On the existence of positive-definite maximum-likelihood estimatesof structured covariance matrices
Fuhrmann, D.R.; Miller, M.I.
Information Theory, IEEE Transactions on
Volume 34, Issue 4, Jul 1988 Page(s):722 - 729
Digital Object Identifier   10.1109/18.9771
Summary:It is shown that a sufficient condition for the likelihood function of a zero-mean Gaussian random vector with covariance R from some class of covariances R to be unbounded above over the set of positive-definite matrices in R is that some singular R_o exists in R whose range space contains the data. The results obtained imply that, for the spectrum estimation problem in which R is the class of Toeplitz covariances and only one long observation vector is available, by constraining the maximum-likelihood estimation problem to the class of Toeplitz matrices with nonnegative definite circulant extensions, a positive-definite solution is guaranteed to exist

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