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L1-consistent estimation of the density of residuals in random design regression models

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  • Devroye, Luc
  • Felber, Tina
  • Kohler, Michael
  • Krzyżak, Adam

Abstract

In this paper we study the problem of estimating the density of the error distribution in a random design regression model, where the error is assumed to be independent of the design variable. Our main result is that the L1 error of the kernel density estimate applied to residuals of a consistent regression estimate converges with probability 1 to 0 regardless of the form of the true density. We demonstrate that this result is in general no longer true if the error distribution and the design variable are dependent.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:1:p:173-179
    DOI: 10.1016/j.spl.2011.09.023
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    References listed on IDEAS

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    1. Michael G. Akritas & Ingrid Van Keilegom, 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, September.
    2. 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.
    3. 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).
    4. 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.
    5. Ahmad, Ibrahim A., 1992. "Residuals density estimation in nonparametric regression," Statistics & Probability Letters, Elsevier, vol. 14(2), pages 133-139, May.
    6. 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.
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    Cited by:

    1. 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.
    2. Györfi, László & Walk, Harro, 2012. "Strongly consistent density estimation of the regression residual," Statistics & Probability Letters, Elsevier, vol. 82(11), pages 1923-1929.

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    More about this item

    Keywords

    Density estimation; L1 error; Residuals; Nonparametric regression; Universal consistency;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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