Estimation of Multivariate Conditional Tail Expectation using Kendall's Process

This paper deals with the problem of estimating the multivariate version of the Conditional-Tail-Expectation, proposed by Di Bernardino et al. [(2013), ?Plug-in Estimation of Level Sets in a Non-Compact Setting with Applications in Multivariable Risk Theory?, ESAIM: Probability and Statistics, (17), 236?256]. We propose a new nonparametric estimator for this multivariate risk-measure, which is essentially based on Kendall's process [Genest and Rivest, (1993), ?Statistical Inference Procedures for Bivariate Archimedean Copulas?, Journal of American Statistical Association, 88(423), 1034?1043]. Using the central limit theorem for Kendall's process, proved by Barbe et al. [(1996), ?On Kendall's Process?, Journal of Multivariate Analysis, 58(2), 197?229], we provide a functional central limit theorem for our estimator. We illustrate the practical properties of our nonparametric estimator on simulations and on two real test cases. We also propose a comparison study with the level sets-based estimator introduced in Di Bernardino et al. [(2013), ?Plug-In Estimation of Level Sets in A Non-Compact Setting with Applications in Multivariable Risk Theory?, ESAIM: Probability and Statistics, (17), 236?256] and with (semi-)parametric approaches