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 errors 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.
Highlights
Data subject to heavy-tailed errors or outliers are commonly encountered in applications which may appear either in response variables or in the predictors
In [36], the authors propose to treat the problem of robust model selection by combining least absolute deviation (LAD) loss and adaptive lasso penalty
This approach is compared with LAD-lasso based on least absolute deviation with adaptive lasso
Summary
Data subject to heavy-tailed errors or outliers are commonly encountered in applications which may appear either in response variables or in the predictors. We consider here the regression problem with responses subject to heavy-tailed errors or outliers In this case, the Ordinary Least Square (OLS) estimator is reputed to be not efficient. In a second attempt, [39] provides a convex optimisation problem leading to a consistent in variable selection estimator He assigns adaptive weights for penalizing differently coefficients in the l1 penalty and calls this new penalty the adaptive lasso. In [36], the authors propose to treat the problem of robust model selection by combining LAD loss and adaptive lasso penalty. They obtain an estimator which is robust against outliers and enjoys a sparse representation.
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