| Abstract | A d-dimensional positive definite correlation matrix R = (rho(ij)) can be parametrized in terms of the correlations rho(i,i+1) for i = 1,..., d - 1, and the partial correlations rho(ij\textbackslashi+1,.... j-1) for j - i >= 2. These ((d)(2)) parameters can independently take values in the interval (- 1, 1). Hence we can generate a random positive definite correlation matrix by choosing independent distributions F-ij, 1 <= i < j <= d, for these ((d)(2)) parameters. We obtain conditions on the F-ij so that the joint density of (rho(ij)) is proportional to a power of det(R) and hence independent of the order of indices defining the sequence of partial correlations. As a special case, we have a simple construction for generating R that is uniform over the space of positive definite correlation matrices. As a byproduct, we determine the volume of the set of correlation matrices in ((d)(2))-dimensional space. To prove our results, we obtain a simple remarkable identity which expresses det(R) as a function of rho(i,i+1) for i = 1,..., d - 1, and p(ij\textbackslashi+1,... j-1) for j - i >= 2. (C) 2005 Elsevier Inc. All rights reserved. |
