Dealing with extremal independence

Abstract

Consider a regularly varying random vector $\varvec{X}$ which is extremally independent.

3.4 Bibliographical notes

Hidden regular variation was introduced formally in [Res02] using the cone ${\mathbb {C}_{d, 2}}$ ($\mathbb {E}_0$ in the notation of [Res02]). The index of hidden regular variation $\tilde{\alpha }$ was anticipated in [LT96] as the tail dependence coefficient $\eta =\alpha /\tilde{\alpha }$. In [MR04] hidden regular variation was introduced relative to a given subcone. In particular, besides $\mathbb {E}_0$, the cone $\mathbb {E}_{00}=({\varvec{0}},\varvec{\infty })$ is used. [MR04, Example 5.1] (cf. Example 3.1.5) gives different indices of regular variation on $\mathbb {E}_0$ and $\mathbb {E}_{00}$. This implies that the minimum is not characterized by the hidden regular variation on ${\mathbb {C}_{d, 2}}=\mathbb {E}_0$ as erroneously stated in [Res02, Theorem 1 and page 304]. The issue is that the function $d(\varvec{a})$ that appears in [Res02, Theorem 1(ii)] may vanish. Indeed, $d(\varvec{a})=\tilde{\varvec{\nu }}_{\varvec{X}}(\{\varvec{x}\in \mathbb {R}^d:\bigwedge _{i=1}^d a_ix_i>1\})$, where at least two components of $\varvec{a}=(a_1,\ldots , a_d)$ are finite. If $d=3$ and $a_1=a_2=a_3=1$, then $d(\varvec{a})=0$.

Conditioning on an extreme event is related to the old problem of the concomitant, that is, the Y observation corresponding to the maximum of the X observation in a bivariate sample $(X_1,Y_1),\dots ,(X_n, Y_n)$. The earlier references were concerned with elliptical distributions with light-tailed radial component. See for instance [EG81] or [Ber92, Theorem 12.4.1] and (Problem 7.7). For regularly varying random variables, [MRR02] seems to be the first article that deals (implicitly) with partial regular variation by considering vague convergence of vectors of non-negative random variables on the cone $(0,\infty )\times [0,\infty )$. A rigorous framework for conditioning on extreme event is [HR07] in the case of both Gumbel and Fréchet domain of attraction. The name “conditional extreme value” has been used for the condition in Definition 3.2.1. We have avoided this terminology since it is ambiguous: the conditioning is on the variable which is in the domain of attraction of an extreme value distribution, and the limiting conditional distribution of the second one, given the first one is extremely large, is not necessarily an extreme value distribution. This terminology has also been used to describe extreme value theory in presence of covariates.

Hidden regular variation and conditioning on extreme event can be studied in a unified manner using regular variation on subcones. The cone $\mathbb {E}_0$ gives hidden regular variation while $(0,\infty )\times [0,\infty )$ yields conditioning on an extreme event in the sense of Section 3.2. See [Res08, MR11, DMR13]. In [DR11] the authors discuss links between hidden regular variation and conditioning on extreme event. As shown in [DR11, Example 4], it is possible to construct random vectors which have hidden regular variation, but not the other property. The term conditional scaling exponent was introduced in [KS15]. Different issues with conditioning on extreme event are discussed in [DJ17]. These issues are mainly due to compactification with a point at $\infty$ and do not arise when vague$^\#$ convergence is used as in the present chapter.

In the framework of conditioning on extreme events, the first results on regular variation of products can be found in [MRR02]. Lemma 2.2 therein corresponds to the case $\kappa =0$ and $b(x)\equiv 1$ in Proposition 3.2.12. Proposition 3.2.12 is adapted from [KS15]. See also [HM11] for a variety of similar results.

Risk measures (including conditional expectations) under hidden regular variation and conditional extreme value model are considered in [DFH18] and [KT19], respectively. Problem 3.7 is adapted from [KT19].