IDENTIFIABILITY OF THE AR PARAMETERS OF AN ARMA PROCESS USING CUMULANTS

We address the problem of estimating the AR order and the AR parameters of a causal, stable, SISO ARMA (p,q) model, excited by an unobservable i.i.d. process; the observed output is corrupted by additive colored Gaussian noise, whose power spectral density is unknown. The ARMA model may be mixed-phase, and have inherent all-pass factors and repeated poles. We show that consistent AR parameter estimates can be obtained via the normal equations based on (p + 1) 1-D slices of the mth-order (m > 2) cumulant. We show via a counterexample that consistent AR estimates cannot, in general, be obtained from a subset of these p + 1 slices. Necessary and sufficient conditions for the existence of a full-rank slice are also derived.