Article
Multivariate Distributions with Fixed Marginals and Correlations
Journal of Applied Probability
(2015)
Abstract
Consider the problem of drawing random variates (X1,…,Xn) from a distribution where the marginal of each Xi is specified, as well as the correlation between every pair Xi and Xj. For given marginals, the Fr\'echet-Hoeffding bounds put a lower and upper bound on the correlation between Xi and Xj. Any achievable correlation between Xi and Xj is a convex combinations of these bounds. The value λ(Xi,Xj)∈[0,1] of this convex combination is called here the convexity parameter of (Xi,Xj), with λ(Xi,Xj)=1 corresponding to the upper bound and maximal correlation. For given marginal distributions functions F1,…,Fn of (X1,…,Xn) we show that λ(Xi,Xj)=λij if and only if there exist symmetric Bernoulli random variables (B1,…,Bn) (that is {0,1}random variables with mean 1/2) such that λ(Bi,Bj)=λij. In addition, we characterize completely the set of convexity parameters for symmetric Bernoulli marginals in two, three and four dimensions.
Disciplines
Publication Date
2015
Citation Information
Nevena Maric and Mark Huber. "Multivariate Distributions with Fixed Marginals and Correlations" Journal of Applied Probability Vol. 52 Iss. 2 (2015) p. 602 - 608 Available at: http://works.bepress.com/nevena-maric/5/