Extreme behavior of bivariate elliptical distributions

Abstract

This paper exploits a stochastic representation of bivariate elliptical distributions in order to obtain asymptotic results which are determined by the tail behavior of the generator. Under certain specified assumptions, we present the limiting distribution of componentwise maxima, the limiting upper copula, and a bivariate version of the classical peaks over threshold result.

Extreme-Value Copulas

“…This extreme-value copula was coined the t-EV copula. Building upon results in [39,51], exactly the same extreme-value attractor is found in [2] for the more general class of (meta-)elliptical distributions whose generator has a regularly varying tail.…”

On Pearson–Kotz Dirichlet distributions

“…Asimit and Jones [3] consider the approximation of the joint conditional excess distribution in a bivariate elliptical setup. As shown therein the approximating distribution is given in terms of the survival function of the univariate t-distribution and some parameter γ > 0, provided that the elliptically symmetric random vector is regularly varying with index γ .…”

“…The approximating distribution function is given in terms of the survival function of Pearson-Kotz Dirichlet distribution and some index γ ∈ (0, ∞) if X is regularly varying with index γ . We consider for simplicity below the case (b) generalizing Theorem 1 of [3] to the k-dimensional setup. A simple formula can be derived for the bivariate Dirichlet framework, see Example 4.…”

“…An elaborate discussion on various t-distributions has been provided by Kotz and Nadarajah [30]. One may also refer to [1,8,35,3,9,28,6,[31][32][33]36] for related discussions.…”

Asymptotic Tail Probabilities for Large Claims Reinsurance of a Portfolio of Dependent Risks

“…For more details, see Asimit and Jones (2007a). Note that we use the fact that h(x) = xh(1/x) holds.…”