TY - JOUR

T1 - Vector quantile regression

T2 - An optimal transport approach

AU - Carlier, Guillaume

AU - Chernozhukov, Victor

AU - Galichon, Alfred

N1 - Funding Information:
Supported in part by INRIA and the ANR Projects ISOTACE (ANR-12-MONU-0013) and OPTIFORM (ANR-12-BS01-0007). Supported in part by an NSF grant. Supported in part by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement 313699.
Publisher Copyright:
© Institute of Mathematical Statistics, 2016.

PY - 2016/6

Y1 - 2016/6

N2 - 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(UZ) 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)Τ f (Z) for f (Z) denoting a known set of transformations of Z where u →β(u)Τ 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. An application to multiple Engel curve estimation is considered.

AB - 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(UZ) 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)Τ f (Z) for f (Z) denoting a known set of transformations of Z where u →β(u)Τ 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. An application to multiple Engel curve estimation is considered.

KW - Monge-Kantorovich-Brenier

KW - Vector conditional quantile function

KW - Vector quantile regression

UR - http://www.scopus.com/inward/record.url?scp=84963541962&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84963541962&partnerID=8YFLogxK

U2 - 10.1214/15-AOS1401

DO - 10.1214/15-AOS1401

M3 - Article

AN - SCOPUS:84963541962

VL - 44

SP - 1165

EP - 1192

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 3

ER -