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978-3-8439-0669-2, Reihe Mathematik
Measuring Multivariate Dependence - an Analytical Approach with Copulas
128 Seiten, Dissertation Carl von Ossietzky Universität Oldenburg (2011), Softcover, B5
The tails of a multivariate distribution are of particular interest in the behavior of highdimensional distributions. Dependencies of extreme events of bivariate distributions can be measured by the tail dependence coefficient. It describes the amount of dependence in the upper right tail or lower left tail of the distribution. We investigate and compare different multivariate generalizations and apply them to the family of Archimedean copulas. This family is characterized by a univariate function, the so-called Archimedean generator. It will be shown that the index of regular variation of the Archimedean generator is essential in all possible generalizations. The results indicate which Archimedean generator fits best to the data or to the desired model. The goodness of fit of a univariate distribution function can be tested using the Anderson-Darling test which has a special emphasis on the fit in the tails of the distribution.
We present a multivariate extension of the Anderson-Darling test statistics for a general weight function. The Karhunen-Loève decomposition of the weighted uniform empirical process enables us to determine the limiting distribution of the test statistics by the eigenvalues of an integral equation. Analogously to the univariate case we reduce the problem to a boundary value problem for a partial differential equation. This test statistics is based on the Rosenblatt-transform which is particular simple and applicable for the family of Archimedean copulas.