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Simultaneous Inference for High-Dimensional Linear Models

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  • Xianyang Zhang
  • Guang Cheng

Abstract

This article proposes a bootstrap-assisted procedure to conduct simultaneous inference for high-dimensional sparse linear models based on the recent desparsifying Lasso estimator. Our procedure allows the dimension of the parameter vector of interest to be exponentially larger than sample size, and it automatically accounts for the dependence within the desparsifying Lasso estimator. Moreover, our simultaneous testing method can be naturally coupled with the margin screening to enhance its power in sparse testing with a reduced computational cost, or with the step-down method to provide a strong control for the family-wise error rate. In theory, we prove that our simultaneous testing procedure asymptotically achieves the prespecified significance level, and enjoys certain optimality in terms of its power even when the model errors are non-Gaussian. Our general theory is also useful in studying the support recovery problem. To broaden the applicability, we further extend our main results to generalized linear models with convex loss functions. The effectiveness of our methods is demonstrated via simulation studies. Supplementary materials for this article are available online.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:jnlasa:v:112:y:2017:i:518:p:757-768
    DOI: 10.1080/01621459.2016.1166114
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    Cited by:

    1. Alexandre Belloni & Victor Chernozhukov & Denis Chetverikov & Christian Hansen & Kengo Kato, 2018. "High-dimensional econometrics and regularized GMM," CeMMAP working papers CWP35/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Kaspar Wuthrich & Ying Zhu, 2019. "Omitted variable bias of Lasso-based inference methods: A finite sample analysis," Papers 1903.08704, arXiv.org, revised Sep 2021.
    3. Xingcai Zhou & Zhaoyang Jing & Chao Huang, 2024. "Distributed Bootstrap Simultaneous Inference for High-Dimensional Quantile Regression," Mathematics, MDPI, vol. 12(5), pages 1-54, February.
    4. Victor Chernozhukov & Wolfgang K. Hardle & Chen Huang & Weining Wang, 2018. "LASSO-Driven Inference in Time and Space," Papers 1806.05081, arXiv.org, revised May 2020.
    5. Jelena Bradic & Yinchu Zhu, 2017. "Comments on: High-dimensional simultaneous inference with the bootstrap," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(4), pages 720-728, December.
    6. Aaron Hudson & Ali Shojaie, 2022. "Covariate-Adjusted Inference for Differential Analysis of High-Dimensional Networks," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(1), pages 345-388, June.
    7. Maur,Jean-Christophe & Nedeljkovic,Milan & Von Uexkull,Jan Erik, 2022. "FDI and Trade Outcomes at the Industry Level—A Data-Driven Approach," Policy Research Working Paper Series 9901, The World Bank.
    8. Xiaorui Zhu & Yichen Qin & Peng Wang, 2023. "Sparsified Simultaneous Confidence Intervals for High-Dimensional Linear Models," Papers 2307.07574, arXiv.org.
    9. Jingxuan Luo & Lili Yue & Gaorong Li, 2023. "Overview of High-Dimensional Measurement Error Regression Models," Mathematics, MDPI, vol. 11(14), pages 1-22, July.
    10. Alexander Giessing & Jianqing Fan, 2020. "Bootstrapping $\ell_p$-Statistics in High Dimensions," Papers 2006.13099, arXiv.org, revised Aug 2020.
    11. Zhan Gao & Ji Hyung Lee & Ziwei Mei & Zhentao Shi, 2024. "On LASSO Inference for High Dimensional Predictive Regression," Papers 2409.10030, arXiv.org.
    12. He, Yi & Jaidee, Sombut & Gao, Jiti, 2023. "Most powerful test against a sequence of high dimensional local alternatives," Journal of Econometrics, Elsevier, vol. 234(1), pages 151-177.
    13. Shi, Chengchun & Lu, Wenbin & Song, Rui, 2019. "A sparse random projection-based test for overall qualitative treatment effects," LSE Research Online Documents on Economics 102107, London School of Economics and Political Science, LSE Library.
    14. Rui Wang & Xingzhong Xu, 2021. "A Bayesian-motivated test for high-dimensional linear regression models with fixed design matrix," Statistical Papers, Springer, vol. 62(4), pages 1821-1852, August.
    15. Victor Chernozhukov & Denis Chetverikov & Kengo Kato & Yuta Koike, 2022. "High-dimensional Data Bootstrap," Papers 2205.09691, arXiv.org.

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