On quantile regression models for multivariate data

The goal of this thesis is to bridge the gap between univariate and multivariate quantiles by extending the study of univariate quantile regression and its generalizations to multivariate responses. The statistical analysis focuses on a multivariate framework where we consider vector-valued quantile functions associated with multivariate distributions, providing inferential procedures and establishing the asymptotic properties of the proposed estimators. We illustrate their applicability in a wide variety of scientific settings, including time series, longitudinal and clustered data. The dissertation is divided into four chapters, each of them focusing on various aspects of multivariate analysis and different data types and structures. The methodologies we propose are supported by theoretical results and illustrated using simulation studies and real-world data.