Correlations and Copulas in Monte Carlo Methods. C. N. Haas, Drexel University, Philadelphia PA 19104; and M. J. Frank, Illino Institute of Technology, Chicago IL 60616
Inputs to Monte Carlo risk assessment computations have in some cases been found to be non independent. Almost invariably, the rank or grade correlation (Spearman correlation) has been used to model this dependency along with the particular marginal distributions for the individual variables. The correlation structure between dependent variables can easily be described using the framework of copulas. In this paper we will demonstrate this approach in which the nature of the bivariate association (including higher order cross moments) is estimated in a distribution free context. However, we will also show that the existence of multiple copulas results in the bivariate relationship not being completely describable by a rank correlation (or for that matter a product moment correlation). Essentially, higher order bivariate moments are not uniquely determined by specifying the correlation. The implications of this for bivariate confidence regions will be discussed by studying the problem using several two parameter copulas that have recently been developed. The problem of parameter estimation will also be discussed using several data sets in the risk analysis literature.