Vector quantile regression: An optimal transport approach

Guillaume Carlier, Victor Chernozhukov, Alfred Galichon

    Research output: Contribution to journalArticle

    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 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.

    Original languageEnglish (US)
    Pages (from-to)1165-1192
    Number of pages28
    JournalAnnals of Statistics
    Volume44
    Issue number3
    DOIs
    StatePublished - Jun 1 2016

    Fingerprint

    Optimal Transport
    Quantile Regression
    Quantile Function
    Scalar
    Random Vector
    Convex function
    Curve Estimation
    Optimal Transportation
    Gradient
    Monotone Map
    Conditional Quantiles
    Quantile regression
    Unit cube
    Transportation Problem
    Nonparametric Model
    Regression Coefficient
    Uniform distribution
    Covariates
    Linear Model
    Monotone

    Keywords

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

    ASJC Scopus subject areas

    • Statistics and Probability
    • Statistics, Probability and Uncertainty

    Cite this

    Carlier, G., Chernozhukov, V., & Galichon, A. (2016). Vector quantile regression: An optimal transport approach. 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.

    In: Annals of Statistics, Vol. 44, No. 3, 01.06.2016, p. 1165-1192.

    Research output: Contribution to journalArticle

    Carlier, G, Chernozhukov, V & Galichon, A 2016, 'Vector quantile regression: An optimal transport approach', Annals of Statistics, vol. 44, no. 3, pp. 1165-1192. https://doi.org/10.1214/15-AOS1401
    Carlier, Guillaume ; Chernozhukov, Victor ; Galichon, Alfred. / Vector quantile regression : An optimal transport approach. In: Annals of Statistics. 2016 ; Vol. 44, No. 3. pp. 1165-1192.
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