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A class of optimal estimators for the covariance operator in reproducing kernel Hilbert spaces

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  • Zhou, Yang
  • Chen, Di-Rong
  • Huang, Wei

Abstract

The covariance operator plays an important role in modern statistical methods and is critical for inference. It is most often estimated by the empirical covariance operator. In spite of its simple and appealing properties, however, this estimator can be improved by a class of shrinkage operators. In this paper, we study shrinkage estimation of the covariance operator in reproducing kernel Hilbert spaces. A data-driven shrinkage estimator enjoying desirable theoretical and computational properties is proposed. The procedure is easily implemented and its numerical performance is investigated through simulations. In finite samples, the estimator outperforms the empirical covariance operator, especially when the data dimension is much larger than the sample size. We also show that the rate of convergence in Hilbert–Schmidt norm is of the order n−1∕2. Furthermore, we establish the minimax optimal rate of convergence over suitable classes of probability measures and demonstrate that these shrinkage operators are all minimax rate-optimal.

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  • Zhou, Yang & Chen, Di-Rong & Huang, Wei, 2019. "A class of optimal estimators for the covariance operator in reproducing kernel Hilbert spaces," Journal of Multivariate Analysis, Elsevier, vol. 169(C), pages 166-178.
    • Handle: RePEc:eee:jmvana:v:169:y:2019:i:c:p:166-178
    DOI: 10.1016/j.jmva.2018.09.003
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    1. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    2. Touloumis, Anestis, 2015. "Nonparametric Stein-type shrinkage covariance matrix estimators in high-dimensional settings," Computational Statistics & Data Analysis, Elsevier, vol. 83(C), pages 251-261.
    3. Fisher, Thomas J. & Sun, Xiaoqian, 2011. "Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1909-1918, May.
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