This paper studies vector quantile regression (VQR), which is a way to model
the dependence of a random vector of interest with respect to a vector of
explanatory variables so to capture the whole conditional distribution, and not
only the conditional mean. The problem of vector quantile regression is
formulated as an optimal transport problem subject to an additional
mean-independence condition. This paper provides a new set of results on VQR
beyond the case with correct specification which had been the focus of previous
work. First, we show that even under misspecification, the VQR problem still
has a solution which provides a general representation of the conditional
dependence between random vectors. Second, we provide a detailed comparison
with the classical approach of Koenker and Bassett in the case when the
dependent variable is univariate and we show that in that case, VQR is
equivalent to classical quantile regression with an additional monotonicity
constraint.

研究向量分位数回归模型及其解决优化输运问题的能力，探讨其在未正确规定情况下的解决方案，说明该模型在解决单变量情况下叶克弗（Koenker）与巴塞特（Bassett）经典方法的等价性，同时还提供其它详细的结果。

超越正确规范的向量分位数回归

Vector quantile regression beyond correct specification

We propose a notion of conditional vector quantile function and a vector
quantile regression. A \emph{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 $u \longmapsto Q_{Y\mid Z}(u,z)$,
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 $Q_{Y\mid Z}(U,z)$ has the distribution of $Y$ conditional on
$Z=z$. Moreover, we have a strong representation, $Y = Q_{Y\mid Z}(U,Z)$ almost
surely, for some version of $U$. The \emph{vector quantile regression} (VQR) is
a linear model for CVQF of $Y$ given $Z$. Under correct specification, the
notion produces strong representation, $Y=\beta \left(U\right) ^\top f(Z)$, 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.

本文提出了条件向量分位函数和向量分位回归的概念。向量分位回归是一种针对给定协变量的条件向量分位函数的线性模型。该模型嵌入了著名的 Monge-Kantorovich 最优运输问题作为其核心的特殊情况。我们考虑了多重 Engel 曲线估计的应用。