Multivariate composite distributions for coefficients in synthetic optimization problems

In most cases, coefficients in synthetic optimization problems are randomly generated based on specified univariate marginal distributions. Additionally, the various types of coefficients are assumed to be mutually independent, even though coefficients in practical problems may be correlated. In this paper, multivariate composite distributions with specified marginal distributions and a specified Pearson product–moment population correlation structure are characterized. The generation of synthetic optimization problems is the principal motivation for characterizing these composite distributions, but they are also useful for many other simulation applications. Type L composite distributions are composed of the extreme-correlation distributions for a multivariate random variable only, while Type U composite distributions are based on the extreme-correlation distributions and the joint distribution under independence. Closed-form composition probabilities for distributions of trivariate random variables are presented. Methods for identifying correlation structures that are amenable to representation by composite distributions are discussed.