On minimum correlation in construction of multivariate distributions

TL;DR

This paper introduces an exact method for generating multivariate samples with specified marginals and correlation, including a novel algorithm for correlated Beta variables in higher dimensions.

Contribution

It presents a new algorithm combining Fréchet-Hoeffding bounds and marginal products for exact multivariate sampling with specified correlations.

Findings

Calculated minimum correlations for common distributions.

Developed the first non-copula algorithm for correlated Beta variables in >2 dimensions.

Demonstrated the algorithm's effectiveness through implementation results.

Abstract

In this paper we present a method for exact generation of multivariate samples with pre-specified marginal distributions and a given correlation matrix, based on a mixture of Fr\'echet-Hoeffding bounds and marginal products. The bivariate algorithm can accommodate any among the theoretically possible correlation coefficients, and explicitly provides a connection between simulation and the minimum correlation attainable for different distribution families. We calculate the minimum correlations in several common distributional examples, including in some that have not been looked at before. As an illustration, we provide the details and results of implementing the algorithm for generating three-dimensional negatively and positively correlated Beta random variables, making it the only non-copula algorithm for correlated Beta simulation in dimensions greater than two. This work has potential for impact in a variety of fields where simulation of multivariate stochastic components is desired.