Abstract
This paper investigates properties of extensions of tail dependence of Archimax copulas to high dimensional analysis in a spatialized framework. Specifically, we propose a characterization of bivariate margins of spatial Archimax processes while spatial multivariate upper and lower tail dependence coefficients are modeled, respectively, for Archimedean copulas and Archimax ones. A property of stability is given using convex transformations of survival copulas in a spatialized Archimedean family.
Highlights
In stochastic multivariate modeling, the use of copulas provides a powerful method of analysing the dependence structure of two or more random variables
Arising naturally in the context of Laplace Transforms, Archimedean copulas form a prominent class of copulas which leads to the construction of multivariate distributions involving one-dimensional generator functions (Charpentier and Segers [4])
The reader which is interested in bivariate statements of Archimedean copulas is referred to Genest and MacKay [10] and the software R and Evanesce [11] implements Archimax models proposed by Joe [2]
Summary
The use of copulas provides a powerful method of analysing the dependence structure of two or more random variables. The copula functions appear implicitly in any multivariate distributions as a structure that allows separating the marginal distributions and the dependence model. Joe [2] and Nelsen [3] are two key textbooks on copulas analysis, providing clear and detailed introductions to copulas and dependence modeling with an emphasis on statistical and mathematical foundations in spatial context. Arising naturally in the context of Laplace Transforms (see Joe [2]), Archimedean copulas form a prominent class of copulas which leads to the construction of multivariate distributions involving one-dimensional generator functions (Charpentier and Segers [4]). An n-dimensional copula C is an Archimedean copula, if there exists a continuous and strictly decreasing convex function φ, the generator of C, in the class of completely monotone functions:
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