On the Existence of Positive-Definite Maximum-Likelihood Estimates of Structured Covariance Matrices

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 R0 exists in R whose range space contains the data. When R is the class of Toeplitz matrices, then the probability that this condition will be satisfied for a single observation can be very close to 1. When R is the class of Toeplitz matrices which have nonnegative definite circulant extensions to some finite period P, then the condition is also necessary, but the probability that one observation vector satisfies this condition is 0 provided the true covariance is nonsingular. The implication is 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.