### 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 ℝ^{d} given covariates Z = z, taking values in ℝ^{k}, is a map u→→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 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}(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.

Original language | English (US) |
---|---|

Pages (from-to) | 1165-1192 |

Number of pages | 28 |

Journal | Annals of Statistics |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - Jun 1 2016 |

### Fingerprint

### Keywords

- Monge-Kantorovich-Brenier
- Vector conditional quantile function
- Vector quantile regression

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Statistics*,

*44*(3), 1165-1192. https://doi.org/10.1214/15-AOS1401

**Vector quantile regression : An optimal transport approach.** / Carlier, Guillaume; Chernozhukov, Victor; Galichon, Alfred.

Research output: Contribution to journal › Article

*Annals of Statistics*, vol. 44, no. 3, pp. 1165-1192. https://doi.org/10.1214/15-AOS1401

}

TY - JOUR

T1 - Vector quantile regression

T2 - An optimal transport approach

AU - Carlier, Guillaume

AU - Chernozhukov, Victor

AU - Galichon, Alfred

PY - 2016/6/1

Y1 - 2016/6/1

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 -