FULLY EFFICIENT ROBUST ESTIMATION, OUTLIER DETECTION AND VARIABLE SELECTION VIA PENALIZED REGRESSION

This paper studies the outlier detection and variable selection problem in linear regression. A mean shift parameter is added to the linear model to reflect the effect of outliers, where an outlier has a nonzero shift parameter. We then apply an adaptive regularization to these shift parameters to shrink most of them to zero. Those observations with nonzero mean shift parameter estimates are regarded as outliers. An L1 penalty is added to the regression parameters to select important predictors. We propose an efficient algorithm to solve this jointly penalized optimization problem and use the extended Bayesian information criteria tuning method to select the regularization parameters, since the number of parameters exceeds the sample size. Theoretical results are provided in terms of high breakdown point, full efficiency, as well as outlier detection consistency. We illustrate our method with simulations and data. Our method is extended to high-dimensional problems with dimension much larger than the sample size.