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On the strong universal consistency of local averaging regression estimates

Author

Listed:
  • Matthias Hansmann

    (Technische Universität Darmstadt)

  • Michael Kohler

    (Technische Universität Darmstadt)

  • Harro Walk

    (Universität Stuttgart)

Abstract

A general result concerning the strong universal consistency of local averaging regression estimates is presented, which is used to extend previously known results on the strong universal consistency of kernel and partitioning regression estimates. The proof is based on ideas from Etemadi’s proof of the strong law of large numbers, which shows that these ideas are also useful in the context of strong laws of large numbers for conditional expectations in $$L_2$$ L 2 .

Suggested Citation

  • Matthias Hansmann & Michael Kohler & Harro Walk, 2019. "On the strong universal consistency of local averaging regression estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(5), pages 1233-1263, October.
    • Handle: RePEc:spr:aistmt:v:71:y:2019:i:5:d:10.1007_s10463-018-0674-9
    DOI: 10.1007/s10463-018-0674-9
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    References listed on IDEAS

    as
    1. Kohler, Michael, 1999. "Universally Consistent Regression Function Estimation Using Hierarchial B-Splines," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 138-164, January.
    2. Harro Walk, 2005. "Strong universal consistency of smooth kernel regression estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(4), pages 665-685, December.
    3. Harro Walk, 2001. "Strong Universal Pointwise Consistency of Recursive Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(4), pages 691-707, December.
    4. Michael Kohler & Adam Krzyżak & Harro Walk, 2003. "Strong consistency of automatic kernel regression estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(2), pages 287-308, June.
    5. Györfi L. & Kohler M. & Walk H., 1998. "Weak And Strong Universal Consistency Of Semi-Recursive Kernel And Partitioning Regression Estimates," Statistics & Risk Modeling, De Gruyter, vol. 16(1), pages 1-18, January.
    6. Györfi, László & Walk, Harro, 1997. "On the strong universal consistency of a recursive regression estimate by Pál Révész," Statistics & Probability Letters, Elsevier, vol. 31(3), pages 177-183, January.
    7. Irle, A., 1997. "On Consistency in Nonparametric Estimation under Mixing Conditions," Journal of Multivariate Analysis, Elsevier, vol. 60(1), pages 123-147, January.
    8. Walk, Harro, 2008. "A universal strong law of large numbers for conditional expectations via nearest neighbors," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1035-1050, July.
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    Cited by:

    1. Matthias Hansmann & Benjamin M. Horn & Michael Kohler & Stefan Ulbrich, 2022. "Estimation of conditional distribution functions from data with additional errors applied to shape optimization," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(3), pages 323-343, April.

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