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Some pathological regression asymptotics under stable conditions

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  • Koenker, Roger
  • Portnoy, Stephen

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.

Suggested Citation

  • Koenker, Roger & Portnoy, Stephen, 2000. "Some pathological regression asymptotics under stable conditions," Statistics & Probability Letters, Elsevier, vol. 50(3), pages 219-228, November.
    • Handle: RePEc:eee:stapro:v:50:y:2000:i:3:p:219-228
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    References listed on IDEAS

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    1. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    2. Nolan, John P., 1998. "Parameterizations and modes of stable distributions," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 187-195, June.
    3. Kooperberg, Charles & Stone, Charles J., 1991. "A study of logspline density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 12(3), pages 327-347, November.
    4. Weron, Rafal, 1996. "On the Chambers-Mallows-Stuck method for simulating skewed stable random variables," Statistics & Probability Letters, Elsevier, vol. 28(2), pages 165-171, June.
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