Vector quantile regression beyond the specified case

This paper studies vector quantile regression (VQR), which models the dependence of a random vector with respect to a vector of explanatory variables with enough flexibility to capture the whole conditional distribution, and not only the conditional mean. The problem of vector quantile regression is formulated as an optimal transport problem subject to an additional mean-independence condition. This paper provides results on VQR beyond the specified case which had been the focus of previous work. We show that even beyond the specified case, the VQR problem still has a solution which provides a general representation of the conditional dependence between random vectors. Keywords: Duality; Optimal transport; Vector quantile regression