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Nonparametric -quantile regression using penalised splines

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  • Monica Pratesi
  • M. Ranalli
  • Nicola Salvati

Abstract

Quantile regression investigates the conditional quantile functions of a response variable in terms of a set of covariates. M-quantile regression extends this idea by a ‘quantile-like’ generalisation of regression based on influence functions. In this work, we extend it to nonparametric regression, in the sense that the M-quantile regression functions do not have to be assumed to have a certain parametric form, but can be left undefined and estimated from the data. Penalised splines are employed to estimate them. This choice makes it easy to move to bivariate smoothing and semiparametric modelling. An algorithm based on iteratively reweighted penalised least squares to actually fit the model is proposed. Quantile crossing is addressed using an a posteriori adjustment to the function fits following He [1]. Simulation studies show the finite sample properties of the proposed estimation technique.

Suggested Citation

  • Monica Pratesi & M. Ranalli & Nicola Salvati, 2009. "Nonparametric -quantile regression using penalised splines," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(3), pages 287-304.
    • Handle: RePEc:taf:gnstxx:v:21:y:2009:i:3:p:287-304
    DOI: 10.1080/10485250802638290
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    References listed on IDEAS

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    1. Chaudhuri, Probal, 1991. "Global nonparametric estimation of conditional quantile functions and their derivatives," Journal of Multivariate Analysis, Elsevier, vol. 39(2), pages 246-269, November.
    2. Thomas Lee & Hee-Seok Oh, 2007. "Robust penalized regression spline fitting with application to additive mixed modeling," Computational Statistics, Springer, vol. 22(1), pages 159-171, April.
    3. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    4. X. He & P. Ng & S. Portnoy, 1998. "Bivariate quantile smoothing splines," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(3), pages 537-550.
    5. Yingxing Li & David Ruppert, 2008. "On the asymptotics of penalized splines," Biometrika, Biometrika Trust, vol. 95(2), pages 415-436.
    6. Hee-Seok Oh & Douglas W. Nychka & Thomas C. M. Lee, 2007. "The Role of Pseudo Data for Robust Smoothing with Application to Wavelet Regression," Biometrika, Biometrika Trust, vol. 94(4), pages 893-904.
    7. Roger Koenker & Ivan Mizera, 2004. "Penalized triograms: total variation regularization for bivariate smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 145-163, February.
    8. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
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    Cited by:

    1. Marco Alfò & Maria Francesca Marino & Maria Giovanna Ranalli & Nicola Salvati & Nikos Tzavidis, 2021. "M‐quantile regression for multivariate longitudinal data with an application to the Millennium Cohort Study," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(1), pages 122-146, January.
    2. Patrick Munyangabo & Anthony Waititu & Anthony Kibira Wanjoya, 2019. "Estimation of Nested Error Non-parametric Unit Level Model," Journal of Statistical and Econometric Methods, SCIENPRESS Ltd, vol. 8(1), pages 1-3.

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