The Annals of Statistics

Vector quantile regression: An optimal transport approach

Guillaume Carlier, Victor Chernozhukov, and Alfred Galichon

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Abstract

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 $\mathbb{R}^{d}$ given covariates $Z=z$, taking values in $\mathbb{R}^{k}$, is a map $u\longmapsto Q_{Y|Z}(u,z)$, 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 $F_{U}$, for instance uniform distribution on a unit cube in $\mathbb{R}^{d}$, the random vector $Q_{Y|Z}(U,z)$ has the distribution of $Y$ conditional on $Z=z$. Moreover, we have a strong representation, $Y=Q_{Y|Z}(U,Z)$ 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, $Y=\beta (U)^{\top}f(Z)$, for $f(Z)$ denoting a known set of transformations of $Z$, where $u\longmapsto\beta(u)^{\top}f(Z)$ is a monotone map, the gradient of a convex function and the quantile regression coefficients $u\longmapsto\beta(u)$ 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.

Article information

Source
Ann. Statist. Volume 44, Number 3 (2016), 1165-1192.

Dates
Received: June 2015
Revised: September 2015
First available in Project Euclid: 11 April 2016

Permanent link to this document
http://projecteuclid.org/euclid.aos/1460381690

Digital Object Identifier
doi:10.1214/15-AOS1401

Mathematical Reviews number (MathSciNet)
MR3485957

Zentralblatt MATH identifier
06590312

Subjects
Primary: 62J99: None of the above, but in this section
Secondary: 62H05: Characterization and structure theory 62G05: Estimation

Keywords
Vector quantile regression vector conditional quantile function Monge–Kantorovich–Brenier

Citation

Carlier, Guillaume; Chernozhukov, Victor; Galichon, Alfred. Vector quantile regression: An optimal transport approach. Ann. Statist. 44 (2016), no. 3, 1165--1192. doi:10.1214/15-AOS1401. http://projecteuclid.org/euclid.aos/1460381690.


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Supplemental materials

  • Supplement to “Vector quantile regression”. In the online supplement [3], we provide additional results for Sections 2 and 3, including a proof of duality for CVQF and Linear VQR, and the measurability claims for Theorem 2.1.