From Copulas and Statistical Depth to Multivariate Quantiles

 The univariate concept of quantile function---the inverse of a distribution function---plays a fundamental role in Probability and Statistics. In dimensions two and higher, however, inverting traditional distribution functions does not lead to any satisfactory notion. In their quest for the Grail of an adequate definition, statisticians developed two extremely fruitful theoretical pathways: copula transforms, where marginal quantiles are privileged over global ones, and depth functions, where a center-outward ordering is substituting the more traditional South-West/North-East one. We show how a recent center-outward redefinition, based on measure transportation ideas, of the concept of distribution function, reconciles and fine-tunes these two approaches, and eventually yields a notion of multivariate quantile matching, in arbitrary dimension d, all the properties that make univariate quantiles a successful and vital tool of statistical inference. 

Marc Hallin is an Emeritus Professor of the Université libre de Bruxelles, where he obtained his PhD in Mathematics and has been teaching Statistics and Mathematical Statistics at the Departments of Mathematics and Economics. He also held invited positions in Lille 1, Paris 6, the Université Catholique de Louvain, and Princeton University. The author of about 240 research papers, he received several awards, among which the Senior Gottfried Noether Award of the American Statistical Association, the Pierre-Simon de Laplace Award of the Société française de Statistique, a Humboldt-Forschungspreis of the Alexander-von-Humboldt Foundation, and the Medal of the Faculty of Mathematics and Physics of Charles University in Prague. He also was the 2017 recipient of the Hermann Otto Hirschfeld Lecture Series at the Humboldt-Universität zu Berlin and the 2018 Mahalanobis Memorial Lecture at the Indian Statistical Institute, and the holder of a Catedra de Excelencia at the Universidad Carlos III de Madrid. A Fellow of the Institute of Mathematical Statistics and the American Statistical Association, he is a member of the Classe des Sciences of the Royal Academy of Belgium.