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Bootstrap Confidence Sets for Spectral Projectors of Sample Covariance

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  • Naumov, A.
  • Spokoiny, V.
  • Ulyanovk, V.

Abstract

Let X1, . . . ,Xn be i.i.d. sample in Rp with zero mean and the covariance matrix . The problem of recovering the projector onto an eigenspace of from these observations naturally arises in many applications. Recent technique from [9] helps to study the asymp- totic distribution of the distance in the Frobenius norm kPr - bP rk2 between the true projector Pr on the subspace of the rth eigenvalue and its empirical counterpart bP r in terms of the effective rank of . This paper offers a bootstrap procedure for building sharp confidence sets for the true projector Pr from the given data. This procedure does not rely on the asymptotic distribution of kPr - bP rk2 and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for finite samples with an explicit error bound for the er- ror of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.

Suggested Citation

  • Naumov, A. & Spokoiny, V. & Ulyanovk, V., 2018. "Bootstrap Confidence Sets for Spectral Projectors of Sample Covariance," IRTG 1792 Discussion Papers 2018-024, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    • Handle: RePEc:zbw:irtgdp:2018024
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    References listed on IDEAS

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    1. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2014. "Central limit theorems and bootstrap in high dimensions," CeMMAP working papers 49/14, Institute for Fiscal Studies.
    2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
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    14. Yan, Ji Gao, 2018. "Complete Convergence and Complete Moment Convergence for Maximal Weighted Sums of Extended Negatively Dependent Random Variables," IRTG 1792 Discussion Papers 2018-040, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    15. Kalkbrener, Michael & Packham, Natalie, 2018. "Correlation Under Stress In Normal Variance Mixture Models," IRTG 1792 Discussion Papers 2018-035, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    16. Chiu, Hsin-Yu & Chiang, Mi-Hsiu & Kuo, Wei-Yu, 2018. "Predicative Ability of Similarity-based Futures Trading Strategies," IRTG 1792 Discussion Papers 2018-045, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    17. Guo, Shaojun & Li, Dong & Li, Muyi, 2018. "Strict Stationarity Testing and GLAD Estimation of Double Autoregressive Models," IRTG 1792 Discussion Papers 2018-049, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".
    18. Koziuk, Andzhey & Spokoiny, Vladimir, 2018. "Toolbox: Gaussian comparison on Eucledian balls," IRTG 1792 Discussion Papers 2018-028, Humboldt University of Berlin, International Research Training Group 1792 "High Dimensional Nonstationary Time Series".

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