How shrinkage can be used for robust methods
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In this work, we propose a new methodology, based on the notion of shrinkage, for outlier detection and robust regression. First, we define robust estimators of the location vector and the covariance matrix in case of multivariate data. Then, a robust Mahalanobis distance can be computed based on these estimators, for the task of outlier detection. Some properties are investigated, such as the affine equivariance and the breakdown value. The performance of the proposal is illustrated through the comparison to other robust techniques from the literature, in a simulation study and with a real example of breast cancer data. The robust alternatives are also reviewed, highlighting their advantages and disadvantages. The performance results as well as the significantly smaller computational time show the advantages of the proposal. With the proposed robust estimators, a robust regression approach is proposed as well. It is compared to the classical Ordinary Least Squares (OLS) approach and the robust alternatives from the literature. A real socio-economic dataset about the Living Environment Deprivation (LED) of areas in Liverpool (UK), is studied. The results from the simulations and the real dataset example show the advantages of the proposed robust estimator in regression. Furthermore, the proposed robust regression method has improved performance compared to other machine learning techniques previously used for this data, with the advantage of interpretability.