VECTOR QUANTILE REGRESSION: AN OPTIMAL TRANSPORT APPROACH

We propose a notion of conditional vector quantile function and a vector quantile regression. A conditional vector quantile function (CVQF) of a random vector Y, taking values in .d given covariates Z = z, taking values in .k, is a map u . QY|Z(u, z), which is monotone, in the sense of being a gradient of a convex function, and such that given that vector U follows a reference non-atomic distribution FU, for instance uniform distribution on a unit cube in .d, the random vector QY|Z(U, z) has the distribution of Y conditional on Z = z. Moreover, we have a strong representation, Y = Qy|z(U, Z) almost surely, for some version of U. The vector quantile regression (VQR) is a linear model for CVQF of Y given Z. Under correct specification, the notion produces strong representation, Y = .(U)T f(Z), for f(Z) denoting a known set of transformations of Z, where u . .(u)T f(Z) is a monotone map, the gradient of a convex function and the quantile regression coefficients u . .(u) have the interpretations analogous to that of the standard scalar quantile regression. As f (Z) becomes a richer class of transformations of Z, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge-Kantorovich's optimal transportation problem at its core as a special case. In the classical case, where Y is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. An application to multiple Engel curve estimation is considered.