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Robust Regression through the Huber's criterion and adaptive lasso penalty

Sophie Lambert-Lacroix 1 Laurent Zwald 2, 3
1 TIMC-BCM - Biologie Computationnelle et Mathématique
TIMC - Techniques de l'Ingénierie Médicale et de la Complexité - Informatique, Mathématiques et Applications, Grenoble - UMR 5525
2 LEAR - Learning and recognition in vision
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
3 SAM - Statistique Apprentissage Machine
LJK - Laboratoire Jean Kuntzmann
Abstract : The Huber's Criterion is a useful method for robust regression. The adaptive least absolute shrinkage and selection operator (lasso) is a popular technique for simultaneous estimation and variable selection. The adaptive weights in the adaptive lasso allow to have the oracle properties. In this paper we propose to combine the Huber's criterion and adaptive penalty as lasso. This regression technique is resistant to heavy-tailed er- rors or outliers in the response. Furthermore, we show that the estimator associated with this procedure enjoys the oracle properties. This approach is compared with LAD-lasso based on least absolute deviation with adaptive lasso. Extensive simulation studies demonstrate satisfactory finite-sample performance of such procedure. A real example is analyzed for illustration purposes.
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Sophie Lambert-Lacroix, Laurent Zwald. Robust Regression through the Huber's criterion and adaptive lasso penalty. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2011, 5, pp.1015-1053. ⟨10.1214/11-EJS635⟩. ⟨hal-00661864⟩

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