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Copula-based multivariate association measures and tail coefficients
Kika, Vojtěch ; Omelka, Marek (advisor) ; Veraverbeke, Noel (referee) ; Fuchs, Sebastian (referee)
The dependence structure of a d-variate random vector X is a very complex notion which is fully described by the distribution of the random vector. Alternatively, it suffices to look into the corresponding copula function of X, as it ignores the marginal distributions of X but still fully describes the dependence structure. However, a copula is a function defined on the d-dimensional hypercube [0, 1]d with values in the interval [0, 1]. As such, it might be too complex for practical use and one would prefer to have tools that can translate the information from the copula function into a simpler indicator. In particular, of interest might be an association measure, that is, a single number that describes the tendency of the components of X to simultaneously take large or small values. Coefficients like Kendall's tau or Spearman's rho, used to measure (strength of) an association between two random variables, were thoroughly studied and described in the middle of 20th century. Requirements on bivariate association measures are well-known. However, generalization of such measures into higher dimensions is not very straightforward and brings discussion on the desirable properties. In addition, bivariate association measures can be often generalized in multiple manners. The same holds true if one wants to...
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