Markov morphisms

Résumé

Our purpose is both conceptual and practical. On the one hand, we discuss the question which properties are basic ingredients of a general conceptual notion of a multivariate quantile. We propose and argue that the object “quantile” should be defined as a Markov morphism which carries over similar algebraic, ordering and topological properties as known for quantile functions on the real line. On the other hand, we also propose a practical quantile Markov morphism which combines a copula standardization and the recent optimal mass transportation method of Chernozhukov et al.(2017). Its empirical counterpart has the advantages of being a bandwidth-free, monotone invariant, a.s. consistent transformation. The proposed approach gives a general and unified framework to quantiles and their corresponding depth areas, for both a continuous or a discrete multivariate distribution.