In this paper, we propose a shrinkage framework for jointly estimating multiple covariance matrices by shrinking the sample covariance matrices towards the pooled sample covariance matrix. This framework allows us to borrow information across different groups. We derive the optimal shrinkage parameters under the Stein and quadratic loss functions, and prove that our derived estimators are asymptotically optimal when the sample size or the number of groups tends to infinity. Simulation studies demonstrate that our proposed shrinkage method performs favorably compared to the existing methods.
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Zongliang Hu’s research was supported by the Natural Science Foundation grant (No. 2019083) of Shenzhen University, and the Foundation and Applied Basic Research Programs (No. 2019A1515110449) of Guangdong Province of China. Zhishui Hu’s research was partially supported by the National Natural Science Foundation of China grant (No. 11671373). Tiejun Tong’s research was supported by the General Research Fund (No. HKBU12303918), the National Natural Science Foundation of China (No. 11671338), and the Initiation Grant for Faculty Niche Research Areas (No. RC-IG-FNRA/17-18/13) of Hong Kong Baptist University. Yuedong Wang’s research was partially supported by the National Science Foundation grant (No. DMS-1507620). The authors also thank the editor, the associate editor, and two reviewers for their constructive comments that have led to a substantial improvement of this paper.
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Hu, Z., Hu, Z., Dong, K. et al. A shrinkage approach to joint estimation of multiple covariance matrices. Metrika (2020). https://doi.org/10.1007/s00184-020-00781-3
- Covariance matrices
- Joint estimation
- Optimal estimator
- Quadratic loss function
- Shrinkage parameter
- Stein loss function