Conditional Quantile Estimation through Optimal Quantization

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).