Modelling Dependence with Copulas - IFOR

10 Conclusions
In this paper we have shown why knowledge of copulas is essential for understanding
dependence among random variables. We have presented copula based measures of
association and dependence concepts such as tail dependence which provide a natural
way of studying dependence. Furthermore, these measures and dependence
concepts have the very nice property of invariance under strictly increasing transforms
of the underlying random variables. This also means that the problem of
finding multivariate models which are consistent with the prespecified margins is
much simplified.
We have presented a large number of copula families and suggested natural multivariate
extensions with interesting properties and dependence structures. Since
much of the literature in this area is basically concerned with the bivariate case,
we have focused on multivariate extensions of bivariate copula families and extensions
of bivariate results to general dimensions. A main aim of this paper has been
constructing efficient algorithms for random variate generation for the presented n-
dimensional copula families. This is interesting from a theoretical point of view but
perhaps more for practical purposes, since it provides a basis for general simulation
tools for modelling dependence in practice. Applications for risk management in
insurance and finance are discussed, where we have used copula based measures of
association which do not suffer from some of the drawbacks of linear correlation.
It should be noted that although the study of copulas and their applications in
statistics is a rather modern phenomenon, a basic knowledge of probability theory
and statistics is enough to understand all results.
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