Rank-based testing in linear models with stable errors

Abstract

Linear models with stable error densities are considered, and their local asymptotic normality with respect to the regression parameter is established. We use this result, combined with Le Cam's third lemma, to obtain local powers and asymptotic relative efficiencies for various classical rank tests (the regression and analysis of variance counterparts of the Wilcoxon, van der Waerden and median tests) under α-stable densities with various values of the skewness parameter and tail index. The same results are used to construct new rank tests, based on ‘stable scores’, achieving parametric optimality at specified stable densities. A Monte Carlo study is conducted to compare their finite-sample relative performances.

Keywords:

• stable distributions
• local asymptotic normality
• rank tests
• asymptotic relative efficiencies

Acknowledgements

Marc Hallin acknowledges the support of the Sonderforschungsbereich “Statistical modelling of non-linear dynamic processes” (SFB 823) of the Deutsche Forschungsgemeinschaft, and a Discovery Grant of the Australian Research Council. The research of Yvik Swan is supported by a Mandat de Chargé de Recherche from the Fonds National de la Recherche Scientifique, Communauté française de Belgique, and that of Thomas Verdebout by a BQR (Bonus Qualité Recherche) of the Université de Lille 2. David Veredas acknowledges the financial support of the IAP P6/07 contract, from the IAP program (Belgian Federal Scientific Policy) “Economic Policy and finance in the global economy”.