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Pair-copula constructions of multiple dependence. (English) Zbl 1165.60009

The authors use a cascade of pair-copulae to model complex patterns of dependence in the tails of non-normal multivariate families. In a sense every joint distribution function contains both a description of the marginal behaviour of the individual variables and a description of their dependency structure. Copulae provide a way of isolating the description of their dependency structure. Each joint density function can be decomposed into a product of pair copulae and marginal densities. For high-dimensionals distributions, there are a significant number of possible pair-copulae constructions. To organize them, a graphical model denoted as ‘the regular vine’ is introduced. The model construction is hierarchical in nature. Various levels of hierarchy correspond to incorporation of more variables in the conditioning sets using pair-copulae as simple building blocks. Assuming conditional independence may reduce the number of levels and hence simplify the construction. The methodology is applied to a financial data set containing daily data about four Norwegian indices. Algorithms that allow inference on the parameters of the pair-copulae on the various levels of the construction are developed. The approach is the first step towards the development of an unsupervised algorithm that explores the space of possible pair-copula models, that also can be applied to huge data sets automatically.

MSC:

60E05 Probability distributions: general theory
91B28 Finance etc. (MSC2000)
91B30 Risk theory, insurance (MSC2010)
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