The realization problem for tail correlation functions
详细信息 查看全文
- 作者：Ulf-Rainer Fiebig ; Kirstin Strokorb ; Martin Schlather
- 关键词：Convex polytope ; Extremal coefficient ; Max ; stable process ; Tail correlation matrix ; Tail dependence matrix ; Tawn ; Molchanov model
- 刊名：Extremes
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：20
- 期：1
- 页码：121-168
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Statistics, general; Quality Control, Reliability, Safety and Risk; Civil Engineering; Hydrogeology; Environmental Management; Statistics for Business/Economics/Mathematical Finance/Insurance;
- 出版者：Springer US
- ISSN：1572-915X
- 卷排序：20
文摘
For a stochastic process {Xt}t∈T with identical one-dimensional margins and upper endpoint τup its tail correlation function (TCF) is defined through \(\chi ^{(X)}(s,t) = \lim _{\tau \to \tau _{\text {up}}} P(X_{s} > \tau \,\mid \, X_{t} > \tau )\). 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 \(\lvert T \rvert \times \lvert T \rvert \) matrices. Several general results reveal its complex geometric structure. Up to \(\lvert T \rvert = 6\) a reduced system of necessary and sufficient conditions for being a TCF is determined. None of these conditions will become obsolete as \(\lvert T \rvert \geq 3\) grows.