One-Step R-Estimation in Linear Models with Stable Errors

Abstract

Classical estimation techniques for linear models either are inconsistent, or perform somewhat poorly under stable error densities; most of them are not even rate-optimal. In this paper, we develop an original R-estimation method and investigate its asymptotic performances under stable densities. Contrary to traditional least squares, the proposed R-estimators, remain root-n consistent (the optimal rate) under the whole family of stable distributions, irrespective of their asymmetry and tail index. While stable-likelihood estimation, due to the absence of a closed form for stable densities, is generally considered unfeasible, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values of tail index alpha and the skewness parameter beta, while preserving root-n consistency under any alpha and beta. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.