Robust Adaptive Lasso method for parameter's estimation and variable selection in high-dimensional sparse models.

High dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. To address this issue different penalized regression procedures have been introduced in the litrature, but these methods cannot cope with the problem of outliers and leverage points in the heavy tailed high dimensional data. For this purppose, a new Robust Adaptive Lasso (RAL) method is proposed which is based on pearson residuals weighting scheme. The weight function determines the compatibility of each observations and downweight it if they are inconsistent with the assumed model. It is observed that RAL estimator can correctly select the covariates with non-zero coefficients and can estimate parameters, simultaneously, not only in the presence of influential observations, but also in the presence of high multicolliearity. We also discuss the model selection oracle property and the asymptotic normality of the RAL. Simulations findings and real data examples also demonstrate the better performance of the proposed penalized regression approach.