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Prediction error bounds for linear regression with the TREX

Author

Listed:
  • Jacob Bien

    (University of Southern California)

  • Irina Gaynanova

    (Texas A&M University)

  • Johannes Lederer

    (University of Washington)

  • Christian L. Müller

    (Simons Foundation)

Abstract

The TREX is a recently introduced approach to sparse linear regression. In contrast to most well-known approaches to penalized regression, the TREX can be formulated without the use of tuning parameters. In this paper, we establish the first known prediction error bounds for the TREX. Additionally, we introduce extensions of the TREX to a more general class of penalties, and we provide a bound on the prediction error in this generalized setting. These results deepen the understanding of the TREX from a theoretical perspective and provide new insights into penalized regression in general.

Suggested Citation

  • Jacob Bien & Irina Gaynanova & Johannes Lederer & Christian L. Müller, 2019. "Prediction error bounds for linear regression with the TREX," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 451-474, June.
    • Handle: RePEc:spr:testjl:v:28:y:2019:i:2:d:10.1007_s11749-018-0584-4
    DOI: 10.1007/s11749-018-0584-4
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    References listed on IDEAS

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    1. Jon A. Wellner, 2017. "The Bennett-Orlicz Norm," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 355-383, August.
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    7. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
    8. Rajen D. Shah & Richard J. Samworth, 2013. "Variable selection with error control: another look at stability selection," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(1), pages 55-80, January.
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    Cited by:

    1. Johannes Lederer & Christian L. Müller, 2022. "Topology Adaptive Graph Estimation in High Dimensions," Mathematics, MDPI, vol. 10(8), pages 1-10, April.
    2. Huang, Shih-Ting & Xie, Fang & Lederer, Johannes, 2021. "Tuning-free ridge estimators for high-dimensional generalized linear models," Computational Statistics & Data Analysis, Elsevier, vol. 159(C).
    3. Pun, Chi Seng & Hadimaja, Matthew Zakharia, 2021. "A self-calibrated direct approach to precision matrix estimation and linear discriminant analysis in high dimensions," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).

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