Vector Quantile Regression

By Guillaume Carlier, Victor Chernozhukov, Alfred Galichon. 2014

63 citations recorded by the source corpus.

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 ℝd given covariates Z=z, taking values in ℝk, 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 ℝd, 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=β(U)⊤f(Z), for f(Z) denoting a known set of transformations of Z, where u↦β(u)⊤f(Z) is a monotone map, the gradient of a convex function, and the quantile regression coefficients u↦β(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. Several applications to diverse problems such as multiple Engel curve estimation, and measurement of financial risk, are considered.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1fb7ed8d9bdb0d8813e6d34bf92b62d96bb4b4ad

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Citations in the corpus

Multivariate goodness-of-fit tests based on Wasserstein distance (2020, background): “The most popular ones are the Cramér–von Mises (Cramér, 1928; von Mises, 1928) and Kolmogorov–Smirnov (Kolmogorov, 1933; Smirnov, 1939) tests, involving distances between the cumulative distribution function of P0 and the empirical one. Originally defined for univariate distributions only, they have”
Vector quantile regression beyond correct specification (2016, methodology): “The approach proposed in [4] is based on optimal transport ideas and can be described as follows.”