AR Identifiability Using Cumulants

We address the problem of estimating the AR order and the AR parameters of a causal, stable, S1S0 ARMA(p,q) model, excited by au unobservable i.i.d. process; the observed output is corrupted by additive colored Gaussian noise. 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 m-th order (m > 2) cumulant. We show via counter-examples that consistent AR estimates cannot, in general, be obtained from a single 1-D slice of the cumulant. Necessary and sufficient conditions for the existence of a fullrank slice are also derived. Extensions to the multi-dimensional, multi-channel and non-causal cases are discussed briefly.