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.


62H10 Multivariate distribution of statistics
Full Text: DOI


[1] Bedford, T.J.; Cooke, R.M., Viness—a new graphical model for dependent random variables, Ann. statist., 30, 4, 1031-1068, (2002) · Zbl 1101.62339
[2] Cooke, R.M., Markov and entropy properties of tree and vines-dependent variables, ()
[3] Cooke, R.M.; Waij, R., Monte Carlo sampling for generalized knowledge dependence with application to human reliability, Risk anal., 6, 335-343, (1986)
[4] Dall’Aglio, G.; Kotz, S.; Salinetti, G., Probability distributions with given marginals; beyond the copulas, (1991), Kulwer Academic Publishers Dordrecht · Zbl 0722.00031
[5] Frank, M.J., On the simultaneous associativity of \(f(x, y)\) and \(x + y - f(x, y)\), Aequationes math., 19, 194-226, (1979) · Zbl 0444.39003
[6] Genest, C.; Rivest, L.P., Statistical inference procedure for bivariate Archimedean Copula’s, J. amer. statist. assoc., 88, 423, 1034-1043, (1993) · Zbl 0785.62032
[7] Ghosh, S.; Henderson, S.G., Properties of the notra method in higher dimensions, (), 263-269
[8] Hoeffding, W., 1940. Masstabinvariante korrelationstheorie. Schrijtfen Math. Inst. Inst. Angew. Math. Univ. Berlin 5, 179-233.
[9] Iman, R.; Conver, W., A distribution-free approach to inducing rank correlation among input variables, Comm. statist. simulation comput., 11, 3, 311-334, (1982) · Zbl 0496.65071
[10] Joe, H., 2006. Range of correlation matrices for dependent random variables with given marginal distributions. In: Advances in Distribution Theory, Order Statistics and Inference, in honor of Barry Arnold. Balakrishnan, N., Castillo, E., Sarabia, J.M. (Eds.), Birkhauser, Boston, pp. 125-142. · Zbl 05196667
[11] Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517
[12] Kurowicka, D.; Cooke, R.M., Conditional and partial correlation for graphical uncertainty models, (), 259-276 · Zbl 0957.62086
[13] Kurowicka, D., Cooke, R.M., 2001. Conditional, partial and rank correlation for elliptical copula; dependence modeling in uncertainty analysis. Proceedings of ESREL 2001.
[14] Kurowicka, D.; Cooke, R.M., A parametrization of positive definite matrices in terms of partial correlation vines, Linear algebra appl., 372, 225-251, (2003) · Zbl 1027.60070
[15] Kurowicka, D.; Cooke, R.M., Completion problem with partial correlation vines, Linear algebra appl., 418, 188-200, (2006) · Zbl 1106.15011
[16] Kurowicka, D.; Misiewicz, J.; Cooke, R.M., Elliptical copulae, (), 209-214
[17] Meeuwissen, A.; Bedford, T.J., Minimally informative distributions with given rank correlation for use in uncertainty analysis, J. statist. comput. simulation, 57, 1-4, 143-175, (1997) · Zbl 0873.62006
[18] Nelsen, R.B., An introduction to copulas, (1999), Springer New York · Zbl 0909.62052
[19] Pearson, K., 1907. Mathematical contributions to the theory of evolution. Biometric, VI. Series. · JFM 38.0290.04
[20] Yule, G.U.; Kendall, M.G., An introduction to the theory of statistics, (1965), Charles Griffin & Co Belmont, CA · JFM 67.0472.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.