## 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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### References:

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