Copula Structure Analysis Based on Robust and Extreme Dependence Measures

In this paper we extend the standard approach of correlation
structure analysis in order to reduce the dimension of
highdimensional statistical data. The classical assumption of a
linear model for the distribution of a random vector is replaced by
the weaker assumption of a model for the copula. For elliptical
copulae a correlation-like structure remains but different margins
and non-existence of moments are possible. Moreover, elliptical
copulae allow also for a copula structure analysis of dependence in
extremes. After introducing the new concepts and deriving some
theoretical results we observe in a simulation study the
performance of the estimators: the theoretical asymptotic behavior
of the statistics can be observed even for a sample of only 100
observations. Finally, we test our method on real financial data
and explain differences between our copula based approach and the
classical approach. Our new method yields a considerable dimension
reduction also in non-linear models.