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Algorithms for Finding Copulas Minimizing Convex Functions of Sums

In this paper, we develop improved rearrangement algorithms to find the dependence structure that minimizes a convex function of the sum of dependent variables with given margins. We propose a new multivariate dependence measure, which can assess the convergence of the rearrangement algorithms and can be used as a stopping rule. We show how to apply these algorithms for example to finding the dependence among variables for which the marginal distributions and the distribution of the sum or the difference are known. As an example, we can find the dependence between two uniformly distributed variables that makes the distribution of the sum of two uniform variables indistinguishable from a normal distribution. Using MCMC techniques, we design an algorithm that converges to the global optimum.