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Conditional Quantile Estimation through Optimal Quantization

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  • Isabelle Charlier
  • Davy Paindaveine

Abstract

In this paper, we use quantization to construct a nonparametric estimator of conditionalquantiles of a scalar response Y given a d-dimensional vector of covariates X. First we focuson the population level and show how optimal quantization of X, which consists in discretizingX by projecting it on an appropriate grid of N points, allows to approximate conditionalquantiles of Y given X. We show that this is approximation is arbitrarily good as N goesto infinity and provide a rate of convergence for the approximation error. Then we turnto the sample case and define an estimator of conditional quantiles based on quantizationideas. We prove that this estimator is consistent for its fixed-N population counterpart. Theresults are illustrated on a numerical example. Dominance of our estimators over local constant/linear ones and nearest neighbor ones is demonstrated through extensive simulationsin the companion paper Charlier et al. (2014).

Suggested Citation

  • Isabelle Charlier & Davy Paindaveine, 2014. "Conditional Quantile Estimation through Optimal Quantization," Working Papers ECARES ECARES 2014-28, ULB -- Universite Libre de Bruxelles.
    • Handle: RePEc:eca:wpaper:2013/161134
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    References listed on IDEAS

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    1. Vlad Bally & Gilles Pagès & Jacques Printems, 2005. "A Quantization Tree Method For Pricing And Hedging Multidimensional American Options," Mathematical Finance, Wiley Blackwell, vol. 15(1), pages 119-168, January.
    2. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    3. P. J. Heagerty & M. S. Pepe, 1999. "Semiparametric estimation of regression quantiles with application to standardizing weight for height and age in US children," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 48(4), pages 533-551.
    4. Fischer, Aurélie, 2010. "Quantization and clustering with Bregman divergences," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2207-2221, October.
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    1. Charlier, Isabelle & Paindaveine, Davy & Saracco, Jérôme, 2015. "Conditional quantile estimation based on optimal quantization: From theory to practice," Computational Statistics & Data Analysis, Elsevier, vol. 91(C), pages 20-39.
    2. Isabelle Charlier & Davy Paindaveine & Jérôme Saracco, 2016. "Multiple-Output Quantile Regression through Optimal Quantization," Working Papers ECARES ECARES 2016-18, ULB -- Universite Libre de Bruxelles.
    3. Isabelle Charlier & Davy Paindaveine & Jérôme Saracco, 2014. "Conditional Quantile Estimation Based on Optimal Quantization: from Theory to Practice," Working Papers ECARES ECARES 2014-39, ULB -- Universite Libre de Bruxelles.
    4. Isabelle Charlier & Davy Paindaveine & Jérôme Saracco, 2014. "QuantifQuantile; an R Package for Performing Quantile Regression through Optimal Quantization," Working Papers ECARES ECARES 2014-40, ULB -- Universite Libre de Bruxelles.

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