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.


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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