Functional, randomized and smoothed multivariate quantile regions

Résumé

A notion of multivariate depth, resp. quantile region, was introduced in [Chernozhukov et al., 2017], based on a mass transportation approach. In [Faugeras and Ruschendorf, 2017], this approach was generalized by dening quantiles as Markov morphisms carrying suitable algebraic, ordering and topological structures over probability measures. In addition, a copula step was added to the mass transportation step. Empirical versions of these depth areas do not give exact level depth regions. In this paper, we introduce randomized depth regions by means of a formulation by depth functions, resp. by randomized quantiles sets. These versions attain the exact level and also provide the corresponding consistency property. We also investigate in the case of continuous marginals a smoothed version of the empirical copula and compare its behavior with the unsmoothed version. Extensive simulations illustrate the resulting randomized depth areas and show that they give a valid representation of the central depth areas of a multivariate distribution, and thus are a valuable tool for their analysis.