Estimation of extreme quantiles conditioning on multivariate critical layers

Abstract

Let Ti:=[Xi|X∈∂L(α)], for i = 1,…,d, where X = (X1,…,Xd) is a risk vector and ∂L(α) is the associated multivariate critical layer at level α∈(0,1). The aim of this work is to propose a non‐parametric extreme estimation procedure for the (1 − pn)‐quantile of Ti for a fixed α and when pn→0, as the sample size n→+∞. An extrapolation method is developed under the Archimedean copula assumption for the dependence structure of X and the von Mises condition for marginal Xi. The main result is the central limit theorem for our estimator for p = pn→0, when n tends towards infinity. A set of simulations illustrates the finite‐sample performance of the proposed estimator. We finally illustrate how the proposed estimation procedure can help in the evaluation of extreme multivariate hydrological risks. Copyright © 2016 John Wiley & Sons, Ltd.

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

“…There are fewer papers on the estimation of multivariate risk measures, due to a number of theoretical and practical reasons. Recently, a conditional return level estimator was proposed by Di Bernardino (2016). An estimation of extreme componentwise excess design realization ( CE ) was also proposed by Di Bernardino and Palacios-Rodríguez (2017).…”