On the impact of predictor geometry on the performance on high-dimensional ridge-regularized generalized robust regression estimators

We study ridge-regularized generalized robust regression estimators, i.e β̂ = argminβ∈Rp 1 n n ∑ i=1 ρi(Yi −X ′ iβ) + τ 2 ‖β‖ , where Yi = i +X ′ iβ0 . in the situation where p/n tends to a finite non-zero limit. Our study here focuses on the situation where the errors i’s are heavy-tailed and Xi’s have an “elliptical-like” distribution. Our assumptions are quite general and we do not require homoskedasticity of i’s for instance. We obtain a characterization of the limit of ‖β̂−β0‖, as well as several other results, including central limit theorems for the entries of β̂.