Estimation of multivariate conditional-tail-expectation using Kendall's process

Abstract

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.