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How Close is the Sample Covariance Matrix to the Actual Covariance Matrix?

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  • Roman Vershynin

    (University of Michigan)

Abstract

Given a probability distribution in ℝ n with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60–72, 1999), which states that N=O(nlog n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535–561, 2010), which guarantees that N=O(n) for subexponential distributions.

Suggested Citation

  • Roman Vershynin, 2012. "How Close is the Sample Covariance Matrix to the Actual Covariance Matrix?," Journal of Theoretical Probability, Springer, vol. 25(3), pages 655-686, September.
  • Handle: RePEc:spr:jotpro:v:25:y:2012:i:3:d:10.1007_s10959-010-0338-z
    DOI: 10.1007/s10959-010-0338-z
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    References listed on IDEAS

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    1. Schäfer Juliane & Strimmer Korbinian, 2005. "A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-32, November.
    2. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
    3. Rothman, Adam J. & Levina, Elizaveta & Zhu, Ji, 2009. "Generalized Thresholding of Large Covariance Matrices," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 177-186.
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