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Efficiently estimating the error distribution in nonparametric regression with responses missing at random

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  • Justin Chown
  • Ursula U. Müller

Abstract

This article considers nonparametric regression models with multivariate covariates and with responses missing at random. We estimate the regression function with a local polynomial smoother. The residual-based empirical distribution function that only uses complete cases, i.e. residuals that can actually be constructed from the data, is shown to be efficient in the sense of Hájek and Le Cam. In the proofs we derive, more generally, the efficient influence function for estimating an arbitrary linear functional of the error distribution; this covers the distribution function as a special case. We also show that the complete case residual-based empirical distribution function admits a functional central limit theorem. This is done by applying the transfer principle for complete case statistics developed by Koul et al. [(2012), 'The Transfer Principle: a Tool for Complete Case Analysis', Annals of Statistics , 40, 3031-3049], which makes it possible to adapt known results for fully observed data to the missing data case. The article concludes with a small simulation study investigating the performance of the complete case residual-based empirical distribution function.

Suggested Citation

  • Justin Chown & Ursula U. Müller, 2013. "Efficiently estimating the error distribution in nonparametric regression with responses missing at random," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(3), pages 665-677, September.
    • Handle: RePEc:taf:gnstxx:v:25:y:2013:i:3:p:665-677
    DOI: 10.1080/10485252.2013.795222
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    References listed on IDEAS

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    1. Efromovich, Sam, 2011. "Nonparametric Regression With Predictors Missing at Random," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 306-319.
    2. Hua Liang & Suojin Wang & Raymond J. Carroll, 2007. "Partially linear models with missing response variables and error-prone covariates," Biometrika, Biometrika Trust, vol. 94(1), pages 185-198.
    3. Müller Ursula U. & Schick Anton & Wefelmeyer Wolfgang, 2007. "Estimating the error distribution function in semiparametric regression," Statistics & Risk Modeling, De Gruyter, vol. 25(1), pages 1-18, January.
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    Cited by:

    1. Xu Guo & Wangli Xu & Lixing Zhu, 2015. "Model checking for parametric regressions with response missing at random," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(2), pages 229-259, April.
    2. Chown, Justin, 2016. "Efficient estimation of the error distribution function in heteroskedastic nonparametric regression with missing data," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 31-39.

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