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Cholesky Decompositions and Estimation of A Covariance Matrix: Orthogonality of Variance–Correlation Parameters
Division of Statistics, Northern Illinois University, DeKalb, Illinois 60115-2854, U.S.A. pourahm{at}math.niu.edu
Received for publication 1 July 2005. Revision received 1 April 2007.
Chen & Dunson ([3]) have proposed a modified Cholesky decomposition of the form 
= D L L'D for a covariance matrix where D is a diagonal matrix with entries proportional to the square roots of the diagonal entries of 
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 LD2L', (Pourahmadi, [13]). 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.
Key Words: Maximum likelihood estimation • Moving average coefficient • Positive-definiteness constraint • Unconstrained parameterization • Variance–correlation separation
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