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L 1 regression estimate and its bootstrap

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Abstract

We study the asymptotic distribution of the L 1 regression estimator under general conditions with matrix norming and possibly non i.i.d. errors. We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties. It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution.

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References

  1. Bassett G, Koenker R. Asymptotic theory of least absolute error regression. J Amer Statist Assoc, 73: 618–622 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bloomfield P, Steiger W L. Least Absolute Deviations: Theory, Applications and Algorithms. Boston: Birkhäuser, 1983

    MATH  Google Scholar 

  3. Bai Z D, Chen X R, Wu Y, et al. Asymptotic normality of minimum L 1 norm estimates in linear models. Sci China Ser A, 33: 449–463 (1990)

    MathSciNet  Google Scholar 

  4. Pollard D. Asymptotics for least absolute deviation regression estimators. Econom Theory, 7: 186–199 (1991)

    Article  MathSciNet  Google Scholar 

  5. Smirnov N V. Limit distributions for the terms of a variational series. Amer Math Soc Transl Ser, 11: 82–143 (1952)

    Google Scholar 

  6. Jureckova J. Asymptotics behavior of M-estimators of location in nonregular cases. Statist Decisions, 1: 323–340 (1983)

    MATH  MathSciNet  Google Scholar 

  7. Bose A, Chatterjee S. Generalised bootstrap in nonregular M estimation problems. Statist Prob Lett, 55: 319–328 (2000)

    Article  MathSciNet  Google Scholar 

  8. Knight K. Limit distributions for L 1 regression estimators under general conditions. Ann Statist, 26: 755–770 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Knight K. Bootstrapping sample quantiles in non-regular cases. Statist Prob Lett, 37: 259–267 (1998)

    Article  MathSciNet  Google Scholar 

  10. Hjort N L, Pollard D. Asymptotics for minimisers of convex processes. Preprint, Department of Statistics, Yale University, 1993

  11. Bickel P J, Freedman D A. Some asymptotic theory for the bootstrap. Ann Statist, 9(6): 1196–1217 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bose A, Chatterjee S. Generalised bootstrap for estimators of minimisers of convex functionals. J Statist Plan Infer, 11: 225–239 (2003)

    Article  MathSciNet  Google Scholar 

  13. Hu F, Kalbfleisch J D. The estimating function bootstrap (with discussion). Canad J Statist, 28: 449–499 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Liu R Y, Singh K. Efficiency and robustness in resampling. Ann Statist, 20(1): 370–384 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bose A, Kushary D. Jackknife and weighted jackknife estimation of the variance of M-estimators in linear regressions. Technical Report No. 96-12, Department of Statistics, Purdue University, 1996

  16. Wu C F J. On the asymptotic properties of the jackknife histogram. Ann Statist, 18: 1438–1452 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  17. Bickel P J, Gotze F, Van Z W R. Resampling fewer than n observations: gains, losses and remedies for losses. Statistica Sinica, 7: 1–31 (1997)

    MATH  MathSciNet  Google Scholar 

  18. Athreya K B, Fukuchi J I. Bootstrapping extremes of i.i.d. random variables. In: Proc of Conf on Extreme Value Theory and its Applications. Maryland: NIST, 1997

    Google Scholar 

  19. Athreya K B, Fukuchi J I. Confidence intervals for endpoints of a C.D.F. via bootstrap. J Statist Plan Infer, 58: 299–320 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Deheuvels P, Mason D, Shorack G. Some results on the influence of extremes on the bootstrap. Ann Inst Henri Poincare, 29: 83–103 (1993)

    MATH  MathSciNet  Google Scholar 

  21. Athreya K B. Bootstrap of the mean in the infinite variance case. Ann Statist, 14: 724–731 (1987)

    Article  MathSciNet  Google Scholar 

  22. Knight K. On the bootstrap of the sample mean in the infinite variance case. Ann Statist, 17: 1168–1175 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  23. Praestgaard J, Wellner J. Exchangeably weighted bootstrap of the general empirical process. Ann Probab, 21: 2053–2086 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  24. Knight K. Asymptotics for L 1 estimators of regression parameters under heteroscedasticity. Canad J Statist, 27: 497–507 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata, 700108, India

    Arup Bose

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  1. Arup Bose
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Correspondence to Arup Bose.

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Dedicated to Professor Zhidong Bai on the occasion of his 65th birthday

This work was supported by J.C. Bose National Fellowship, Government of India

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Bose, A. L 1 regression estimate and its bootstrap. Sci. China Ser. A-Math. 52, 1251–1261 (2009). https://doi.org/10.1007/s11425-009-0087-6

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  • DOI: https://doi.org/10.1007/s11425-009-0087-6

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