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Rate optimal estimation and confidence intervals for high-dimensional regression with missing covariates

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  • Wang, Yining
  • Wang, Jialei
  • Balakrishnan, Sivaraman
  • Singh, Aarti

Abstract

We consider the problems of estimation and of constructing component-wise confidence intervals in a sparse high-dimensional linear regression model when some covariates of the design matrix are missing completely at random. We analyze a variant of the Dantzig selector for estimating the regression model and we use a de-biasing argument to construct component-wise confidence intervals. We also complement our mathematical study in the supplementary materials with extensive simulations on synthetic and semi-synthetic data that show the accuracy of our asymptotic predictions for finite sample sizes.

Suggested Citation

  • Wang, Yining & Wang, Jialei & Balakrishnan, Sivaraman & Singh, Aarti, 2019. "Rate optimal estimation and confidence intervals for high-dimensional regression with missing covariates," Journal of Multivariate Analysis, Elsevier, vol. 174(C).
    • Handle: RePEc:eee:jmvana:v:174:y:2019:i:c:s0047259x18304238
    DOI: 10.1016/j.jmva.2019.06.004
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    References listed on IDEAS

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    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    2. Cai, Tony & Liu, Weidong & Luo, Xi, 2011. "A Constrained â„“1 Minimization Approach to Sparse Precision Matrix Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 594-607.
    3. Cun-Hui Zhang & Stephanie S. Zhang, 2014. "Confidence intervals for low dimensional parameters in high dimensional linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 76(1), pages 217-242, January.
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    Cited by:

    1. Yining Wang & Xi Chen & Xiangyu Chang & Dongdong Ge, 2021. "Uncertainty Quantification for Demand Prediction in Contextual Dynamic Pricing," Production and Operations Management, Production and Operations Management Society, vol. 30(6), pages 1703-1717, June.
    2. Fan, Jinlin & Zhang, Yaowu & Zhu, Liping, 2022. "Independence tests in the presence of measurement errors: An invariance law," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    3. Li, Mengyan & Li, Runze & Ma, Yanyuan, 2021. "Inference in high dimensional linear measurement error models," Journal of Multivariate Analysis, Elsevier, vol. 184(C).
    4. Adel Javanmard & Jason D. Lee, 2020. "A flexible framework for hypothesis testing in high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 685-718, July.

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