The realization problem for tail correlation functions

Fiebig, Ulf-Rainer, Strokorb, Kirstin ORCID: https://orcid.org/0000-0001-8748-3014 and Schlather, Martin 2017. The realization problem for tail correlation functions. Extremes 20 , pp. 121-168. 10.1007/s10687-016-0250-8

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Abstract

For a stochastic process {Xt}t∈T with identical one-dimensional margins and upper endpoint τup its tail correlation function (TCF) is defined through χ(X)(s,t)=limτ→τupP(Xs>τ∣Xt>τ)χ(X)(s,t)=limτ→τupP(Xs>τ∣Xt>τ) . It is a popular bivariate summary measure that has been frequently used in the literature in order to assess tail dependence. In this article, we study its realization problem. We show that the set of all TCFs on T×T coincides with the set of TCFs stemming from a subclass of max-stable processes and can be completely characterized by a system of affine inequalities. Basic closure properties of the set of TCFs and regularity implications of the continuity of χ are derived. If T is finite, the set of TCFs on T×T forms a convex polytope of |T|×|T||T|×|T| matrices. Several general results reveal its complex geometric structure. Up to |T|=6|T|=6 a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as |T|≥3|T|≥3 grows.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Convex polytope; Extremal coefficient; Max-stable process; Tail correlation matrix; Tail dependence matrix; Tawn-Molchanov model
Additional Information:	PDF uploaded in accordance with publisher's policies at http://www.sherpa.ac.uk/romeo/issn/1386-1999/ (accessed 6.1.17).
Publisher:	Springer Verlag
ISSN:	1386-1999
Funders:	DFG (RTG1023)
Date of First Compliant Deposit:	6 January 2017
Date of Acceptance:	27 March 2016
Last Modified:	05 Dec 2024 08:00
URI:	https://orca.cardiff.ac.uk/id/eprint/97230

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