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Penalised robust estimators for sparse and high-dimensional linear models

Author

Listed:
  • Umberto Amato

    (Italian National Research Council)

  • Anestis Antoniadis

    (Université Joseph Fourier
    University of Cape Town)

  • Italia De Feis

    (Italian National Research Council)

  • Irene Gijbels

    (KU Leuven)

Abstract

We introduce a new class of robust M-estimators for performing simultaneous parameter estimation and variable selection in high-dimensional regression models. We first explain the motivations for the key ingredient of our procedures which are inspired by regularization methods used in wavelet thresholding in noisy signal processing. The derived penalized estimation procedures are shown to enjoy theoretically the oracle property both in the classical finite dimensional case as well as the high-dimensional case when the number of variables p is not fixed but can grow with the sample size n, and to achieve optimal asymptotic rates of convergence. A fast accelerated proximal gradient algorithm, of coordinate descent type, is proposed and implemented for computing the estimates and appears to be surprisingly efficient in solving the corresponding regularization problems including the case for ultra high-dimensional data where $$p \gg n$$ p ≫ n . Finally, a very extensive simulation study and some real data analysis, compare several recent existing M-estimation procedures with the ones proposed in the paper, and demonstrate their utility and their advantages.

Suggested Citation

  • Umberto Amato & Anestis Antoniadis & Italia De Feis & Irene Gijbels, 2021. "Penalised robust estimators for sparse and high-dimensional linear models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 1-48, March.
    • Handle: RePEc:spr:stmapp:v:30:y:2021:i:1:d:10.1007_s10260-020-00511-z
    DOI: 10.1007/s10260-020-00511-z
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    Cited by:

    1. Umberto Amato & Anestis Antoniadis & Italia Feis & Irène Gijbels, 2022. "Penalized wavelet estimation and robust denoising for irregular spaced data," Computational Statistics, Springer, vol. 37(4), pages 1621-1651, September.
    2. Thompson, Ryan, 2022. "Robust subset selection," Computational Statistics & Data Analysis, Elsevier, vol. 169(C).
    3. Daniela De Canditiis & Italia De Feis, 2021. "Anomaly Detection in Multichannel Data Using Sparse Representation in RADWT Frames," Mathematics, MDPI, vol. 9(11), pages 1-26, June.

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