This is a short course of the Mathematics for daTa scieNce study plan

Marc Hallin

Marc Hallin
(Department of Mathematics, Université libre de Bruxelles, Belgium)
Bio Marc Hallin holds a PhD in Mathematics from the Université libre de Bruxelles (1976). A Fellow of the Institute of Mathematical Statistics (I.M.S.) and the American Statistical Association (A.S.A.), he is an elected member of the International Statistical Institute (I.S.I.), and member of the Classe des Sciences of the Royal Academy of Belgium. The author of about 260 papers in the Annals of Statistics, Bernoulli, JASA, the Journal of Econometrics, JRSS B, ..., he is the recipient of several prestigious awards among which the Gottfried Noether Distinguished Scholar Award (American Statistical Association), the Prix Pierre-Simon de Laplace (Société française de Statistique), the Mahalanobis Memorial Award and Lecture (Indian Statistical Institute), the Medal of the Faculty of Mathematics and Physics (Charles University, Prague), and the Hermann Otto Hirschfeld Award and Lecture (Humboldt-Universität zu Berlin). His research interests in mathematical statistics include semiparametric and rank-based inference, statistical depth, quantile-oriented models, and measure transportation. He also had significant contributions in time series and time series econometrics, with special focus on factor models in the analysis of high-dimensional time series.

Course Description

  1. Introduction: Measure Transportation in a Nutshell. From Monge and Kantorovich to Brenier and McCann, a user-friendly introduction to measure transportation.
  2. The long quest for multivariate quantiles and ranks. Quantiles are a fundamental concept in probability and an essential tool in statistics, from descriptive to inferential. Still, until recently, and despite half a century of attempts (motivating the development of copula, Tukey depth, spatial and geometric quantiles), no fully satisfactory and fully agreed-upon definition of the concept is available beyond the well-understood case of univariate variables and distributions. The need for such a definition is particularly critical for variables taking values in \(R^d\), for directional variables (values on the hypersphere), and, more generally, for variables with values on manifolds. Unlike the real line, indeed, no canonical ordering is available on these domains. We show how measure transportation brings a solution to this long-standing problem by characterizing distribution-specific (data-driven, in the empirical case) orderings and center-outward quantile functions that satisfy all the properties expected from such concepts while reducing, in the case of real-valued variables, to the classical univariate notions.
  3. Multivariate ranks and multivariate rank tests. Distribution functions and ranks are dual to the concepts of quantile functions and empirical quantiles, respectively: measure-transportation-based quantiles, by duality, characterize center-outward distribution functions (in populations) and center-outward ranks (in the sample). These ranks can be used to construct distribution-free tests and R-estimators for a variety of problems: two-sample location, MANOVA, multiple-output regression, vector independence, etc., extending the classical theory of Hájek to a multivariate setting.
  4. Multiple-output quantile regression and vector quantile autoregression. Among the most powerful applications of classical quantiles is quantile regression, introduced in a pathbreaking paper by Koenker and Bassett (Econometrica 1978). Unlike traditional mean regression, which deals with the dependence of the expected value of a variable of interest on a set of covariates, quantile regression models the dependence of the entire conditional distribution via its quantiles. Due to the lack of an appropriate concept of multivariate quantiles, and despite many attempts (based on marginal quantiles or statistical depth), quantile regression so far has been limited to single-output regression. The related concept of quantile autoregression similarly was limited to univariate time series. Thanks to the measure-transportation-based concept of center-outward quantiles, these powerful methods are extended to multiple-output regression and vector autoregressions.

Main References

  • Hallin, M., del Barrio, E., Cuesta-Albertos, J., and Matrán, C. (2021). Center-outward distribution and quantile functions, ranks, and signs in \(R^d\): a measure transportation approach, Annals of Statistics 49, 1139-1165.
  • Hallin, M. (2022). Measure transportation and statistical decision theory, Annual Review of Statistics and its Applications 9, 401-424.
  • Hallin, M., Hlubinka, D., and Hudecova, S. (2022). Fully distribution-free center-outward rank tests for multiple-output regression and MANOVA, Journal of the American Statistical Association 118, 1923-1939.
  • Shi, H., Hallin, M., Drton, M., and Han, F. (2022). On universally consistent and fully distribution-free rank tests of vector independence, Annals of Statistics 50, 1933-1959.
  • Shi, H., Hallin, M., Drton, M., and Han, F. (2024). Distribution-free tests of multivariate independence based on center-outward quadrant, Spearman, Kendall, and van der Waerden statistics, Bernoulli 31, 106-129.
  • Hallin, M., Liu, H., and Verdebout, Th. (2024). Nonparametric measure-transportation-based methods for directional data, Journal of the Royal Statistical Society Series B 86, 1172–1196.
  • del Barrio, E., Gonzalez-Sanz, A., and Hallin, M. (2025). Nonparametric multiple-output center-outward quantile regression, Journal of the American Statistical Association 120, 818–832.
  • Hallin, M. and Liu, H. (2025). Quantiles and quantile regression on Riemannian manifolds: a measure-transportation-based approach. Available at arXiv:2410.15711.
  • Gonzalez-Sanz, A., Hallin, M., and Yao, Y. (2025). Nonparametric Vector Quantile Autoregression. Available at arXiv:2510.03166.

Schedule

  • TBA (the course will probably be delivered on the week starting November, 23, 2026

Details