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 YY, taking values in RdRd given covariates Z=zZ=z, taking values in RkRk, is a map u⟼QY|Z(u,z)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 UU follows a reference non-atomic distribution FUFU, for instance uniform distribution on a unit cube in RdRd, the random vector QY|Z(U,z)QY|Z(U,z) has the distribution of YY conditional on Z=zZ=z. Moreover, we have a strong representation, Y=QY|Z(U,Z)Y=QY|Z(U,Z) almost surely, for some version of UU. The vector quantile regression (VQR) is a linear model for CVQF of YY given ZZ. Under correct specification, the notion produces strong representation, Y=β(U)⊤f(Z)Y=β(U)⊤f(Z), for f(Z)f(Z) denoting a known set of transformations of ZZ, where u⟼β(u)⊤f(Z)u⟼β(u)⊤f(Z) is a monotone map, the gradient of a convex function and the quantile regression coefficients u⟼β(u)u⟼β(u) have the interpretations analogous to that of the standard scalar quantile regression. As f(Z)f(Z) becomes a richer class of transformations of ZZ, 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 YY 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.

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