Cholesky decompositions and estimation of a covariance matrix: orthogonality of variance-correlation parameters

Chen & Dunson (2003) have proposed a modified Cholesky decomposition of the form Sigma = DLL'D for a covariance matrix where D is a diagonal matrix with entries proportional to the square roots of the diagonal entries of Sigma and L is a unit lower-triangular matrix solely determining its correlation matrix. This total separation of variance and correlation is definitely a major advantage over the more traditional modified Cholesky decomposition of the form (LDL)-L-2' (Pourahmadi, 1999). We show that, though the variance and correlation parameters of the former decomposition are separate, they are not asymptotically orthogonal and that the estimation of the new parameters could be more demanding computationally. We also provide statistical interpretation for the entries of L and D as certain moving average parameters and innovation variances and indicate how the existing likelihood procedures can be employed to estimate the new parameters.