Identifiability of general ARMA processes using linear cumulant-based estimators

Cumulants of order greater than two may be used to estimate the parameters of non-Gaussian ARMA processes, observed in additive colored Gaussian noise. The ARMA model may be causal or non-causal, mixed phase, and have inherent all-pass factors and repeated poles. Recent counterexamples to existing algorithms, based on third-order cumulants, have given rise to questions of uniqueness and consistency of the parameter estimates. Two new order and parameter estimation algorithms are derived, and consistency of the estimators is established. These algorithms involve the solution of sets of linear equations, and may be based on cumulants of any order greater than three. It is demonstrated that the counterexamples do not apply to the new methods.