Some novel aspects of quantile regression: local stationarity, random forests and optimal transportation

This paper is written for a Festschrift in honour of Professor Marc Hallin and it proposes some developments on quantile regression. We connect our investigation to Marc's scientific production and we present some theoretical and methodological advances for quantiles estimation in non standard settings. We split our contributions in two parts. The first part is about conditional quantiles estimation for nonstationary time series: our approach combines local stationarity for Markov processes with quantile regression. The second part is about conditional quantiles estimation for the analysis of multivariate independent data in the presence of possibly large dimensional covariates: our procedure combines optimal transport theory with quantile regression forests. Monte Carlo studies illustrate numerically the performance of our methods and compare them to extant methods. The codes needed to replicate our results are available on our $\mathtt{GitHub}$ pages.