A Multivariate Quantile Based on Kendall Ordering

We introduce the Kendall multivariate quantiles, which are a transformation of orthant quantiles by the Kendall function. Each quantile is then a set of vectors with some advantageous properties, compared to the standard orthant quantile: i) it induces a total order ii) the probability level of the quantile is consistent with the probability measure of the set of the dominated vectors, iii) the multivariate quantiles based on the distribution function or on the survival function have vectors in common which conciliate both upper-and lower-orthant approaches. Definition and properties of the Kendall multivariate quantiles are illustrated using Archimedean copulas.