Some pathological regression asymptotics under stable conditions

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.