positive definite.\
\
[**Choice of weight matrix A:**]{} For optimization problem in (3.5), the weights are chosen in following way:\
Let $\mathscr{E}$ be the set of sorted diagonal elements of the sample covariance matrix $S$.\
**i)** Let $k$ be largest index of $\mathscr{E}$ such that $k^{th}$ elements of $\mathscr{E}$ is less than $1$. For $i \le k, ~a_i=\mathscr{E}_{i}$. For $k<i \le p,~ a_i=1/\mathscr{E}_{i}.$\
**ii)** $A=\text{diag}(a_1,a_2,\cdots,a_p),~\text{where}~ a_j=a_j/\sum_{i=1}^p a_i.$ Such choice of weights allows more shrinkage of extreme sample eigenvalues than the ones in center of eigen-spectrum.

Inverse Covariance Matrix Estimation:
-------------------------------------

To get an expression of inverse covariance matrix estimate, we replace $K$ by $\hat{R}_K^{-1}$ in (4.2), where $\hat{R}_K$ is a JPEN estimator of correlation matrix. We chose $(\lambda,\gamma) \in \hat{\mathscr{S}}^{{K}}_2$. For a given $\gamma$, we chose $\lambda>0$ satisfying:\
$$\begin{aligned}
\sigma_{min}\{(\hat{R}_K^{-1}+\gamma t_1 I)-\frac{\lambda}{2}*sign(\hat{R}_K^{-1}+\gamma t_1 I)\}>0\end{aligned}$$ which can be simplified to $$\begin{aligned}
\lambda < \frac{\sigma_{min}(\hat{R}_K^{-1}+\gamma t_1 I)}{C_{12}~\sigma_{max}(\text{sign}(\hat{R}_K^{-1}))}. \end{aligned}$$

Computational Complexity
------------------------

The JPEN estimator $\hat{\Sigma}_{K}$ has computational complexity of $O(p^2)$ as there are at most $3p^2$ multiplications for computing the estimator $\hat{\Sigma}_{K}$. The other existing algorithm Glasso ([@Freidman:11]), PDSCE ([@Roth1:26]) have computational complexity of $O(p^3)$. We compare the computational timing of our algorithm to some other existing algorithms Glasso (@Freidman:11), PDSCE (@Roth1:26). 