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Adaptive estimation with soft thresholding penalties

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  • Jean–Michel Loubes
  • Sara Van De Geer

Abstract

We show that various robust nonparametric regression estimators, such as the least absolute deviations estimator, can be made adaptive (up to logarithmic factors), by adding a soft thresholding type penalty to the loss function. As an example, we consider the situation where the roughness of the regression function is described by a single parameter p. The theory is complemented with a simulation study.

Suggested Citation

  • Jean–Michel Loubes & Sara Van De Geer, 2002. "Adaptive estimation with soft thresholding penalties," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(4), pages 453-478, November.
    • Handle: RePEc:bla:stanee:v:56:y:2002:i:4:p:453-478
    DOI: 10.1111/1467-9574.t01-1-00212
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    Cited by:

    1. Jhong, Jae-Hwan & Koo, Ja-Yong, 2019. "Simultaneous estimation of quantile regression functions using B-splines and total variation penalty," Computational Statistics & Data Analysis, Elsevier, vol. 133(C), pages 228-244.
    2. Anne Vanhems & Jean-Michel Loubes, 2004. "Saturation spaces for regularization methods in inverse problems," Econometric Society 2004 North American Summer Meetings 380, Econometric Society.
    3. Laan Mark J. van der & Dudoit Sandrine & Vaart Aad W. van der, 2006. "The cross-validated adaptive epsilon-net estimator," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 1-23, December.

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