Some pathological regression asymptotics under stable conditions

Abstract

We consider a simple through-the-origin linear regression example introduced by Rousseeuw, van Aelst and Hubert (J. Amer. Stat. Assoc., 94 (1994) 419–434). It is shown that the conventional least squares and least absolute error estimators converge in distribution without normalization and consequently are inconsistent. A class of weighted median regression estimators, including the maximum depth estimator of Rousseeuw and Hubert (J. Amer. Stat. Assoc., 94 (1999) 388–402), are shown to converge at rate n −1. Finally, the maximum likelihood estimator is considered, and we observe that there exist estimators that converge at rate n −2. The results illustrate some interesting, albeit somewhat pathological, aspects of stable-law convergence.