Algorithms for Finding Copulas Minimizing Convex Functions of Sums

TL;DR

This paper introduces advanced rearrangement algorithms and a new dependence measure to identify dependence structures that minimize convex functions of sums, with applications including matching sum distributions to normality.

Contribution

The paper presents novel rearrangement algorithms and a multivariate dependence measure for optimizing dependence structures under given marginals.

Findings

Algorithms successfully find dependence structures minimizing convex functions.

The new dependence measure effectively assesses convergence and guides stopping.

Application example matches sum distribution to a normal distribution.

Abstract

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.