A Multivariate Quantile Based on Kendall Ordering

Abstract

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 in[1]duces 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.

Keywords

• multivariate quantile
• copula
• Archimedean copula
• Kendall distribution
• orthant quantile