Measure Transportation and Statistical Decision Theory

Abstract

Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered. As a consequence, extending to a multivariate context fundamental univariate statistical tools such as quantiles, signs, and ranks is anything but obvious. Tentative definitions have been proposed in the literature but do not enjoy the basic properties (e.g., distribution-freeness of ranks, their independence with respect to the order statistic, their independence with respect to signs) they are expected to satisfy. Based on measure transportation ideas, new concepts of distribution and quantile functions, ranks, and signs have been proposed recently that, unlike previous attempts, do satisfy these properties. These ranks, signs, and quantiles have been used, quite successfully, in several inference problems and have triggered, in a short span of time, a number of applications: fully distribution-free testing for multiple-output regression, MANOVA, and VAR models; R-estimation for VARMA parameters; distribution-free testing for vector independence; multiple-output quantile regression; nonlinear independent component analysis; and so on. Expected final online publication date for the Annual Review of Statistics, Volume 9 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Estimation of extreme quantiles from heavy-tailed distributions with neural networks

“…Our further work will consist in extending our NN approximations of extreme quantiles to other risk measures such as expected shortfall, or expectiles and then implementing the associated estimators. To complete the current theoretical analysis which ensures accurate approximation in the univariate case, our further work will be dedicated to investigate (in the non-conditional case) multivariate extreme quantile estimation basing on recent characterizations through optimal transport [42].…”

Nonparametric Multiple-Output Center-Outward Quantile Regression

“…Center-outward quantile functions define nested closed regions C P (τ ) and continuous contours C P (τ ) indexed by τ ∈ ([0, 1) such that, for any absolutely continuous P, P C P (τ ) = τ irrespective of P. Unlike the previous concepts, thus, and unlike the depth-based ones (as proposed in Hallin, Paindaveine and Šiman (2010) or Kong and Mizera (2012)), these measuretransportation-based quantiles do satisfy the essential property that the P-probability contents of the resulting quantile regions do not depend on P. Moreover, the corresponding quantile regions are not necessarily convex and, as shown in Figure 1, they are able to cap- ture the "shape" of the underlying distribution. We refer to Hallin (2022) for a survey of measure-transportation-based center-outward quantiles, the dual concepts of multivariate ranks and signs, and their many applications in inference problems (Ghosal and Sen (2019) Motivated by this long list of successful applications, we are proposing in this paper a novel and meaningful solution, based on the concept of center-outward quantiles, to the problem of nonparametric multiple-output quantile regression. Namely, for a pair of multidimensional random variables (X, Y) with values in R m × R d (Y the variable of interest, X the vector of covariates) and joint distribution 1 P, we define (Section 2.2) the center-outward quantile map…”

A note on the regularity of optimal-transport-based center-outward distribution and quantile functions

“…Given this considerations it cannot be a surprise the large amount of literature devoted to different extensions of the univariate distribution functions to higher dimension and the related concept of ranks. These generalizations come from different motivations related to depth, elliptic models or componentwise extensions among other (Hallin et al (2015(Hallin et al ( , 2010; Hallin (2022)). However, since the univariate distribution function is deeply related with the natural ordering of R, all of them have had to deal with the same problem, the lack of canonical order in dimension higher than one.…”

“…From the point of view of statistical inference, none of the extensions before Chernozhukov et al (2017) and was able to produce a meaningful version of mutivariate ranks. These last two works introduced a multivariate extension of the distribution function which, in turn, allows to construct semiparametrically efficient statistical procedures for multivariate data Hallin (2022). The idea for this extension is based on the mass transportation characterization of the the distribution function Fit is the optimal transport map from the associated probability measure P to the uniform distribution on [0, 1], or, more generally, the unique gradient of a convex map that pushes forward P to the uniform on [0,1].…”