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Strongly consistent density estimation of the regression residual

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  • Györfi, László
  • Walk, Harro
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    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Statistics & Probability Letters.

    Volume (Year): 82 (2012)
    Issue (Month): 11 ()
    Pages: 1923-1929

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    Handle: RePEc:eee:stapro:v:82:y:2012:i:11:p:1923-1929

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    Related research

    Keywords: Regression residual; Nonparametric kernel density estimation; Nonparametric regression estimation; Heteroscedastic regression;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Harro Walk, 2005. "Strong universal consistency of smooth kernel regression estimates," Annals of the Institute of Statistical Mathematics, Springer, vol. 57(4), pages 665-685, December.
    2. 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.
    3. Kohler, Michael, 1999. "Universally Consistent Regression Function Estimation Using Hierarchial B-Splines," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 138-164, January.
    4. 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.
    5. Ahmad, Ibrahim A., 1992. "Residuals density estimation in nonparametric regression," Statistics & Probability Letters, Elsevier, vol. 14(2), pages 133-139, May.
    6. Michael G. Akritas, 2001. "Non-parametric Estimation of the Residual Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 28(3), pages 549-567.
    7. 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.
    8. Sam Efromovich, 2007. "Optimal nonparametric estimation of the density of regression errors with finite support," Annals of the Institute of Statistical Mathematics, Springer, vol. 59(4), pages 617-654, December.
    9. 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 18, January.
    10. Cheng, Fuxia, 2002. "Consistency of error density and distribution function estimators in nonparametric regression," Statistics & Probability Letters, Elsevier, vol. 59(3), pages 257-270, October.
    11. 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.
    12. 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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