}$ has similar rate of convergence as that of 