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Approximating multivariate distributions with vines

Bedford, T.J. and Daneshkhah, A. (2010) Approximating multivariate distributions with vines. Working paper. Management Science Working Series. (In Press)

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Abstract

In a series of papers, Bedford and Cooke used vine (or pair-copulae) as a graphical tool for representing complex high dimensional distributions in terms of bivariate and conditional bivariate distributions or copulae. In this paper, we show that how vines can be used to approximate any given multivariate distribution to any required degree of approximation. This paper is more about the approximation rather than optimal estimation methods. To maintain uniform approximation in the class of copulae used to build the corresponding vine we use minimum information approaches. We generalised the results found by Bedford and Cooke that if a minimal information copula satis¯es each of the (local) constraints (on moments, rank correlation, etc.), then the resulting joint distribution will be also minimally informative given those constraints, to all regular vines. We then apply our results to modelling a dataset of Norwegian financial data that was previously analysed in Aas et al. (2009).