B1713
Title: Multivariate geometric quantiles: PDE aspects, Kolmogorov's distance, and linear universality
Authors: Dimitri Konen - University of Warwick (United Kingdom) [presenting]
Abstract: The concept of "geometric quantiles and cdf" is one of the most popular approaches to defining a multivariate analogue of traditional quantiles and cdf in dimension one. A horizon tour is provided for some recent advances in geometric quantiles. Among others, it was shown that, in any dimension d, the geometric cdf of an arbitrary probability measure P is related to P through a (potentially fractional) linear PDE of order d. Surprisingly, this link displays different behaviours when d is odd or even. Then, it is explained how this puzzling result, in fact, allows one to show that the multivariate geometric cdf characterizes weak convergence of probability measures, thus providing a multivariate counterpart to Kolmogorov's distance in dimension one. In addition to being easily computable in virtually any dimension, this distance is finer than the popular Wasserstein distance. Finally, it is proven that, although the multivariate geometric cdf has conceptual disadvantages and advantages, this concept is essentially unique in the class of admissible linear cdf's in any dimension.
