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Bootstrapping $\ell_p$-Statistics in High Dimensions

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  • Alexander Giessing
  • Jianqing Fan

Abstract

This paper considers a new bootstrap procedure to estimate the distribution of high-dimensional $\ell_p$-statistics, i.e. the $\ell_p$-norms of the sum of $n$ independent $d$-dimensional random vectors with $d \gg n$ and $p \in [1, \infty]$. We provide a non-asymptotic characterization of the sampling distribution of $\ell_p$-statistics based on Gaussian approximation and show that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance under mild conditions on the covariance structure of the data. As an application of the general theory we propose a bootstrap hypothesis test for simultaneous inference on high-dimensional mean vectors. We establish its asymptotic correctness and consistency under high-dimensional alternatives, and discuss the power of the test as well as the size of associated confidence sets. We illustrate the bootstrap and testing procedure numerically on simulated data.

Suggested Citation

  • Alexander Giessing & Jianqing Fan, 2020. "Bootstrapping $\ell_p$-Statistics in High Dimensions," Papers 2006.13099, arXiv.org, revised Aug 2020.
    • Handle: RePEc:arx:papers:2006.13099
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    References listed on IDEAS

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    1. Jianqing Fan & Shaojun Guo & Ning Hao, 2012. "Variance estimation using refitted cross‐validation in ultrahigh dimensional regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(1), pages 37-65, January.
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    3. Marco Avella-Medina & Heather S Battey & Jianqing Fan & Quefeng Li, 2018. "Robust estimation of high-dimensional covariance and precision matrices," Biometrika, Biometrika Trust, vol. 105(2), pages 271-284.
    4. 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.
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    6. Xianyang Zhang & Guang Cheng, 2017. "Simultaneous Inference for High-Dimensional Linear Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 757-768, April.
    7. 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.
    8. Lam, Clifford & Fan, Jianqing, 2009. "Sparsistency and rates of convergence in large covariance matrix estimation," LSE Research Online Documents on Economics 31540, London School of Economics and Political Science, LSE Library.
    9. Chen, Song Xi & Qin, Yingli, 2010. "A Two Sample Test for High Dimensional Data with Applications to Gene-set Testing," MPRA Paper 59642, University Library of Munich, Germany.
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