Bandwidth Selection for High-Dimensional Covariance Matrix Estimation

Abstract

The banding estimator of Bickel and Levina and its tapering version of Cai, Zhang, and Zhou are important high-dimensional covariance estimators. Both estimators require a bandwidth parameter. We propose a bandwidth selector for the banding estimator by minimizing an empirical estimate of the expected squared Frobenius norms of the estimation error matrix. The ratio consistency of the bandwidth selector is established. We provide a lower bound for the coverage probability of the underlying bandwidth being contained in an interval around the bandwidth estimate. Extensions to the bandwidth selection for the tapering estimator and threshold level selection for the thresholding covariance estimator are made. Numerical simulations and a case study on sonar spectrum data are conducted to demonstrate the proposed approaches. Supplementary materials for this article are available online.

• Banding estimator
• Large p, small n
• Ratio-consistency
• Tapering estimator
• Thresholding estimator.

Additional information

Notes on contributors

Yumou Qiu

Yumou Qiu is Assistant Professor, Department of Statistics, University of Nebraska-Lincoln, NE 68583-0963 (E-mail: [email protected]). Song Xi Chen is Chair Professor, Department of Business Statistics and Econometrics, Guanghua School of Management and Center for Statistical Science, Peking University, Beijing 100651, China, and Professor, Department of Statistics, Iowa State University, Ames, IA 50011-1210 (E-mail: [email protected]).We thank the editors, the AE and three anonymous referees for constructive comments and suggestions which have improved the presentation of the article. We also thank Feng Yi and Professor Hui Zou for sharing the code of their work. The research was partially supported by NSF grant DMS-1309210 and National Natural Science Foundation of China key grant 11131002.

Song Xi Chen

Yumou Qiu is Assistant Professor, Department of Statistics, University of Nebraska-Lincoln, NE 68583-0963 (E-mail: [email protected]). Song Xi Chen is Chair Professor, Department of Business Statistics and Econometrics, Guanghua School of Management and Center for Statistical Science, Peking University, Beijing 100651, China, and Professor, Department of Statistics, Iowa State University, Ames, IA 50011-1210 (E-mail: [email protected]).We thank the editors, the AE and three anonymous referees for constructive comments and suggestions which have improved the presentation of the article. We also thank Feng Yi and Professor Hui Zou for sharing the code of their work. The research was partially supported by NSF grant DMS-1309210 and National Natural Science Foundation of China key grant 11131002.