Chessboard Distributions and Random Vectors with Speci ed

There is a growing need for the ability to specify and generate correlated random variables as primitive inputs to stochastic models. Motivated by this need, several authors have explored the generation of random vectors with speciied marginals, together with a speciied covariance matrix, through the use of a transformation of a multivariate normal random vector. A covariance matrix is said to be feasible for a given set of marginal distributions if a random vector exists with these characteristics. We develop a computational approach for establishing whether a given covariance matrix is feasible for a given set of marginals. The approach is used to rigorously establish that there are sets of marginals with feasible covariance matrix that the normal transformation technique referred to above cannot match. An important feature of our analysis is that we show that for almost any covariance matrix (in a certain precise sense), our computational procedure either explicitly provides a construction of a random vector with the required properties, or establishes that no such random vector exists. We also provide two new methodologies that may be used to deal with the situation where one cannot exactly match the desired marginals and covariance matrix using the normal transformation technique. The new methodologies possess certain advantages over other approaches that have been suggested in the past.