The mass transportation approach to multivariate quantiles in Chernozhukov et al. (2017) was modified in Faugeras and Rüschendorf (2017) by a two steps procedure. In the first step, a mass transportation problem from a spherical reference measure to the copula is solved and combined in the second step with a marginal quantile transformation in the sample space. Also, generalized quantiles given by suitable Markov morphisms are introduced there. In the present paper, this approach is further extended by a functional approach in terms of membership functions, and by the introduction of randomized quantile regions. In addition, in the case of continuous marginals, a smoothed version of the empirical quantile regions is obtained by smoothing the empirical copula. All three extended approaches give empirical quantile ares of exact level and improved stability. The resulting depth areas give a valid representation of the central quantile areas of a multivariate distribution and provide a valuable tool for their analysis.
Copula; Depth area; Mass transportation; Vector quantiles;
Olivier Faugeras, and Ludger Rüschendorf, “Functional, randomized and smoothed multivariate quantile regions”, TSE Working Paper, n. 19-1039, September 2019, revised June 2021.
Olivier Faugeras, and Ludger Rüschendorf, “Functional, randomized and smoothed multivariate quantile regions”, Journal of Multivariate Analysis, vol. 186, n. 104802, November 2021.
Journal of Multivariate Analysis, vol. 186, n. 104802, November 2021