On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity

✍ Scribed by Seokhoon Yun

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 393 KB
• Volume: 63
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1997.1706

✦ Synopsis

The paper gives sufficient conditions for domains of attraction of multivariate extreme value distributions. Under the assumption of absolute continuity of a multivariate distribution, the criteria enable one to examine, by using limits of some rescaled conditional densities, whether the distribution belongs to the domain of attraction of some multivariate extreme value distribution. If this is the case, the criteria also determine how to construct such an extreme value distribution. Unlike the criterion given by de Haan and Resnick [1987, Stochastic Process. Appl. 25 83 93], the criteria are easily applicable even when the marginal tails are not Pareto-like. 1997 Academic Press 1. INTRODUCTION Characterizing domains of attraction of univariate extreme value distributions is one of the classical problems in extreme value theory and was completely solved by Gnedenko [3]. There have been a number of attempts to extend this to the multivariate case. These include Berman [1], Rvaceva [14], de Haan and Resnick [4, 5, 7], Marshall and Olkin [12], de Haan and Omey [6], and de Haan et al. [8]. In particular, de Haan and Resnick [7] derived a criterion for domains of attraction of multivariate extreme value distributions in terms of densities. Their method is convenient to apply when marginal tails are Pareto-like. It is however clumsy for other cases, particularly when functional forms of marginal distribution functions cannot be explicitly obtained like gamma or normal distributions. In this paper we give sufficient conditions for domains of attraction of multivariate extreme value distributions which can be easily applied to most absolutely continuous multivariate distributions.