Limitations of the ARMA w-slice method using a linear combination of higher-order cumulants

The paper analyses autoregressive moving-average (ARMA) system identification method. This method belongs to higher-order statistical methods of a linear algebra type, showing a unique feature that the method works for any kind of model, i.e. MA, AR, or ARMA, and that the model´s order (p, q) need not be known in advance. Our analyses of the ARMA approach proved that there is a class of systems not being identifiable. All these systems having poles s_i; i = 1,..., p, and at least one zero of type of (s_i1 s_i2 ... Si_k-1)^-1; i₁, i₂,..., i_k-1 ϵ (1,..., p) cannot be identified by ARMA w-slices using k^th-order cumulante, no matter whether with single cumulante, linear combination of cumulante, 1-D slices, or multidimensional slices. The analytical result is backed by simulations. Finally, we propose a procedure of verification of ARMA identifiability and an extension of ARMA w-slice in order to assure the identifiability.