## The $$t$$ copula and related copulas.(English)Zbl 1104.62060

Summary: The $$t$$ copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate $$t$$ distribution is used as a starting point to construct two new copulas, the skewed $$t$$ copula and the grouped $$t$$ copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the $$t$$ extreme value copula is the limiting copula of componentwise maxima of $$t$$ distributed random vectors; the $$t$$ lower tail copula is the limiting copula of bivariate observations from a $$t$$ distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas, respectively.

### MSC:

 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62G32 Statistics of extreme values; tail inference 62E20 Asymptotic distribution theory in statistics 62H10 Multivariate distribution of statistics 62H20 Measures of association (correlation, canonical correlation, etc.) 62H12 Estimation in multivariate analysis
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