β Scribed by Roger Koenker; Stephen Portnoy
No coin nor oath required. For personal study only.
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.
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