Correction to "On the existence of positive-definite maximum-likelihood estimates of structured covariance matrices"

A correction to, and a more complete proof of, Theorem 1 of Fuhrmann and Miller (see ibid., vol.34, no.7, p.722-29, 1988) is given. We consider the existence of solutions to the maximum-likelihood (ML) structured covariance estimation problem. In particular, we are interested in determining the circumstances under which the log likelihood is unbounded above for a given covariance constraint set R and a data set x/sub 1/...x/sub M/. Theorem 1 states that, provided certain conditions are met by the constraint set R, the log likelihood is unbounded above if and only if there exists some singular matrix R/sub 0/ in R whose range space contains the data. We refer to the latter occurrence as the failure condition of the ML covariance estimation problem. This result has a certain intuitive appeal; however, one must take care in specifying the conditions on R for this to be precisely correct. In the statement of Theorem 1, two restrictions are given, one for sufficiency and one for necessity. Both restrictions as stated are in error. In this article we propose modified restrictions on the space of covariances R which will correct these errors.