A simple approach to the generation of uniformly distributed random variables with prescribed correlations

In a simulation study we encountered the problem of generating uniformly on (0, 1) distributed random variables with a prescribed correlation matrix. This article reports our solution to this problem We show that the following simple device leads to uniformly on (0, 1) distributed random variables, which have a correlation matrix that is for practical purposes sufficiently close to the prescribed one: Let (X-1,...,X-m) be a normal vector with the prescribed correlation matrix S = (rho(ij)). Such a vector can easily be generated. Suppose for simplicity that each X-i is standard normal and denote by Phi the standard normal distribution function. It turns out that the vector (V-1,...,V-m) := (Phi(X1()),..., Phi(X-m)) with uniformly on (0,1) distributed components V-i then has a correlation matrix S', whose entries are for practical purposes sufficiently close to that of the target matrix S. This is a consequence of the grade correlation (or Spearman's rho) of normal vectors. Suppose, in addition, that the matrix (S) over tilde := 2 sin(pi/6S) = (2 sin(pi/6 rho(ij))) is positive semidefinite. Then the vector ((V) over tilde(1), ...., (V) over tilde(m)) := (Phi((X) over tilde(1)),..., Phi((X) over tilde(1),..., Phi((X) over tilde(m))), with (X) over tilde(i) being standard normal and ((X) over tilde(1),..., (X) over tilde(m)) being normal with correlation matrix (S) over tilde, has exact correlation matrix S. The matrix (S) over tilde is, however, not in general positive semidefinite.