Behaviour of the NORTA Method for Correlated Random Vector Generation as the Dimension Increases

The NORTA method is a fast general-purpose method for generating samples of a random vector with given marginal distributions and given correlation matrix. It is known that there exist marginal distributions and correlation matrices that the NORTA method cannot match, even though a random vector with the prescribed qualities exists. In this case we say that the correlation matrix is NORTA defective (for the given marginals). We investigate this problem as the dimension of the random vector increases. Simulation results show that the problem rapidly becomes acute, in the sense that an increasingly large proportion of correlation matrices are NORTA defective. Simulation results also show that if one is willing to settle for a correlation matrix that is “close” to the desired one, then NORTA performs well with increasing dimension. As part of our analysis we develop a method for sampling symmetric positive definite correlation matrices uniformly (in the Lebesgue measure sense) from the set of all such matrices. This procedure can be used more generally for sampling uniformly from the space of all symmetric positive definite matrices with diagonal elements fixed at positive values.