## 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) |
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Pages (from-to) | 1165-1192 |

Number of pages | 28 |

Journal | Annals of Statistics |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2016 |

## Keywords

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

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty