$\hat{\Sigma}_S$ in Frobenius norm.

Let $(\lambda, \gamma) \in \hat{\mathscr{S}}^S_1$, and let $\hat{\Sigma}_S$ be as defined in (3.3). Under Assumptions A0, A1, A2, $$\|\hat{\Sigma}_S-\Sigma_0\|_F=O_P \Big ( \sqrt{\frac{(s+p) log~p}{n}} \Big )$$

As noted in @Roth1:26 the worst part of convergence here comes from estimating the diagonal entries.

### Weighted JPEN Estimator for the Covariance Matrix Estimation

A modification of estimator $\hat{R}_{K}$ is obtained by adding positive weights to the term $(\sigma_i(R)-\bar{\sigma}_R)^2$. This leads to weighted eigenvalues variance penalty with larger weights amounting to greater shrinkage towards the center and vice versa. Note that the choice of the weights allows one to use any prior structure of the eigenvalues (if known) in estimating the covariance matrix. The weighted JPEN correlation matrix estimator $\hat{R}_A$ is given by : $$\hat{R}_A=\operatorname*{arg\,min}_{R=R^T,tr(R)=p|(\lambda, \gamma) \in \hat{\mathscr{S}}^{K,A}_1}~~\Big[  ||R-K||^2_F +  \lambda \|R^-\|_1  + \gamma\sum_{i=1}^{p} a_i\{\sigma_i(R)-\bar{\sigma}_R\}^2\Big],$$ where $$\begin{aligned}
\hat{\mathscr{S}}^{K,A}_1& = \Big \{(\lambda,\gamma): \lambda \asymp \gamma \asymp \sqrt{\frac{log~p}{n}}, \lambda \le \frac{(2~\sigma_{min}(K))(1+\gamma ~max(A_{ii})^{-1})}{(1+\gamma~ min(A_{ii}))^{-1}p}+ \frac{\gamma ~min(A_{ii})}{p} \Big \},\end{aligned}$$ and $A=\text{diag}(A_{11},A_{22},\cdots A_{pp})$ with $A_{ii}=a_i$. The proposed covariance matrix estimator is $\hat{\Sigma}_{K,A}=(S^{+})^{1/2}\hat{R}_A (S^{+})^{1/2}$. The optimization problem in (3.5) is convex and yields a positive definite estimator for each $(\lambda,\gamma) \in \hat{\mathscr{S}}^{K,A}_1$. A simple excercise shows that the estimator $\hat{\Sigma}_{K,A}$ has same rate of convergence as 