On the worst and least possible asymptotic dependence

JOURNAL OF MULTIVARIATE ANALYSIS

卷:144

Article

Asimit, Alexandru V.¹  Gerrard, Russell¹ 

[1] City Univ London, Cass Business Sch, London EC1Y 8TZ, England

关键词: Asymptotic dependence/independence;    Copula;    Extreme Value Theory;    Gumbel tail;    Regular variation;    Risk measure;   

DOI  :  10.1016/j.jmva.2015.11.004

来源: Elsevier

【 摘 要 】

Multivariate extremes behave very differently under asymptotic dependence as compared to asymptotic independence. In the bivariate setting, we are able to characterise the extreme behaviour of the asymptotic dependent case by using the concept of the copula. As a result, we are able to identify the properties of the boundary cases, that are asymptotic independent but still have some asymptotic dependent features. These situations are the most problematic in statistical extreme, and, for this reason, distinguishing between asymptotic dependence and asymptotic independence represents a difficult problem. We propose a simple test to resolve this issue which is an alternative to the procedure based on the classical coefficient of tail dependence. In addition, we are able to identify the worst/least asymptotic dependence (in the presence of asymptotic dependence) that maximises/minimises the probability of a given extreme region if tail dependence parameter is fixed. It is found that the perfect extreme association is not the worst asymptotic dependence, which is consistent with the existing literature. We are able to find lower and upper bounds for some risk measures of functions of random variables. A particular example is the sum of random variables, for which a vivid academic effort has been noticed in the last decade, where bounds for a sum of random variables are sought. It is numerically shown that our approach provides a great improvement of the existing methods, which reiterates the sensible conclusion that any additional piece of information on dependence would help to reduce the spread of these bounds. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free