Extremes for multivariate expectiles

Abstract

In [? ], a new family of vector-valued risk measures called multivariate expectiles is introduced. In this paper, we focus on the asymptotic behavior of these measures in a multivariate regular variations context. For models with equivalent tails, we propose an estimator of extreme multivariate expectiles, in the Fréchet attraction domain case, with asymptotic independence, or for comonotonic margins.

Semi-parametric estimation of multivariate extreme expectiles

“…Proposition 1 (Maume-Deschamps et al [42], Proposition 3.3). Assume that X has regular varying multivariate distribution in the sense of Definition 1 and that Assumption 1 holds.…”

“…The notion of extreme expectiles has been explored in the univariate case by Bellini and Bignozzi [5], Mao et al [39], Bellini and Di Bernardino [6], Daouia et al [16,17] and in the multivariate case by Maume-Deschamps et al [41,42] and Herrmann et al [31], for example. In the latter works, the authors discuss situations for which expectiles can be solved for explicitly or for moderate risk levels α.…”

“…For more information on the concept of regular variation and how it relates to MEEs, see Section 1 of Maume-Deschamps et al [42]. Drawing further from Maume-Deschamps et al [42], the focus here is on marginal distributions with asymptotically equivalent tails. This can be stated in the following assumption.…”

“…where C is the survival copula associated to X (see, for instance, Equation (1.4) or (1.5) in Maume-Deschamps et al [42]). Pairwise upper tail dependencies will be of particular importance in this paper, so we denote by λ ik (x i , x k ), the upper tail dependence function for the random pair (X i , X k ).…”

“…and the bivariate relationship L(x, y) = x + y − λ(x, y). The following proposition from Maume-Deschamps et al [42] connects all these concepts to MEEs by providing an alternative system of equations when α → 1.…”