July, 2015

Report Number:

826

Abstract:

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 $\betaHat$.

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