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Strongly consistent density estimation of the regression residual

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  • Györfi, László
  • Walk, Harro

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. For heteroscedastic regression, we prove the strong universal (density-free) L1-consistency of a recursive and a nonrecursive kernel density estimate based on a regression estimate.

Suggested Citation

  • Györfi, László & Walk, Harro, 2012. "Strongly consistent density estimation of the regression residual," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1923-1929.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:11:p:1923-1929
    DOI: 10.1016/j.spl.2012.06.021
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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.
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    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. 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.
    8. Müller Ursula U. & Schick Anton & Wefelmeyer Wolfgang, 2007. "Estimating the error distribution function in semiparametric regression," Statistics & Risk Modeling, De Gruyter, vol. 25(1/2007), pages 1-18, January.
    9. 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).
    10. 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.
    11. Neumeyer, Natalie & Van Keilegom, Ingrid, 2010. "Estimating the error distribution in nonparametric multiple regression with applications to model testing," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1067-1078, May.
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