Identifiability of the AR parameters of an ARMA process using cumulants

The problem of estimating the autoregressive (AR)-order and the AR parameters of a causal, stable, single input single output (SISO) autoregressive moving average (ARMA) (p,q) model, excited by an unobservable i.i.d. process, is addressed. 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. It is shown 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. It is shown 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.<
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