Geodesic
Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles
Mathematica Applicanda, Tome 45 (2017) no. 1, pp. 21-63.

Voir la notice de l'article provenant de la source Annales Societatis Mathematicae Polonae Series

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. (2015). Its empirical counterpart has the advantages of being a bandwidth-free, monotone invariant, a.s. consistent transformation. The proposed the approach gives a general and unified framework to quantiles and their corresponding depth areas, for both a continuous or a discrete multivariate distribution.
DOI : 10.14708/ma.v45i1.2921
Classification : 62G15, 62E20
Mots-clés : Statistical depth, vector quantiles, Markov morphism, copula, Mass transportation 
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Olivier Paul Faugeras; Ludger Rüschendorf. Markov morphisms: a combined copula and mass transportation approach to multivariate quantiles. Mathematica Applicanda, Tome 45 (2017) no. 1, pp.  21-63. doi : 10.14708/ma.v45i1.2921. http://geodesic.mathdoc.fr/articles/10.14708/ma.v45i1.2921/

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