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Asymptotic theory for maximum deviations of sample covariance matrix estimates

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  • Xiao, Han
  • Wu, Wei Biao

Abstract

We consider asymptotic distributions of maximum deviations of sample covariance matrices, a fundamental problem in high-dimensional inference of covariances. Under mild dependence conditions on the entries of the data matrices, we establish the Gumbel convergence of the maximum deviations. Our result substantially generalizes earlier ones where the entries are assumed to be independent and identically distributed, and it provides a theoretical foundation for high-dimensional simultaneous inference of covariances.

Suggested Citation

  • Xiao, Han & Wu, Wei Biao, 2013. "Asymptotic theory for maximum deviations of sample covariance matrix estimates," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2899-2920.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:7:p:2899-2920
    DOI: 10.1016/j.spa.2013.03.012
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    References listed on IDEAS

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    6. Cai, Tony & Liu, Weidong, 2011. "Adaptive Thresholding for Sparse Covariance Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 672-684.
    7. Ravi Jagannathan & Tongshu Ma, 2003. "Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps," Journal of Finance, American Finance Association, vol. 58(4), pages 1651-1684, August.
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    Cited by:

    1. Butucea, Cristina & Zgheib, Rania, 2016. "Sharp minimax tests for large Toeplitz covariance matrices with repeated observations," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 164-176.
    2. Jiayu Lai & Xiaoyi Wang & Kaige Zhao & Shurong Zheng, 2023. "Block-diagonal test for high-dimensional covariance matrices," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 32(1), pages 447-466, March.
    3. Tiefeng Jiang & Junshan Xie, 2020. "Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2380-2400, December.
    4. Tiefeng Jiang & Yongcheng Qi, 2015. "Likelihood Ratio Tests for High-Dimensional Normal Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 988-1009, December.

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