View Submission - EcoSta2018

A0698
**Title: **High dimensional covariance matrix estimation by penalizing the matrix-logarithm transformed likelihood
**Authors: **Xiaohang Wang - BNU-HKBU United International College (China) **[presenting]**

**Abstract: **It is well known that when the dimension of the data becomes very large, the sample covariance matrix $S$ will not be a good estimator of the population covariance matrix $\Sigma$. Using such estimator, one typical consequence is that the estimated eigenvalues from S will be distorted. Many existing methods tried to solve the problem, and examples of which include regularizing $\Sigma$ by thresholding or banding. We estimate $\Sigma$ by maximizing the likelihood using a new penalization on the matrix logarithm of $\Sigma$ (denoted by A) of the form: $\|\boldsymbol{A}-m\boldsymbol{I}\|_{F}^{2}=\sum_{i}(\log(d_{i})-m)^{2}$, where $d_i$ is the ith eigenvalue of $\Sigma$. This penalty aims at shrinking the estimated eigenvalues of $A$ toward the mean eigenvalue $m$. The merits of our method are that it guarantees $\Sigma$ to be non-negative definite and is computational efficient. The simulation study and applications on portfolio optimization and classification of genomic data show that the proposed method outperforms existing methods.