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Sparsity Oriented Importance Learning for High-Dimensional Linear Regression

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  • Chenglong Ye
  • Yi Yang
  • Yuhong Yang

Abstract

With now well-recognized nonnegligible model selection uncertainty, data analysts should no longer be satisfied with the output of a single final model from a model selection process, regardless of its sophistication. To improve reliability and reproducibility in model choice, one constructive approach is to make good use of a sound variable importance measure. Although interesting importance measures are available and increasingly used in data analysis, little theoretical justification has been done. In this article, we propose a new variable importance measure, sparsity oriented importance learning (SOIL), for high-dimensional regression from a sparse linear modeling perspective by taking into account the variable selection uncertainty via the use of a sensible model weighting. The SOIL method is theoretically shown to have the inclusion/exclusion property: When the model weights are properly around the true model, the SOIL importance can well separate the variables in the true model from the rest. In particular, even if the signal is weak, SOIL rarely gives variables not in the true model significantly higher important values than those in the true model. Extensive simulations in several illustrative settings and real-data examples with guided simulations show desirable properties of the SOIL importance in contrast to other importance measures. Supplementary materials for this article are available online.

Suggested Citation

  • Chenglong Ye & Yi Yang & Yuhong Yang, 2018. "Sparsity Oriented Importance Learning for High-Dimensional Linear Regression," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(524), pages 1797-1812, October.
  • Handle: RePEc:taf:jnlasa:v:113:y:2018:i:524:p:1797-1812
    DOI: 10.1080/01621459.2017.1377080
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    Cited by:

    1. Qin, Yichen & Wang, Linna & Li, Yang & Li, Rong, 2023. "Visualization and assessment of model selection uncertainty," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    2. Liao, Jun & Zou, Guohua, 2020. "Corrected Mallows criterion for model averaging," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    3. Zishu Zhan & Yang Li & Yuhong Yang & Cunjie Lin, 2023. "Model averaging for semiparametric varying coefficient quantile regression models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(4), pages 649-681, August.
    4. Yuchen Chen & Yuhong Yang, 2021. "The One Standard Error Rule for Model Selection: Does It Work?," Stats, MDPI, vol. 4(4), pages 1-25, November.
    5. Peng, Jingfu & Yang, Yuhong, 2022. "On improvability of model selection by model averaging," Journal of Econometrics, Elsevier, vol. 229(2), pages 246-262.

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