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Rate of convergence of the density estimation of regression residual

Author

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  • Györfi László
  • Walk Harro

    (Universität Stuttgart, Department of Mathematics, Stuttgart, Deutschland)

Abstract

Consider the regression problem with a response variable Y and with a d-dimensional feature vector X. For the regression function m(x) = E{Y|X = x}, this paper investigates methods for estimating the density of the residual Y − m(X) from independent and identically distributed data. If the density is twice differentiable and has compact support then we bound the rate of convergence of the kernel density estimate. It turns out that for d ≤ 3 and for partitioning regression estimates, the regression estimation error has no influence on the rate of convergence of the density estimate.

Suggested Citation

  • Györfi László & Walk Harro, 2013. "Rate of convergence of the density estimation of regression residual," Statistics & Risk Modeling, De Gruyter, vol. 30(1), pages 55-74, March.
    • Handle: RePEc:bpj:strimo:v:30:y:2013:i:1:p:55-74:n:3
    DOI: 10.1524/strm.2013.1127
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    References listed on IDEAS

    as
    1. Neumeyer, N. & Van Keilegom, I., 2010. "Estimating the error distribution in nonparametric multiple regression with applications to model testing," LIDAM Reprints ISBA 2010006, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Harro Walk, 2002. "Almost Sure Convergence Properties of Nadaraya-Watson Regression Estimates," International Series in Operations Research & Management Science, in: Moshe Dror & Pierre L’Ecuyer & Ferenc Szidarovszky (ed.), Modeling Uncertainty, chapter 0, pages 201-223, Springer.
    3. Devroye, Luc & Felber, Tina & Kohler, Michael & Krzyżak, Adam, 2012. "L1-consistent estimation of the density of residuals in random design regression models," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 173-179.
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    5. 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.
    6. 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.
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    8. Ahmad, Ibrahim A., 1992. "Residuals density estimation in nonparametric regression," Statistics & Probability Letters, Elsevier, vol. 14(2), pages 133-139, May.
    9. Müller, Ursula U. & Schick, Anton & Wefelmeyer, Wolfgang, 2009. "Estimating the error distribution function in nonparametric regression with multivariate covariates," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 957-964, April.
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