## The Annals of Statistics

### Vector quantile regression: An optimal transport approach

#### 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
https://projecteuclid.org/euclid.aos/1460381690

Digital Object Identifier
doi:10.1214/15-AOS1401

Mathematical Reviews number (MathSciNet)
MR3485957

Zentralblatt MATH identifier
1381.62239

#### 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. https://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.