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

TL;DR

This paper introduces a novel one-step R-estimation method for linear models with stable errors, achieving root-n consistency and efficiency across a wide range of stable distributions, outperforming traditional techniques.

Contribution

The paper presents an original R-estimation approach that remains consistent and efficient under all stable error densities, including asymmetric and heavy-tailed cases.

Findings

Estimates are root-n consistent across all stable distributions.

Method achieves parametric efficiency bounds for specified tail index and skewness.

Simulations demonstrate superior finite-sample performance.

Abstract

Classical estimation techniques for linear models either are inconsistent, or perform rather poorly, under $\alpha$-stable error densities; most of them are not even rate-optimal. In this paper, we propose an original one-step 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 parametric stable-likelihood estimation, due to the absence of a closed form for stable densities, is quite cumbersome, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values $(\alpha_0, \ b_0)$ of the tail index $\alpha$ and skewness parameter $b$, while preserving root-$n$ consistency under any $(\alpha, \ b)$ as well as under usual light-tailed densities. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.