A0274
Title: Quantiles and quantile regression on Riemannian manifolds: A measure-transportation-based approach
Authors: Marc Hallin - Universite Libre de Bruxelles (Belgium)
Hang Liu - University of Science and Technology of China (China) [presenting]
Thomas Verdebout - Universite Libre de Bruxelles (Belgium)
Abstract: Increased attention has been given recently to the statistical analysis of variables with values on manifolds. A natural but nontrivial problem in that context is: ``can we define quantile concepts for such variables?'' We are proposing a solution to that problem for compact Riemannian manifolds without boundaries; typical examples are polyspheres, hyperspheres, and toroidal manifolds equipped with Riemannian metric. Our concept of quantile functions comes along with a concept of distribution function and, in the empirical case, ranks and signs. The absence of a canonical ordering is offset by resorting to the data-driven ordering induced by optimal transports. Statistical inference applications, from GOF to distribution-free rank-based testing, are without number. Of particular importance is the case of quantile regression with directional or toroidal multiple output, which is given special attention. Theoretical properties, such as the uniform convergence of the empirical distribution and conditional (and unconditional) quantile functions and distribution-freeness of ranks and signs, are established. Extensive simulations are carried out to illustrate these novel concepts.
