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

July, 2015
Report Number: 
Noureddine El Karoui
We study ridge-regularized generalized robust regression estimators, i.e  $$ \betaHat=\argmin_{\beta \in \mathbb{R}^p} \frac{1}{n}\sum_{i=1}^n \rho_i(Y_i-X_i\trsp \beta)+\frac{\tau}{2}\norm{\beta}^2\;, \text{ where } Y_i=\eps_i+X_i\trsp \beta_0\;. $$ in the situation  where $p/n$ tends to a finite non-zero limit.    Our study here focuses on the situation where the errors $\eps_i$'s are heavy-tailed and $X_i$'s have an ``elliptical-like" distribution. Our assumptions are quite general and we do not require homoskedasticity of $\eps_i$'s for instance.    We obtain a characterization of the limit of $\norm{\betaHat-\beta_0}$, as well as several other results, including central limit theorems for the entries of $\betaHat$.