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Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data

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  • Kohler, Michael
  • Krzyzak, Adam
  • Walk, Harro

Abstract

Estimation of regression functions from independent and identically distributed data is considered. The L2 error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates besides smoothness assumptions on the regression function and moment conditions on Y also boundedness assumptions on X are made. In this article we consider partitioning and nearest neighbor estimates and show that by replacing the boundedness assumption on X by a proper moment condition the same rate of convergence can be shown as for bounded data.

Suggested Citation

  • Kohler, Michael & Krzyzak, Adam & Walk, Harro, 2006. "Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data," Journal of Multivariate Analysis, Elsevier, vol. 97(2), pages 311-323, February.
    • Handle: RePEc:eee:jmvana:v:97:y:2006:i:2:p:311-323
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    References listed on IDEAS

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    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. 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.
    3. 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.
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    Cited by:

    1. Leluc, Rémi & Portier, François & Zhuman, Aigerim & Segers, Johan, 2023. "Speeding up Monte Carlo Integration: Control Neighbors for Optimal Convergence," LIDAM Discussion Papers ISBA 2023019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Charlier, Isabelle & Paindaveine, Davy & Saracco, Jérôme, 2015. "Conditional quantile estimation based on optimal quantization: From theory to practice," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 20-39.
    3. Liitiäinen, Elia & Corona, Francesco & Lendasse, Amaury, 2010. "Residual variance estimation using a nearest neighbor statistic," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 811-823, April.
    4. Sancetta, A., 2007. "Online Forecast Combination for Dependent Heterogeneous Data," Cambridge Working Papers in Economics 0718, Faculty of Economics, University of Cambridge.
    5. Kohler, Michael, 2014. "Optimal global rates of convergence for noiseless regression estimation problems with adaptively chosen design," Journal of Multivariate Analysis, Elsevier, vol. 132(C), pages 197-208.
    6. P. G. Ferrario & H. Walk, 2012. "Nonparametric partitioning estimation of residual and local variance based on first and second nearest neighbours," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 1019-1039, December.
    7. Paola Gloria Ferrario, 2018. "Partitioning estimation of local variance based on nearest neighbors under censoring," Statistical Papers, Springer, vol. 59(2), pages 423-447, June.
    8. Cattaneo, Matias D. & Farrell, Max H., 2013. "Optimal convergence rates, Bahadur representation, and asymptotic normality of partitioning estimators," Journal of Econometrics, Elsevier, vol. 174(2), pages 127-143.
    9. Kohler, Michael & Krzyżak, Adam, 2013. "Optimal global rates of convergence for interpolation problems with random design," Statistics & Probability Letters, Elsevier, vol. 83(8), pages 1871-1879.

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