  We consider the problem of simultaneous variable selection and estimation of
the corresponding regression coefficients in an ultra-high dimensional linear
regression models, an extremely important problem in the recent era. The
adaptive penalty functions are used in this regard to achieve the oracle
variable selection property along with easier computational burden. However,
the usual adaptive procedures (e.g., adaptive LASSO) based on the squared error
loss function is extremely non-robust in the presence of data contamination
which are quite common with large-scale data (e.g., noisy gene expression data,
spectra and spectral data). In this paper, we present a regularization
procedure for the ultra-high dimensional data using a robust loss function
based on the popular density power divergence (DPD) measure along with the
adaptive LASSO penalty. We theoretically study the robustness and the
large-sample properties of the proposed adaptive robust estimators for a
general class of error distributions; in particular, we show that the proposed
adaptive DPD-LASSO estimator is highly robust, satisfies the oracle variable
selection property, and the corresponding estimators of the regression
coefficients are consistent and asymptotically normal under easily verifiable
set of assumptions. Numerical illustrations are provided for the mostly used
normal error density. Finally, the proposal is applied to analyze 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.
