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Variable selection in high-dimensional linear models: partially faithful distributions and the pc -simple algorithm

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  • P. Bühlmann
  • M. Kalisch
  • M. H. Maathuis

Abstract

We consider variable selection in high-dimensional linear models where the number of covariates greatly exceeds the sample size. We introduce the new concept of partial faithfulness and use it to infer associations between the covariates and the response. Under partial faithfulness, we develop a simplified version of the pc algorithm (Spirtes et al., 2000), which is computationally feasible even with thousands of covariates and provides consistent variable selection under conditions on the random design matrix that are of a different nature than coherence conditions for penalty-based approaches like the lasso. Simulations and application to real data show that our method is competitive compared to penalty-based approaches. We provide an efficient implementation of the algorithm in the R-package pcalg. Copyright 2010, Oxford University Press.

Suggested Citation

  • P. Bühlmann & M. Kalisch & M. H. Maathuis, 2010. "Variable selection in high-dimensional linear models: partially faithful distributions and the pc -simple algorithm," Biometrika, Biometrika Trust, vol. 97(2), pages 261-278.
    • Handle: RePEc:oup:biomet:v:97:y:2010:i:2:p:261-278
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    File URL: http://hdl.handle.net/10.1093/biomet/asq008
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    Cited by:

    1. Wang, Jia & Cai, Xizhen & Li, Runze, 2021. "Variable selection for partially linear models via Bayesian subset modeling with diffusing prior," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    2. He, Kevin & Kang, Jian & Hong, Hyokyoung G. & Zhu, Ji & Li, Yanming & Lin, Huazhen & Xu, Han & Li, Yi, 2019. "Covariance-insured screening," Computational Statistics & Data Analysis, Elsevier, vol. 132(C), pages 100-114.
    3. Dong, Yuexiao & Yu, Zhou & Zhu, Liping, 2020. "Model-free variable selection for conditional mean in regression," Computational Statistics & Data Analysis, Elsevier, vol. 152(C).
    4. Liu, Jingyuan & Lou, Lejia & Li, Runze, 2018. "Variable selection for partially linear models via partial correlation," Journal of Multivariate Analysis, Elsevier, vol. 167(C), pages 418-434.
    5. Fan, Jianqing & Liao, Yuan, 2012. "Endogeneity in ultrahigh dimension," MPRA Paper 38698, University Library of Munich, Germany.
    6. Peter Bühlmann, 2013. "Causal statistical inference in high dimensions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 357-370, June.
    7. Randy C. S. Lai & Jan Hannig & Thomas C. M. Lee, 2015. "Generalized Fiducial Inference for Ultrahigh-Dimensional Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(510), pages 760-772, June.

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