c_0(p)$ and JPEN has the same rate of convergence as that of Bickel and Levina’s estimator.\
**3.** The operator norm rate of convergence is much faster than Frobenius norm. This is due to the fact that Frobenius norm convergence is in terms of all eigenvalues of the covariance matrix whereas the operator norm gives the convergence of the estimators in terms of the largest eigenvalue.\
**4.** Our proposed estimator is applicable to estimate any non-negative definite covariance matrix.\
Note that the estimator $\hat{\Sigma}_K$ is obtained by regularization of sample correlation matrix in (2.4). In some application it is desirable to directly regularize the sample covariance matrix. The JPEN estimator of the covariance matrix based on regularization of sample covariance matrix is obtained by solving the following optimization problem: $$\hat{\Sigma}_S=\operatorname*{arg\,min}_{\Sigma=\Sigma^T,tr(\Sigma)=tr(S)|(\lambda, \gamma) \in \hat{\mathscr{S}}^{S}_1}~~\Big[  ||\Sigma-S||^2_F +  \lambda \|\Sigma^-\|_1  + \gamma\sum_{i=1}^{p} \{\sigma_i(\Sigma)-\bar{\sigma}_{\Sigma}\}^2\Big],$$ where $$\begin{aligned}
\hat{\mathscr{S}}^S_1& = \Big \{(\lambda,\gamma): \lambda,\gamma >0, \lambda \asymp \gamma \asymp \sqrt{\frac{log~ p}{n}},  \forall \epsilon >0, \sigma_{min}\{(S+\gamma t I)-\frac{\lambda}{2}*sign(S+\gamma t I)\}>\epsilon \},\end{aligned}$$ and $S$ is sample covariance matrix. The minimization in (3.3) over $\Sigma$ is for fixed $ (\lambda, \gamma) \in \hat{\mathscr{S}}^{S}_1$. The estimator $\hat{\Sigma}_S$ is positive definite and well-conditioned. Theorem 3.2 gives the rate of convergence of the estimator 