Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours
Abstract
Quantiles are a fundamental concept in probability and theoretical statistics and a daily tool in their applications. While the univariate concept of quantiles is quite clear and well understood, its multivariate extension is more problematic. After half a century of continued efforts and many proposals, two concepts, essentially, are emerging: the so-called (relabeled) geometric quantiles, extending the characterization of univariate quantiles as minimizers of an L1 loss function involving the check functions, and the more recent center-outward quantiles based on measure transportation ideas. These two concepts yield distinct families of quantile regions and quantile contours. Our objective here is to present a comparison of their main theoretical properties and a numerical investigation of their differences.
Cite
@article{arxiv.2401.02499,
title = {Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours},
author = {Marc Hallin and Dimitri Konen},
journal= {arXiv preprint arXiv:2401.02499},
year = {2024}
}