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Universally Consistent Regression Function Estimation Using Hierarchial B-Splines

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  • Kohler, Michael

Abstract

Estimation of multivariate regression functions from i.i.d. data is considered. We construct estimates by empiricalL2-error minimization over data-dependent spaces of polynomial spline functions. For univariate regression function estimation these spaces are spline spaces with data-dependent knot sequences. In the multivariate case, we use so-called hierarchical spline spaces which are defined as linear span of tensor product B-splines with nested knot sequences. The knot sequences of the chosen B-splines depend locally on the data. Â We show the strongL2-consistency of the estimators without any condition on the underlying distribution. The estimators are similar to histogram regression estimators using data-dependent partitions and partitioning regression estimators based on local polynomial fits. The main difference is that the estimators considered here are smooth functions, which seems to be desirable especially in the case that the regression function to be estimated is smooth.

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  • Kohler, Michael, 1999. "Universally Consistent Regression Function Estimation Using Hierarchial B-Splines," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 138-164, January.
    • Handle: RePEc:eee:jmvana:v:68:y:1999:i:1:p:138-164
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    References listed on IDEAS

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    1. Marta Horvath & Gábor Lugosi, 1996. "A data-dependent skeleton estimate and a scale-sensitive dimension for classification," Economics Working Papers 199, Department of Economics and Business, Universitat Pompeu Fabra.
    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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    1. 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.
    2. Györfi, László & Walk, Harro, 2012. "Strongly consistent density estimation of the regression residual," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1923-1929.
    3. Kohler, Michael & Máthé, Kinga & Pintér, Márta, 2002. "Prediction from Randomly Right Censored Data," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 73-100, January.
    4. Michael Kohler, 2002. "Universal Consistency of Local Polynomial Kernel Regression Estimates," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 54(4), pages 879-899, December.
    5. 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.
    6. 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.

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