On nonexistence of the maximum likelihood estimate in blind multichannel identification

In this paper, a blind multichannel identification problem for which the maximum likelihood estimate (MLE) does not exist is considered. More specifically, the likelihood function associated with this problem turns out to have no maximum but only saddle points. This interesting instance of nonexistence of the MLE for a practically relevant problem was first presented in the statistical literature on errors-in-variables regression (M. Solari, 1969). New insights into this result are presented in this paper, along with a direct proof based on the indefiniteness of the Hessian matrix.