achieve positive definiteness but it is still unclear whether the iterative procedure proposed in @Roth:25 actually finds the right solution to the corresponding optimization problem. In another interesting paper, @Xue:31 proposed an estimator of covariance matrix as a minimizer of penalized quadratic loss function over set of positive definite matrices. In their paper, the authors solve a positive definite constrained optimization problem and establish the consistency of estimator. The resulting estimator is sparse and positive definite but whether it overcomes the over-dispersion of the eigen-spectrum of sample covariance matrix, is hard to justify. @Maurya:19 proposed a joint convex penalty as function of $\ell_1$ and trace norm (defined as sum of singular values of a matrix) for inverse covariance matrix estimation based on penalized likelihood approach.

In this paper, we propose the JPEN (Joint PENalty) estimators for covariance and inverse covariance matrices estimation and derive an explicit rate of convergence in both the operator and Frobenius norm. The JPEN estimators achieves minimax rate of convergence under operator norm for the underlying class of sparse covariance and inverse covariance matrices and hence is optimal. For more details see section $\S3$. One of the major advantage of the proposed estimators is that the 