Simultaneous estimation of the intermediate correlation matrix for arbitrary marginal densities

A popular approach to the simulation of multivariate, non-normal data in the social sciences is to define a multivariate normal distribution first, and then alter its lower-dimensional marginals to achieve the shape of the distribution intended by the researchers. A consequence of this process is that the correlation structure is altered, so further methods are needed to specify an intermediate correlation matrix in the multivariate normal distribution step. Most of the techniques available in the literature estimate this intermediate correlation matrix bivariately (i.e., correlation by correlation), risking the possibility of generating a non-positive definite matrix. The present article addresses this issue by offering an algorithm that estimates all elements of the intermediate correlation matrix simultaneously, through stochastic approximation. A small simulation study demonstrates the feasibility of the present method to induce the correlation structure both in simulated and empirical data.