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-TailExpectation, proposed by Cousin and Di Bernardino (2012). We propose a new non-parametric estimator for this multivariate risk-measure, which is essentially based on the Kendall's process (see Genest and Rivest, 1993). Using the Central Limit Theorem for the Kendall's process, proved by Barbe et al. (1996), we provide a functional Central Limit Theorem for our estimator. We illustrate the practical properties of our estimator on simulations. A real case in environmental framework is also analyzed. The performances of our new estimator are compared to the ones of the level sets-based estimator, previously proposed in Di Bernardino et al. (2011).

Estimation of the multivariate conditional tail expectation for extreme risk levels: Illustration on environmental data sets

“…observations. This assumption was validated for the present data set by Di Bernardino and Prieur (2014). The length of the data set is n = 125.…”

“…Making the “best choice” for ( T n ) n ≥ 1 is not trivial (see Di Bernardino et al, ). Recently, Di Bernardino and Prieur () proposed a nonparametric estimator for based on the estimation of the Kendall's process. For this estimator, they provide a functional central limit theorem without requiring the calibration of extra parameters or sequences.…”

“…Multivariate CTE under study In the following, we deal with a version of the multivariate CTE, previously proposed by Di Bernardino, Laloë, Maume-Deschamps, and Prieur (2013). See also Cousin and Di Bernardino (2014) and Di Bernardino and Prieur (2014). Let I = {1, … , d}.…”

“…Making the "best choice" for (T n ) n ≥ 1 is not trivial (see Di Bernardino et al, 2013). Recently, Di Bernardino and Prieur (2014) proposed a nonparametric estimator for E […”

Nonparametric estimation of multivariate quantiles

“…There, they lead to multivariate extensions of risk measures such as value at risk and expected shortfall. References in this field using a copula approach are, for example, Cousin & Di Bernardino (2013) and Di Bernardino & Prieur (2014).…”