Robust adaptive variable selection in ultra-high dimensional linear regression models

We consider the problem of simultaneous variable selection and parameter estimation in an ultra-high dimensional linear regression model. The adaptive penalty functions are used in this regard to achieve the oracle variable selection property with simpler assumptions and lesser computational burden. Noting the non-robust nature of the usual adaptive procedures (e.g. adaptive LASSO) based on the squared error loss function against data contamination, quite frequent with modern large-scale data sets (e.g. noisy gene expression data, spectra and spectral data), in this paper, we present a new adaptive regularization procedure using a robust loss function based on the density power divergence (DPD) measure under a general class of error distributions. We theoretically prove that the proposed adaptive DPD-LASSO estimator of the regression coefficients is highly robust, consistent, asymptotically normal and leads to robust oracle-consistent variable selection under easily verifiable assumptions. Numerical illustrations are provided for the mostly used normal and heavy-tailed error densities. Finally, the proposal is applied to analyse an interesting spectral dataset, in the field of chemometrics, regarding the electron-probe X-ray microanalysis (EPXMA) of archaeological glass vessels from the 16th and 17th centuries.