Capturing the multivariate extremal index: Bounds and interconnections

Ehlert, Andree ; Schlather, Martin

Document Type: Article
Year of publication: 2008
The title of a journal, publication series: Extremes : Statistical Theory and Applications in Science, Engineering and Economics
Volume: 11
Issue number: 4
Page range: 353-377
Place of publication: Dordrecht [u.a.]
Publishing house: Springer Science + Business Media
ISSN: 1386-1999
Publication language: English
Institution: School of Business Informatics and Mathematics > Mathematische Statistik (Schlather 2012-)
Subject: 510 Mathematics
Keywords (English): Multivariate extremal index function, Dependence function, Adjusted extremal coefficient, Max-stable process, Upper bound, Lower bound
Abstract: The multivariate extremal index function is a direction specific extension of the well-known univariate extremal index. Since statistical inference on this function is difficult it is desirable to have a broad characterization of its attributes. We extend the set of common properties of the multivariate extremal index function and derive sharp bounds for the entire function given only marginal dependence. Our results correspond to certain restrictions on the two dependence functions defining the multivariate extremal index, which are opposed to Smith and Weissman’s (1996) conjecture on arbitrary dependence functions. We show further how another popular dependence measure, the extremal coefficient, is closely related to the multivariate extremal index. Thus, given the value of the former it turns out that the above bounds may be improved substantially. Conversely, we specify improved bounds for the extremal coefficient itself that capitalize on marginal dependence, thereby approximating two views of dependence that have frequently been treated separately. Our results are completed with example processes.

Dieser Datensatz wurde nicht während einer Tätigkeit an der Universität Mannheim veröffentlicht, dies ist eine Externe Publikation.

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