Multiple-Output Quantile Regression through Optimal Quantization

Charlier et al. (2015a,b) developed a new nonparametric quantile regression method based on the concept of optimal quantization and showed that the resulting estimators often dominate their classical, kernel-type, competitors. The construction, however, remains limited to single-output quantile regression. In the present work, we therefore extend the quantization-based quantile regression method to the multiple-output context. We show how quantization allows to approximate the population multiple-output regression quantiles introduced in Hallin et al. (2015), which are conditional versions of the location multivariate quantiles from Hallin et al. (2010). We prove that this approximation becomes arbitrarily accurate as the size of the quantization grid goes to infinity. We also consider a sample version of the proposed quantization-based quantiles and establish their weak consistency for their population version. Through simulations, we compare the performances of the proposed quantization-based estimators with their local constant and local bilinear kernel competitors from Hallin et al. (2015). We also compare the corresponding sample quantile regions. The results reveal that the proposed quantization-based estimators, which are local constant in nature, outperform their kernel counterparts and even often dominate their local bilinear kernel competitors.