Abstract
To perform multiple regression, the least squares estimator is commonly used. However, this estimator is not robust to outliers. Therefore, robust methods such as S-estimation have been proposed. These estimators flag any observation with a large residual as an outlier and downweight it in the further procedure. However, a large residual may be caused by an outlier in only one single predictor variable, and downweighting the complete observation results in a loss of information. Therefore, we propose the shooting S-estimator, a regression estimator that is especially designed for situations where a large number of observations suffer from contamination in a small number of predictor variables. The shooting S-estimator combines the ideas of the coordinate descent algorithm with simple S-regression, which makes it robust against componentwise contamination, at the cost of failing the regression equivariance property.
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Notes
The signal-to-noise ratio equals \(\frac{\sqrt{{{\varvec{\beta }}}'\Sigma {{\varvec{\beta }}}}}{\sigma }\).
The expected value of the number of contaminated rows is \(n(1-(1-\epsilon )^p)\) for a cellwise contamination level \(\epsilon \).
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Acknowledgments
We gratefully acknowledge support from the GOA/12/014 Project of the Research Fund KU Leuven. We thank the referees for their constructive comments, and in particular the third anonymous referee who corrected some flaws in the first version of the paper and who made many suggestions for improving the write up of the paper.
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Öllerer, V., Alfons, A. & Croux, C. The shooting S-estimator for robust regression. Comput Stat 31, 829–844 (2016). https://doi.org/10.1007/s00180-015-0593-7
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DOI: https://doi.org/10.1007/s00180-015-0593-7