| We propose a LASSO-type penalized regression method for simultaneous variable selection and outlier detection in high dimensional linear regression. We apply a mean-shift model to incorporate the coefficients associated with the potential outliers by expressing them as different intercept terms. The sparsity assumption is imposed on both X-covariates and the outlier indicator variables. With suitable penalty factors between X-covariates and the outlier indicators, we show that the proposed method selects a model of the correct order of dimensionality, under the sparse Riesz condition on the correlation of design variables and a joint sparse Reisz condition on the augmented design matrix. We also show that the estimation/prediction of the selected model can be controlled at a level determined by the sizes of the true model, the outliers and the thresholding level. Moreover, the estimation has a positive breakdown point when both the dimension p and the sample size n tend to infinity, and p >> n. We also provide a generalized version for the estimator by adjusting the penalty weight factor. Finally, we apply the proposed method to analyze an aircraft landing performance data set, for identifying the precursors for undesirable landing performance and reducing the risk of runway overruns. |
