A NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE OF THE MAXIMUM-LIKELIHOOD ESTIMATE IN AUTOREGRESSIVE MODELS

This correspondence concerns the existence problem of a maximum likelihood estimate (MLE) for the parameters of a pth order autoregressive (AR) model from n greater-than-or-equal-to 1 independent records of length m of a complex time series. It is shown that, for almost all such sets of observations, the MLE exists if and only if the n records cannot be exactly fitted by complex undamped sinusoids using the same set of p distinct frequencies.