An Introduction to Copulas
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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 [-12,12]. 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 conditional rank correlations on a vine and correlation matrix of corresponding distribution is complex, and depends on the copula used. Some results for the elliptical copulae are given.