Regression extremiles are of great practical importance in risk management as they satisfy the coherency axiom and take the severity of tail losses into account. Yet the existing work mainly focuses on the univariate extremile regression in the low-dimensional framework. High-dimensional data subject to heavy-tailed phenomena are commonly encountered in various scientific fields and pose new challenges for extremile regression. In this article, we propose a (penalized) robust linear extremile regression model (remire) in the multivariate setting, incorporating the Huber loss function in place of the squared loss to enhance robustness for high-dimensional heavy-tailed data. In the regularized framework, we adopt the folded concave penalty for variable selection, which is implemented via a local adaptive majorize-minimization algorithm. Theoretically, we establish the convergence properties of the penalized remire estimator. The proposed method exhibits desirable properties and performs well in finite samples in terms of coefficient estimation and model selection, as demonstrated through comprehensive numerical studies. We further illustrate its practical utility through an application to childhood malnutrition data.
