June 2016 Vector quantile regression: An optimal transport approach
Guillaume Carlier, Victor Chernozhukov, Alfred Galichon
Ann. Statist. 44(3): 1165-1192 (June 2016). DOI: 10.1214/15-AOS1401

## 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.

## Citation

Guillaume Carlier. Victor Chernozhukov. Alfred Galichon. "Vector quantile regression: An optimal transport approach." Ann. Statist. 44 (3) 1165 - 1192, June 2016. https://doi.org/10.1214/15-AOS1401

## Information

Received: 1 June 2015; Revised: 1 September 2015; Published: June 2016
First available in Project Euclid: 11 April 2016

zbMATH: 1381.62239
MathSciNet: MR3485957
Digital Object Identifier: 10.1214/15-AOS1401

Subjects:
Primary: 62J99
Secondary: 62G05 , 62H05

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