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Sampling algorithms for generating joint uniform distributions using the Vine-Copula method. (English) Zbl 1161.62363

Summary: An \(n\)-dimensional joint uniform distribution is defined as a distribution whose one-dimensional marginals are uniform on some interval \(I\). This interval is taken to be \([0,1]\) or, when more convenient \([-1/2,1/2]\). The specification of joint uniform distributions in a way which captures intuitive dependence structures and also enables sampling routines is considered. The question whether every \(n\)-dimensional correlation matrix can be realized by a joint uniform distribution remains open. It is known, however, that the rank correlation matrices realized by the joint normal family are sparse in the set of correlation matrices.
A joint uniform distribution is obtained by specifying conditional rank correlations on a regular vine and a copula is chosen to realize the conditional bivariate distributions corresponding to the edges of the vine. In this way a distribution is sampled which corresponds exactly to the specification. The relation between the conditional rank correlations on a vine and the correlation matrix of the corresponding distribution is complex, and depends on the copulas used. Some results for the elliptical copulae are given.

MSC:

62H10 Multivariate distribution of statistics
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