Remire: Robust extremile regression in high dimensions

Unconditional extremiles offer significant advantages over quantiles in risk management, as they satisfy the coherency axioms and provide various interpretative benefits. However, the application of extremile regression in multivariate conditional settings remains underexplored. In this paper, we propose a (penalized) robust linear extremile regression model (remire), incorporating the Huber loss function in place of the squared loss to enhance robustness against heavy-tailed errors. For high-dimensional data, we introduce a variable selection method using a folded concave penalty, and design an iteratively reweighted l1-penalized procedure for estimation. Each iteration’s estimation is solved via a local adaptive majorize-minimization algorithm. The proposed method exhibits desirable properties and performs well in finite samples, as demonstrated through comprehensive numerical studies. We further illustrate its practical utility with a real data application focused on childhood malnutrition.