analysis problems.

Proposed Estimators
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Let $S$ be the sample covariance matrix. Consider the following optimization problem. $$\hat{\Sigma}_{\lambda,\gamma}=\operatorname*{arg\,min}_{\Sigma=\Sigma^T,tr(\Sigma)=tr(S)}~~\Big[  ||\Sigma-S||^2_2 +  \lambda \|{\Sigma^-}\|_1  + \gamma\sum_{i=1}^{p} \big \{\sigma_i(\Sigma)-\bar{\sigma}_{\Sigma} \big \}^2\Big],$$ where $\bar{\sigma}_\Sigma$ is the mean of eigenvalues of $\Sigma$, $\lambda$ and $\gamma$ are some positive constants. Note that by penalty function $\|{\Sigma^-}\|_1$, we only penalize off-diagonal elements of $\Sigma$. The eigenvalues variance penalty term for eigen-spectrum shrinkage is chosen from the following points of interest: i) It is easy to interpret and ii) this choice of penalty function yields a very fast optimization algorithm. By constraint $tr(\Sigma)=tr(S)$, the total variation in $\hat{\Sigma}_{\lambda,\gamma}$ is same as that in sample covariance matrix $S$, however the eigenvalues of $\hat{\Sigma}_{\lambda,\gamma} $ are well-conditioned than those of $S$. From here onwards we suppress the dependence of $\lambda, \gamma $ on $\hat{\Sigma    }$ and denote $\hat{\Sigma}_{\lambda,\gamma} $ by $\hat{\Sigma}$.\
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For $\gamma=0$, the solution to (2.1) is the standard soft-thresholding estimator for quadratic loss function and its solution is given by (see $\S4$ for derivation of this estimator): $$\begin{aligned}
\begin{split}
\hat{\Sigma}_{ii}& =s_{ii} \\
\hat{\Sigma}_{ij}& =\text{sign}(s_{ij})\max\Big (|s_{ij}|-\frac{\lambda}{2},0\Big), ~~~~~~~~~~~~~i \neq j.
\end{split}\end{aligned}$$ It is clear from this expression that a sufficiently large value of $\lambda$ will result in sparse covariance 